Tuesday, September 10th Intro to CS4400/CS5400

• General plan for how the course will go.

• Administrative stuff. (Mostly going over the web pages.)

  https://pl.barzilay.org/

Tuesday, September 10th Intro to Programming Languages

PLAI §1

• •

  Why should we care about programming languages? (Any examples of big projects without a little language?)

• •

  What defines a language?

  • syntax
  • semantics
  • libraries (runtime)
  • idioms (community)
• •

  How important is each of these?

  • •

    libraries give you the run-time support, not an important part of the language itself. (BTW, the line between “a library” and “part of the language” is less obvious than it seems.)

  • •

    idioms originate from both language design and culture. They are often misleading. For example, JavaScript programmers will often write:

    function explorer_move() { doThis(); }
    function mozilla_move() { doThat(); }
    if (isExplorer)
      document.onmousemove = explorer_move;
    else
      document.onmousemove = mozilla_move;

    or

    if (isExplorer)
      document.onmousemove = function() { doThis(); };
    else
      document.onmousemove = function() { doThat(); };

    or

    document.onmousemove =
      isExplorer ? function() { ... }
                : function() { ... };

    or

    document.onmousemove =
      isExplorer ? () => { doThis(); } : () => { doThat(); };

    or

    document.onmousemove = isExplorer ? doThis : doThat;

    How many JavaScript programmers will know what this does:

    function foo(n) {
      return function(m) { return m+n; };
    }

    or this:

    n => m => m+n;
    (x,y) => s => s(x,y);

    or, what seems fishy in this? —

    const foo = (x,y) => bar(x,y)

    Yet another example:

    let x = "";
    while (foo())
      x += whatever();

    How would you expect this code perform? How do you think it does in the reality of many uses of JS by people who are not really programmers?

  • •

    Compare:

    • • a[25]+5 (Java: exception)
    • • (+ (vector-ref a 25) 5) (Racket: exception)
    • • a[25]+5 (JavaScript: exception (or NaN))
    • • a[25]+5 (Python: exception)
    • • $a[25]+5 (Perl: 5)
    • • a[25]+5 (C: BOOM)
    • 🡆 syntax is mostly in the cosmetics department; semantics is the real thing.
  • •

    Another example:

    • • a + 1 > a (Python: always true)
    • • (> (+ a 1) a) (Racket: always true)
    • • a + 1 > a (C: sometimes true)
• •

  How should we talk about semantics?

  • A few well-known formalisms for semantics.

  • We will use programs to explain semantics: the best explanation is a program.

  • Ignore possible philosophical issues with circularity (but be aware of them). (Actually, they are solved: Scheme has a formal explanation that can be taken as a translation from Scheme to logic, which means that things that we write can be translated to logic.)

  • We will use Racket for many reasons (syntax, functional, practical, simple, formal, statically typed, environment).

Tuesday, September 10th Intro to Racket

• General layout of the parts of Racket:

  • The Racket language is (mostly) in the Scheme family, or more generally in the Lisp family;

  • Racket: the core language implementation (language and runtime), written in C;

  • The actual language(s) that are available in Racket have lots of additional parts that are implemented in Racket itself;

  • GRacket: a portable Racket GUI extension, written in Racket too;

  • DrRacket: a GRacket application (also written in Racket);

  • Our language(s)…

• Documentation: the Racket documentation is your friend (But beware that some things are provided in different forms from different places).

Tuesday, September 10th Side-note: “Goto Statement Considered Harmful”

A review of “Goto Statement Considered Harmful”, by E.W. DIJKSTRA

This paper tries to convince us that the well-known goto statement should be eliminated from our programming languages or, at least (since I don’t think that it will ever be eliminated), that programmers should not use it. It is not clear what should replace it. The paper doesn’t explain to us what would be the use of the if statement without a goto to redirect the flow of execution: Should all our postconditions consist of a single statement, or should we only use the arithmetic if, which doesn’t contain the offensive goto?

And how will one deal with the case in which, having reached the end of an alternative, the program needs to continue the execution somewhere else?

The author is a proponent of the so-called “structured programming” style, in which, if I get it right, gotos are replaced by indentation. Structured programming is a nice academic exercise, which works well for small examples, but I doubt that any real-world program will ever be written in such a style. More than 10 years of industrial experience with Fortran have proved conclusively to everybody concerned that, in the real world, the goto is useful and necessary: its presence might cause some inconveniences in debugging, but it is a de facto standard and we must live with it. It will take more than the academic elucubrations of a purist to remove it from our languages.

Publishing this would waste valuable paper: Should it be published, I am as sure it will go uncited and unnoticed as I am confident that, 30 years from now, the goto will still be alive and well and used as widely as it is today.

Confidential comments to the editor: The author should withdraw the paper and submit it someplace where it will not be peer reviewed. A letter to the editor would be a perfect choice: Nobody will notice it there!

Tuesday, September 10th Quick Intro to Racket

Racket syntax: similar to other Sexpr-based languages.

Reminder: the parens can be compared to C/etc function call parens — they always mean that some function is applied. This is the reason why (+ (1) (2)) won’t work: if you use C syntax that is +(1(), 2()) but 1 isn’t a function so 1() is an error.

An important difference between syntax and semantics: A good way to think about this is the difference between the string 42 stored in a file somewhere (two ASCII values), and the number 42 stored in memory (in some representation). You could also continue with the above example: there is nothing wrong with “murder” — it’s just a word, but murder is something you’ll go to jail for.

The evaluation function that Racket uses is actually a function that takes a piece of syntax and returns (or executes) its semantics.

---

define expressions are used for creating new bindings, do not try to use them to change values. For example, you should not try to write something like (define x (+ x 1)) in an attempt to mimic x = x+1. It will not work.

---

There are two boolean values built in to Racket: #t (true) and #f (false). They can be used in if statements, for example:

because (< 2 3) evaluates to #t. As a matter of fact, any value except for #f is considered to be true, so:

---

Note: Racket is a functional language — so everything has a value.

This means that the expression

has no meaning when test evaluates to #f. This is unlike Pascal/C where statements do something (side effect) like printing or an assignment — here an if statement with no alternate part will just do nothing if the test is false… Racket, however, must return some value — it could decide on simply returning #f (or some unspecified value) as the value of

as some implementations do, but Racket just declares it a syntax error. (As we will see in the future, Racket has a more convenient when with a clearer intention.)

---

Well, almost everything is a value…

There are certain things that are part of Racket’s syntax — for example if and define are special forms, they do not have a value! More about this shortly.

(Bottom line: much more things do have a value, compared with other languages.)

---

cond is used for a if … else if … else if … else … sequence. The problem is that nested ifs are inconvenient. For example,

In C/Java/Whatever, you’d write:

(Side question: why isn’t there a return statement in Racket?)

But trying to force Racket code to look similar:

is more than just bad taste — the indentation rules are there for a reason, the main one is that you can see the structure of your program at a quick glance, and this is no longer true in the above code. (Such code will be penalized!)

So, instead of this, we can use Racket’s cond statement, like this:

Note that else is a keyword that is used by the cond form — you should always use an else clause (for similar reasons as an if, to avoid an extra expression evaluation there, and we will need it when we use a typed language). Also note that square brackets are read by DrRacket like round parens, it will only make sure that the paren pairs match. We use this to make code more readable — specifically, there is a major difference between the above use of [] from the conventional use of (). Can you see what it is?

The general structure of a cond:

---

Example for using an if expression, and a recursive function:

Use this to show the different tools, especially:

• special objects that cannot be used
• syntax-checker
• stepper
• submission tool (installing, registering and submitting)

An example of converting it to tail recursive form:

---

Additional notes about homework submissions:

• Begin every function with clear documentation: a type followed by a purpose statement.

• Document the function when needed, and according to the guidelines above and in the style guide.

• After the function, always have a few test cases — they should cover your complete code (make sure to include possible corner cases). Later on, we will switch to testing the whole file through it’s “public interface”, instead of testing each function.

Tuesday, September 10th Lists & Recursion

Lists are a fundamental Racket data type.

A list is defined as either:

1. the empty list (null, empty, or '()),

2. a pair (cons cell) of anything and a list.

As simple as this may seem, it gives us precise formal rules to prove that something is a list.

• Why is there a “the” in the first rule?

Examples:

List operations — predicates:

We can derive list? from the above rules:

or better:

But why can’t we define list? more simply as

The difference between the above definition and the proper one can be observed in the full Racket language, not in the student languages (where there are no pairs with non-list values in their tails).

List operations — destructors for pairs (cons cells):

Traditionally called car, cdr.

Also, any c<x>r combination for <x> that is made of up to four as and/or ds — we will probably not use much more than cadr, caddr etc.

---

Example for recursive function involving lists:

Use different tools, esp:

• syntax-checker
• stepper

How come we could use list as an argument — use the syntax checker

Main idea: lists are a recursive structure, so functions that operate on lists should be recursive functions that follow the recursive definition of lists.

Another example for list function — summing a list of numbers

Also show how to implement rcons, using this guideline.

---

More examples:

Define reverse — solve the problem using rcons.

rcons can be generalized into something very useful: append.

• How would we use append instead of rcons?

• How much time will this take? Does it matter if we use append or rcons?

Redefine reverse using tail recursion.

• Is the result more complex? (Yes, but not too bad because it collects the elements in reverse.)

Tuesday, September 10th Some Style

When you have some common value that you need to use in several places, it is bad to duplicate it. For example:

What’s bad about it?

• It’s longer than necessary, which will eventually make your code less readable.

• It’s slower — by the time you reach the last case, you have evaluated the two sequences three times.

• It’s more prone to bugs — the above code is short enough, but what if it was longer so you don’t see the three occurrences on the same page? Will you remember to fix all places when you debug the code months after it was written?

In general, the ability to use names is probably the most fundamental concept in computer science — the fact that makes computer programs what they are.

We already have a facility to name values: function arguments. We could split the above function into two like this:

But instead of the awkward solution of coming up with a new function just for its names, we have a facility to bind local names — let. In general, the syntax for a let special form is

For example,

But note that the bindings are done “in parallel”, for example, try this:

(Note that “in parallel” is quoted here because it’s not really parallelism, but just a matter of scopes: the RHSs are all evaluated in the surrounding scope!)

Using this for the above problem:

---

• Some notes on writing code (also see the style-guide in the handouts section)

• Code quality will be graded to in this course!

• Use abstractions whenever possible, as said above. This is bad:

  (define (how-many a b c)
    (cond
      [(> (* b b) (* 4 a c)) 2]
      [(= (* b b) (* 4 a c)) 1]
      [(< (* b b) (* 4 a c)) 0]))
  
  (define (what-kind a b c)
    (cond
      [(= a 0) 'degenerate]
      [(> (* b b) (* 4 a c)) 'two]
      [(= (* b b) (* 4 a c)) 'one]
      [(< (* b b) (* 4 a c)) 'none]))

• But don’t over abstract: (define one 1) or (define two "two")

• Always do test cases, you might want to comment them, but you should always make sure your code works. Use DrRacket’s covergae features to ensure complete coverage.

• Do not under-document, but also don’t over-document.

• INDENTATION! (Let DrRacket decide; get used to its rules) –> This is part of the culture that was mentioned last time, but it’s done this way for good reason: decades of programming experience have shown this to be the most readable format. It’s also extremely important to keep good indentation since programmers in all Lisps don’t count parens — they look at the structure.

• As a general rule, if should be either all on one line, or the condition on the first and each consequent on a separate line. Similarly for define — either all on one line or a newline after the object that is being define (either an identifier or a an identifier with arguments).

• Another general rule: you should never have white space after an open-paren, or before a close paren (white space includes newlines). Also, before an open paren there should be either another open paren or white space, and the same goes for after a closing paren.

• Use the tools that are available to you: for example, use cond instead of nested ifs (definitely do not force the indentation to make a nested if look like its C counterpart — remember to let DrRacket indent for you).

  Another example — do not use (+ 1 (+ 2 3)) instead of (+ 1 2 3) (this might be needed in extremely rare situations, only when you know your calculus and have extensive knowledge about round-off errors).

  Another example — do not use (cons 1 (cons 2 (cons 3 null))) instead of (list 1 2 3).

  Also — don’t write things like:

  (if (< x 100) #t #f)

  since it’s the same as just

  (< x 100)

  A few more of these:

  (if x #t y)  --same-as-->  (or x y)            ; (almost)
  (if x y #f)  --same-as-->  (and x y)            ; (exacly same)
  (if x #f #t)  --same-as-->  (not x)              ; (almost)

  (Actually the first two are almost the same, for example, (and 1 2) will return 2, not #t.)

• Use these as examples for many of these issues:

  (define (interest x)
    (* x (cond
      [(and (> x 0) (<= x 1000)) 0.04]
      [(and (> x 1000) (<= x 5000)) 0.045]
      [else 0.05])))
  
  (define (how-many a b c)
    (cond ((> (* b b) (* (* 4 a) c))
          2)
          ((< (* b b) (* (* 4 a) c))
          0)
          (else
          1)))
  
  (define (what-kind a b c)
    (if (equal? a 0) 'degenerate
        (if (equal? (how-many a b c) 0) 'zero
            (if (equal? (how-many a b c) 1) 'one
                'two)
            )
        )
    )
  
  (define (interest deposit)
    (cond
    [(< deposit 0) "invalid deposit"]
    [(and (>= deposit 0) (<= deposit 1000)) (* deposit 1.04) ]
    [(and (> deposit 1000) (<= deposit 5000)) (* deposit 1.045)]
    [(> deposit 5000) (* deposit 1.05)]))
  
  (define (interest deposit)
    (if (< deposit 1001) (* 0.04 deposit)
      (if (< deposit 5001) (* 0.045 deposit)
      (* 0.05 deposit))))
  
  (define (what-kind a b c) (cond ((= 0 a) 'degenerate)
                            (else (cond ((> (* b b)(*(* 4 a) c)) 'two)
                                (else (cond ((= (* b b)(*(* 4 a) c)) 'one)
                                (else 'none)))))));

Tuesday, September 10th Tail calls

You should generally know what tail calls are, but here’s a quick review of the subject. A function call is said to be in tail position if there is no context to “remember” when you’re calling it. Very roughly, this means that function calls that are not nested as argument expressions of another call are tail calls. Pay attention that we’re talking about function calls, not, for example, being nested in an if expression since that’s not a function. (The same holds for cond, and, or.)

This definition is something that depends on the context, for example, in an expression like

both calls to foo are tail calls, but they’re tail calls of this expression and therefore apply to this context. It might be that this code is inside another call, as in

and the foo calls are now not in tail position. The main feature of all Scheme implementations including Racket (and including Javascript) WRT tail calls is that calls that are in tail position of a function are said to be “eliminated”. That means that if we’re in an f function, and we’re about to call g in tail position and therefore whatever g returns would be the result of f too, then when Racket does the call to g it doesn’t bother keeping the f context — it won’t remember that it needs to “return” to f and will instead return straight to its caller. In other words, when you think about a conventional implementation of function calls as frames on a stack, Racket will get rid of a stack frame when it can.

You can also try this with any code in DrRacket: hovering over the paren that starts a function call will show a faint pinkish arrow showing the tail-call chain from there for call that are actually tail calls. This is a simple feature since tail calls are easily identifiable by just looking at the syntax of a function.

Another way to see this is to use DrRacket’s stepper to step through a function call. The stepper is generally an alternative debugger, where instead of visualizing stack frames it assembles an expression that represents these frames. Now, in the case of tail calls, there is no room in such a representation to keep the call — and the thing is that in Racket that’s perfectly fine since these calls are not kept on the call stack.

Note that there are several names for this feature:

• “Tail recursion”. This is a common way to refer to the more limited optimization of only tail-recursive functions into loops. In languages that have tail calls as a feature, this is too limited, since they also optimize cases of mutual recursion, or any case of a tail call.

• “Tail call optimization”. In some languages, or more specifically in some compilers, you’ll hear this term. This is fine when tail calls are considered only an “optimization” — but in Racket’s case (as well as Scheme), it’s more than just an optimization: it’s a language feature that you can rely on. For example, a tail-recursive function like (define (loop) (loop)) must run as an infinite loop, not just optimized to one when the compiler feels like it.

• “Tail call elimination”. This is the so far the most common proper name for the feature: it’s not just recursion, and it’s not an optimization.

When should you use tail calls?

Often, people who are aware of tail calls will try to use them always. That’s not always a good idea. You should generally be aware of the tradeoffs when you consider what style to use. The main thing to remember is that tail-call elimination is a property that helps reducing space use (stack space) — often reducing it from linear space to constant space. This can obviously make things faster, but usually the speedup is just a constant factor since you need to do the same number of iterations anyway, so you just reduce the time spent on space allocation.

Here is one such example that we’ve seen:

In this case the first (recursive) version version consumes space linear to the length of the list, whereas the second version needs only constant space. But if you consider only the asymptotic runtime, they are both O(length(l)).

A second example is a simple implementation of map:

In this case, both the asymptotic space and the runtime consumption are the same. In the recursive case we have a constant factor for the stack space, and in the iterative one (the tail-call version) we also have a similar factor for accumulating the reversed list. In this case, it is probably better to keep the first version since the code is simpler. In fact, Racket’s stack space management can make the first version run faster than the second — so optimizing it into the second version is useless.

Tuesday, September 10th Sidenote on Types

Note: this is all just a side note for a particularly hairy example. You don’t need to follow all of this to write code in this class! Consider this section a kind of an extra type-related puzzle to read trough, and maybe get back to it much later, after we cover typechecking.

Types can become interestingly complicated when dealing with higher-order functions. Specifically, the nature of the type system used by Typed Racket makes it have one important weakness: it often fails to infer types when there are higher-order functions that operate on polymorphic functions.

For example, consider how map receives a function and a list of some type, and applies the function over this list to accumulate its output, so it’s a polymorphic function with the following type:

But Racket’s map is actually more flexible that that: it can take more than a single list input, in which case it will apply the function on the first element in all lists, then the second and so on. Narrowing our vision to the two-input-lists case, the type of map then becomes:

Now, here’s a hairy example — what is the type of this function:

Begin by what we know — both maps, call them map1 and map2, have the double- and single-list types of map respectively, here they are, with different names for types:

Now, we know that map2 is the first argument to map1, so the type of map1s first argument should be the type of map2:

From here we can conclude that

If we use these equations in map1’s type, we get:

Now, foo’s two arguments are the 2nd and 3rd arguments of map1, and its result is map1s result, so we can now write our “estimated” type of foo:

This should help you understand why, for example, this will cause a type error:

and why this is valid:

But…!

There’s a big “but” here which is that weakness of Typed Racket that was mentioned. If you try to actually write such a defninition in #lang pl (which is based on Typed Racket), you will first find that you need to explicitly list the type variable that are needed to make it into a generic type. So the above becomes:

But not only does that not work — it throws an obscure type error. That error is actually due to TR’s weakness: it’s a result of not being able to infer the proper types. In such cases, TR has two mechanisms to “guide it” in the right direction. The first one is inst, which is used to instantiate a generic (= polymorphic) type some actual type. The problem here is with the second map since that’s the polymorphic function that is given to a higher-order function (the first map). If we provide the types to instantiate this, it will work fine:

Now, you can use this definition to run the above example:

This example works fine, but that’s because we wrote the list argument explicitly. If you try to use the exact example above,

you’d run into the same problem again, since this also uses a polymorphic function (list) with a higher-order one (map). Indeed, an inst can make this work for this too:

The second facility is ann, which can be used to annotate an expression with the type that you expect it to have.

(Note: this is not type casting! It’s using a different type which is also applicable for the given expression, and having the type checker validate that this is true. TR does have a similar cast form, which is used for a related but different cases.)

This tends to be more verbose than inst, but is sometimes easier to follow, since the expected type is given explicitly. The thing about inst is that it’s kind of “applying” a polymorphic (All (A B) ...) type, so you need to know the order of the A B arguments, which is why in the above we use (inst map Y X) rather than (inst map X Y).

Again, remember that this is all not something that you need to know. We will have a few (very rare) cases where we’ll need to use inst, and in each of these, you’ll be told where and how to use it.

Tuesday, September 10th Side-note: Names are important

An important “discovery” in computer science is that we don’t need names for every intermediate sub-expression — for example, in almost any language we can write something like:

instead of

Such languages are put in contrast to assembly languages, and were all put under the generic label of “high level languages”.

(Here’s an interesting idea — why not do the same for function values?)

Tuesday, September 17th BNF, Grammars, the AE Language

Getting back to the theme of the course: we want to investigate programming languages, and we want to do that using a programming language.

The first thing when we design a language is to specify the language. For this we use BNF (Backus-Naur Form). For example, here is the definition of a simple arithmetic language:

Explain the different parts. Specifically, this is a mixture of low-level (concrete) syntax definition with parsing.

We use this to derive expressions in some language. We start with <AE>, which should be one of these:

• a number <num>
• an <AE>, the text “+”, and another <AE>
• the same but with “-”

<num> is a terminal: when we reach it in the derivation, we’re done. <AE> is a non-terminal: when we reach it, we have to continue with one of the options. It should be clear that the + and the - are things we expect to find in the input — because they are not wrapped in <>s.

We could specify what <num> is (turning it into a <NUM> non-terminal):

But we don’t — why? Because in Racket we have numbers as primitives and we want to use Racket to implement our languages. This makes life a lot easier, and we get free stuff like floats, rationals etc.

To use a BNF formally, for example, to prove that 1-2+3 is a valid <AE> expression, we first label the rules:

and then we can use them as formal justifications for each derivation step:

This would be one way of doing this. Alternatively, we can can visualize the derivation using a tree, with the rules used at the nodes.

These specifications suffer from being ambiguous: an expression can be derived in multiple ways. Even the little syntax for a number is ambiguous — a number like 123 can be derived in two ways that result in trees that look different. This ambiguity is not a “real” problem now, but it will become one very soon. We want to get rid of this ambiguity, so that there is a single (= deterministic) way to derive all expressions.

There is a standard way to resolve that — we add another non-terminal to the definition, and make it so that each rule can continue to exactly one of its alternatives. For example, this is what we can do with numbers:

Similar solutions can be applied to the <AE> BNF — we either restrict the way derivations can happen or we come up with new non-terminals to force a deterministic derivation trees.

As an example of restricting derivations, we look at the current grammar:

and instead of allowing an <AE> on both sides of the operation, we force one to be a number:

Now there is a single way to derive any expression, and it is always associating operations to the right: an expression like 1+2+3 can only be derived as 1+(2+3). To change this to left-association, we would use this:

But what if we want to force precedence? Say that our AE syntax has addition and multiplication:

We can do that same thing as above and add new non-terminals — say one for “products”:

Now we must parse any AE expression as additions of multiplications (or numbers). First, note that if <AE> goes to <PROD> and that goes to <num>, then there is no need for an <AE> to go to a <num>, so this is the same syntax:

Now, if we want to still be able to multiply additions, we can force them to appear in parentheses:

Next, note that <AE> is still ambiguous about additions, which can be fixed by forcing the left hand side of an addition to be a factor:

We still have an ambiguity for multiplications, so we do the same thing and add another non-terminal for “atoms”:

And you can try to derive several expressions to be convinced that derivation is always deterministic now.

But as you can see, this is exactly the cosmetics that we want to avoid — it will lead us to things that might be interesting, but unrelated to the principles behind programming languages. It will also become much much worse when we have a real language rather such a tiny one.

Is there a good solution? — It is right in our face: do what Racket does — always use fully parenthesized expressions:

To prevent confusing Racket code with code in our language(s), we also change the parentheses to curly ones:

But in Racket everything has a value — including those +s and -s, which makes this extremely convenient with future operations that might have either more or less arguments than 2 as well as treating these arithmetic operators as plain functions. In our toy language we will not do this initially (that is, + and - are second order operators: they cannot be used as values). But since we will get to it later, we’ll adopt the Racket solution and use a fully-parenthesized prefix notation:

(Remember that in a sense, Racket code is written in a form of already-parsed syntax…)

Tuesday, September 17th Simple Parsing

On to an implementation of a “parser”:

Unrelated to what the syntax actually looks like, we want to parse it as soon as possible — converting the concrete syntax to an abstract syntax tree.

No matter how we write our syntax:

• 3+4 (infix),
• 3 4 + (postfix),
• +(3,4) (prefix with args in parens),
• (+ 3 4) (parenthesized prefix),

we always mean the same abstract thing — adding the number 3 and the number 4. The essence of this is basically a tree structure with an addition operation as the root and two leaves holding the two numerals.

With the right data definition, we can describe this in Racket as the expression (Add (Num 3) (Num 4)) where Add and Num are constructors of a tree type for syntax, or in a C-like language, it could be something like Add(Num(3),Num(4)).

Similarly, the expression (3-4)+7 will be described in Racket as the expression:

Important note: “expression” was used in two different ways in the above — each way corresponds to a different language, and the result of evaluating the second “expression” is a Racket value that represents the first expression.

To define the data type and the necessary constructors we will use this:

• Note — Racket follows the tradition of Lisp which makes syntax issues almost negligible — the language we use is almost as if we are using the parse tree directly. Actually, it is a very simple syntax for parse trees, one that makes parsing extremely easy.

  [This has an interesting historical reason… Some Lisp history — M-expressions vs. S-expressions, and the fact that we write code that is isomorphic to an AST. Later we will see some of the advantages that we get by doing this. See also “The Evolution of Lisp”, section 3.5.1. Especially the last sentence:

  Therefore we expect future generations of Lisp programmers to continue to reinvent Algol-style syntax for Lisp, over and over and over again, and we are equally confident that they will continue, after an initial period of infatuation, to reject it. (Perhaps this process should be regarded as a rite of passage for Lisp hackers.)

  And an interesting & modern counter-example of this here.]

To make things very simple, we will use the above fact through a double-level approach:

• we first “parse” our language into an intermediate representation — a Racket list — this is mostly done by a modified version of Racket’s read function that uses curly {} braces instead of round () parens,

• then we write our own parse function that will parse the resulting list into an instance of the AE type — an abstract syntax tree (AST).

This is achieved by the following simple recursive function:

This function is pretty simple, but as our languages grow, they will become more verbose and more difficult to write. So, instead, we use a new special form: match, which is matching a value and binds new identifiers to different parts (try it with “Check Syntax”). Re-writing the above code using match:

And finally, to make it more uniform, we will combine this with the function that parses a string into a sexpr so we can use strings to represent our programs:

Tuesday, September 17th The match Form

The syntax for match is

The value is matched against each pattern, possibly binding names in the process, and if a pattern matches it evaluates the result expression. The simplest form of a pattern is simply an identifier — it always matches and binds that identifier to the value:

Another simple pattern is a quoted symbol, which matches that symbol. For example:

will evaluate to "yes" if foo is the symbol x, and to "no" otherwise. Note that else is not a keyword here — it happens to be a pattern that always succeeds, so it behaves like an else clause except that it binds else to the unmatched-so-far value.

Many patterns look like function application — but don’t confuse them with applications. A (list x y z) pattern matches a list of exactly three items and binds the three identifiers; or if the “arguments” are themselves patterns, match will descend into the values and match them too. More specifically, this means that patterns can be nested:

As seen above, there is also a cons pattern that matches a non-empty list and then matches the first part against the head for the list and the second part against the tail of the list.

In a list pattern, you can use ... to specify that the previous pattern is repeated zero or more times, and bound names get bound to the list of respective matching. One simple consequent is that the (list hd tl ...) pattern is exactly the same as (cons hd tl), but being able to repeat an arbitrary pattern is very useful:

A few more useful patterns:

Note that the foo: patterns are all specific to our #lang pl, they are not part of #lang racket or #lang typed/racket.

The patterns are tried one by one in-order, and if no pattern matches the value, an error is raised.

Note that ... in a list pattern can follow any pattern, including all of the above, and including nested list patterns.

Here are a few examples — you can try them out with #lang pl untyped at the top of the definitions window. This:

matches x against a pattern that accepts only a list of symbols, and binds syms to those symbols. If you want to match only a list of, say, one or more symbols, then just add one before the ...-ed pattern variable:

which will match such a non-empty list, where the whole list (on the right hand side) is (cons sym syms).

Here’s another example that matches a list of any number of lists, where each of the sub-lists begins with a symbol and then has any number of numbers. Note how the n and s bindings get values for a list of all symbols and a list of lists of the numbers:

Here is a quick example for how or is used with two literal alternatives, how and is used to name a specific piece of data, and how or is used with a binding:

Tuesday, September 17th The define-type Form

The class language that we’re using, #lang pl, is based on Typed Racket: a statically-typed dialect of Racket. It is not exactly the same as Typed Racket — it is restricted in many ways, and extended in a few ways. (You should therefore try to avoid looking at the Typed Racket documentation and expect things to be the same in #lang pl.)

The most important extension is define-type, which is the construct we will be using to create new user-defined types. In general, such definitions looks like what we just used:

This defines a new type called AE, an AE? predicate for this type, and a few variants for this type: Num, Add, and Sub in this case. Each of these variant names is a constructor, taking in arguments with the listed types, where these types can include the newly defined type itself in (the very common) case we’re defining a recursive type. The return type is always the newly defined type, AE here.

To summarize, this definition gives us a new AE type, and three constructors, as if we wrote the following type declarations:

• (: Num : Number -> AE)
• (: Add : AE AE -> AE)
• (: Sub : AE AE -> AE)

The newly defined types are known as “disjoint unions”, since values in these types are disjoint — there is no overlap between the different variants. As we will see, this is what makes this such a useful construct for our needs: the compiler knows about the variants of each newly defined type, which will make it possible for it to complain if we extend a type with more variants but not update all uses of the type.

Furthermore, since the return types of these constructors are all the new type itself, there is no way for us to write code that expects just one of these variants. We will use a second form, cases, to handle these values.

Tuesday, September 17th The cases Form

A define-type declaration defines only what was described above: one new type name and a matching predicate, and a few variants as constructor functions. Unlike HtDP, we don’t get predicates for each of the variants, and we don’t get accessor functions for the fields of the variants.

The way that we handle the new kind of values is with cases: this is a form that is very similar to match, but is specific to instances of the user-defined type.

Many students find it confusing to distinguish match and cases since they are so similar. Try to remember that match is for primitive Racket values (we’ll mainly use them for S-expression values), while cases is for user-defined values. The distinction between the two forms is unfortunate, and doesn’t serve any purpose. It is just technically difficult to unify the two.

For example, code that handles AE values (as defined above) can look as follows:

As you can see, we need to have patterns for each of the listed variants (and the compiler will throw an error if some are missing), and each of these patterns specifies bindings that will get the field values contained in a given variant object.

We can also use nested patterns:

but this is a feature that we will not use too often.

The final clause in a cases form can be an else clause, which serves as a fallback in case none of the previous clauses matched the input value. However, using an else like this is strongly discouraged! The problem with using it is that it effectively eliminates the advantage in getting the type-checker to complain when a type definition is extended with new variants. Using these else clauses, we can actually mimic all of the functionality that you expect in HtDP-style code, which demonstrates that this is equivalent to HtDP-style definitions. For example:

Important reminder: this is code that you should not write! Doing so will lead to code that is more fragile than just using cases, since you’d be losing the protection the compiler gives you in the form of type errors on occurrences of cases that need to be updated when a type is extended with new variants. You would therefore end up writing a bunch of boiler-plate code only to end up with lower-quality code. The core of the problem is in the prevalent use of else which gives up that protection.

In these examples the else clause is justified because even if AE is extended with new variants, functions like Add? and Add-left should not be affected and treat the new variants as they treat all other non-Add instances. (And since else is inherent to these functions, using them in our code is inherently a bad idea.) We will, however, have a few (very few!) places where we’ll need to use else — but this will always be done only on some specific functionality rather than a wholesale approach of defining a different interface for user-defined types.

Tuesday, September 17th Semantics (= Evaluation)

PLAI §2

Back to BNF — now, meaning.

An important feature of these BNF specifications: we can use the derivations to specify meaning (and meaning in our context is “running” a program (or “interpreting”, “compiling”, but we will use “evaluating”)). For example:

To do this a little more formally:

Note the completely different roles of the two +s and -s. In fact, it might have been more correct to write:

or even using a marker to denote meta-holes in these strings:

but we will avoid pretending that we’re doing that kind of string manipulation. (For example, it will require specifying what does it mean to return <num> for $<num> (involves string->number), and the fragments on the right side mean that we need to specify these as substring operations.)

Note that there’s a similar kind of informality in our BNF specifications, where we assume that <foo> refers to some terminal or non-terminal. In texts that require more formal specifications (for example, in RFC specifications), each literal part of the BNF is usually double-quoted, so we’d get

An alternative popular notation for eval(X) is ⟦X⟧:

Is there a problem with this definition? Ambiguity:

Depending on the way the expression is parsed, we can get either a result of 2 or -4:

Again, be very aware of confusing subtleties which are extremely important: We need parens around a sub-expression only in one side, why? — When we write:

we have two expressions, but one stands for an input syntax, and one stands for a real mathematical expression.

In a case of a computer implementation, the syntax on the left is (as always) an AE syntax, and the real expression on the right is an expression in whatever language we use to implement our AE language.

Like we said earlier, ambiguity is not a real problem until the actual parse tree matters. With eval it definitely matters, so we must not make it possible to derive any syntax in multiple ways or our evaluation will be non-deterministic.

---

Quick exercise:

We can define a meaning for <digit>s and then <num>s in a similar way:

Is this exactly what we want? — Depends on what we actually want…

• First, there’s a bug in this code — having a BNF derivation like

  <NUM> ::= <digit> | <digit> <NUM>

  is unambiguous, but makes it hard to parse a number. We get:

  eval(123) = 10*eval(1) + eval(23)
            = 10*1 + 10*eval(2) + eval(3)
            = 10*1 + 10*2 + 3
            = 33

  Changing the order of the last rule works much better:

  <NUM> ::= <digit> | <NUM> <digit>

  and then:

  eval(<NUM> <digit>) = 10*eval(<NUM>) + eval(<digit>)

• As a concrete example see how you would make it work with 107, which demonstrates why compositionality is important.

• Example for free stuff that looks trivial: if we were to define the meaning of numbers this way, would it always work? Think an average language that does not give you bignums, making the above rules fail when the numbers are too big. In Racket, we happen to be using an integer representation for the syntax of integers, and both are unlimited. But what if we wanted to write a Racket compiler in C or a C compiler in Racket? What about a C compiler in C, where the compiler runs on a 64 bit machine, and the result needs to run on a 32 bit machine?

Tuesday, September 17th Side-note: Compositionality

The example of

being a language that is easier to write an evaluator for leads us to an important concept — compositionality. This definition is easier to write an evaluator for, since the resulting language is compositional: the meaning of an expression — for example 123 — is composed out of the meaning of its two parts, which in this BNF are 12 and 3. Specifically, the evaluation of <NUM> <digit> is 10 * the evaluation of the first, plus the evaluation of the second. In the <digit> <NUM> case this is more difficult — the meaning of such a number depends not only on the meaning of the two parts, but also on the <NUM> syntax:

This this case this can be tolerable, since the meaning of the expression is still made out of its parts — but imperative programming (when you use side effects) is much more problematic since it is not compositional (at least not in the obvious sense). This is compared to functional programming, where the meaning of an expression is a combination of the meanings of its subexpressions. For example, every sub-expression in a functional program has some known meaning, and these all make up the meaning of the expression that contains them — but in an imperative program we can have a part of the code be x++ — and that doesn’t have a meaning by itself, at least not one that contributes to the meaning of the whole program in a direct way.

(Actually, we can have a well-defined meaning for such an expression: the meaning is going from a world where x is a container of some value N, to a world where the same container has a different value N+1. You can probably see now how this can make things more complicated. On an intuitive level — if we look at a random part of a functional program we can tell its meaning, so building up the meaning of the whole code is easy, but in an imperative program, the meaning of a random part is pretty much useless.)

Tuesday, September 17th Implementing an Evaluator

Now continue to implement the semantics of our syntax — we express that through an eval function that evaluates an expression.

We use a basic programming principle — splitting the code into two layers, one for parsing the input, and one for doing the evaluation. Doing this avoids the mess we’d get into otherwise, for example:

This is messy because it combines two very different things — syntax and semantics — into a single lump of code. For this particular kind of evaluator it looks simple enough, but this is only because it’s simple enough that all we do is replace constructors by arithmetic operations. Later on things will get more complex, and bundling the evaluator with the parser will be more problematic. (Note: the fact that we can replace constructors with the run-time operators mean that we have a very simple, calculator-like language, and that we can, in fact, “compile” all programs down to a number.)

If we split the code, we can easily include decisions like making

syntactically invalid. (Which is not, BTW, what Racket does…) (Also, this is like the distinction between XML syntax and well-formed XML syntax.)

An additional advantage is that by using two separate components, it is simple to replace each one, making it possible to change the input syntax, and the semantics independently — we only need to keep the same interface data (the AST) and things will work fine.

Our parse function converts an input syntax to an abstract syntax tree (AST). It is abstract exactly because it is independent of any actual concrete syntax that you type in, print out etc.

Tuesday, September 17th Implementing The AE Language

Back to our eval — this will be its (obvious) type:

which leads to some obvious test cases:

which from now on we will write using the new test form that the #lang pl language provides:

Note that we’re testing only at the interface level — only running whole functions. For example, you could think about a test like:

but the details of parsing and of the constructor names are things that nobody outside of our evaluator cares about — so we’re not testing them. In fact, we shouldn’t even mention parse in these tests, since it is not part of the public interface of our users; they only care about using it as a compiler-like black box. (This is sometimes called “integration tests”.) We’ll address this shortly.

Like everything else, the structure of the recursive eval code follows the recursive structure of its input. In HtDP terms, our template is:

In this case, filling in the gaps is very simple

We now further combine eval and parse into a single run function that evaluates an AE string.

This function becomes the single public entry point into our code, and the only thing that should be used in tests that verify our interface:

The resulting full code is:

(Note that the tests are done with a test form, which we mentioned above.)

For anyone who thinks that Racket is a bad choice, this is a good point to think how much code would be needed in some other language to do the same as above.

Tuesday, September 24th Intro to Typed Racket

The plan:

• Why Types?

• Why Typed Racket?

• What’s Different about Typed Racket?

• Some Examples of Typed Racket for Course Programs

Types

• Who has used a (statically) typed language?

• Who has used a typed language that’s not Java?

Typed Racket will be both similar to and very different from anything you’ve seen before.

Why types?

• Types help structure programs.

• Types provide enforced and mandatory documentation.

• Types help catch errors.

Types will help you. A lot.

Structuring programs

• Data definitions

  ;; An AE is one of:  ; \
  ;;  (make-Num Number) ;  > HtDP
  ;;  (make-Add AE AE)  ; /
  
  (define-type AE      ; \
    [Num number?]      ;  > Predicates =~= contracts (PLAI)
    [Add AE? AE?])      ; /    (has names of defined types too)
  
  (define-type AE      ; \
    [Num Number]        ;  > Typed Racket (our PL)
    [Add AE AE])        ; /

• Data-first

  The structure of your program is derived from the structure of your data.

  You have seen this in Fundamentals with the design recipe and with templates. In this class, we will see it extensively with type definitions and the (cases …) form. Types make this pervasive — we have to think about our data before our code.

• A language for describing data

  Instead of having an informal language for describing types in contract lines, and a more formal description of predicates in a define-type form, we will have a single, unified language for both of these. Having such a language means that we get to be more precise and more expressive (since the typed language covers cases that you would otherwise dismiss with some hand waving, like “a function”).

Why Typed Racket?

Racket is the language we all know, and it has the benefits that we discussed earlier. Mainly, it is an excellent language for experimenting with programming languages.

• Typed Racket allows us to take our Racket programs and typecheck them, so we get the benefits of a statically typed language.

• Types are an important programming language feature; Typed Racket will help us understand them.

[Also: the development of Typed Racket is happening here in Northeastern, and will benefit from your feedback.]

How is Typed Racket different from Racket

• Typed Racket will reject your program if there are type errors! This means that it does that at compile-time, before any code gets to run.

• Typed Racket files start like this:

  #lang typed/racket
  ;; Program goes here.

  but we will use a variant of the Typed Racket language, which has a few additional constructs:

  #lang pl
  ;; Program goes here.

• Typed Racket requires you to write the contracts on your functions.

  Racket:

  ;; f : Number -> Number
  (define (f x)
    (* x (+ x 1)))

  Typed Racket:

  #lang pl
  (: f : Number -> Number)
  (define (f x)
    (* x (+ x 1)))

  [In the “real” Typed Racket the preferred style is with prefix arrows:

  #lang typed/racket
  (: f (-> Number Number))
  (define (f x) : Number
    (* x (+ x 1)))

  and you can also have the type annotations appear inside the definition:

  #lang typed/racket
  (define (f [x : Number]) : Number
    (* x (+ x 1)))

  but we will not use these form.]

• As we’ve seen, Typed Racket uses types, not predicates, in define-type.

  (define-type AE
    [Num Number]
    [Add AE AE])

  versus

  (define-type AE
    [Num number?]
    [Add AE? AE?])

• There are other differences, but these will suffice for now.

Examples

More interesting examples

• Typed Racket is designed to be a language that is friendly to the kind of programs that people write in Racket. For example, it has unions:

  (: foo : (U String Number) -> Number)
  (define (foo x)
    (if (string? x)
      (string-length x)
      ;; at this point it knows that `x' is not a
      ;; string, therefore it must be a number
      (+ 1 x)))

  This is not common in statically typed languages, which are usually limited to only disjoint unions. For example, in OCaml you’d write this definition:

  type string_or_number = Str of string | Int of int ;;
  let foo x = match x with Str s -> String.length s
                        | Int i -> i+1 ;;

  And use it with an explicit constructor:

  foo (Str "bar") ;;
  foo (Int 3) ;;

• Note that in the Typed Racket case, the language keeps track of information that is gathered via predicates — which is why it knows that one x is a String, and the other is a Number.

• Typed Racket has a concept of subtypes — which is also something that most statically typed languages lack. In fact, the fact that it has (arbitrary) unions means that it must have subtypes too, since a type is always a subtype of a union that contains this type.

• Another result of this feature is that there is an Any type that is the union of all other types. Note that you can always use this type since everything is in it — but it gives you the least information about a value. In other words, Typed Racket gives you a choice: you decide which type to use, one that is very restricted but has a lot of information about its values to a type that is very permissive but has almost no useful information. This is in contrast to other type system (HM systems) where there is always exactly one correct type.

  To demonstrate, consider the identity function:

  (define (id x) x)

  You could use a type of (: id : Integer -> Integer) which is very restricted, but you know that the function always returns an integer value.

  Or you can make it very permissive with a (: id : Any -> Any), but then you know nothing about the result — in fact, (+ 1 (id 2)) will throw a type error. It does return 2, as expected, but the type checker doesn’t know the type of that 2. If you wanted to use this type, you’d need to check that the result is a number, eg:

  (let ([x (id 123)]) (if (number? x) (+ x 10) 999))

  This means that for this particular function there is no good specific type that we can choose — but there are polymorphic types. These types allow propagating their input type(s) to their output type. In this case, it’s a simple “my output type is the same as my input type”:

  (: id : (All (A) A -> A))

  This makes the output preserve the same level of information that you had on its input.

• Another interesting thing to look at is the type of error: it’s a function that returns a type of Nothing — a type that is the same as an empty union: (U). It’s a type that has no values in it — it fits error because it is a function that doesn’t return any value, in fact, it doesn’t return at all. In addition, it means that an error expression can be used anywhere you want because it is a subtype of anything at all.

• An else clause in a cond expression is almost always needed, for example:

  (: digit-num : Number -> (U Number String))
  (define (digit-num n)
    (cond [(<= n 9)    1]
          [(<= n 99)  2]
          [(<= n 999)  3]
          [(<= n 9999) 4]
          [(>  n 9999) "a lot"]))

  (and if you think that the type checker should know what this is doing, then how about

  (> (* n 10) (/ (* (- 10000 1) 20) 2))

  or

  (>= n 10000)

  for the last test?)

• In some rare cases you will run into one limitation of Typed Racket: it is difficult (that is: a generic solution is not known at the moment) to do the right inference when polymorphic functions are passed around to higher-order functions. For example:

  (: call : (All (A B) (A -> B) A -> B))
  (define (call f x)
    (f x))
  (call rest (list 4))

  In such cases, we can use inst to instantiate a function with a polymorphic type to a given type — in this case, we can use it to make it treat rest as a function that is specific for numeric lists:

  (call (inst rest Number) (list 4))

  In other rare cases, Typed Racket will infer a type that is not suitable for us — there is another form, ann, that allows us to specify a certain type. Using this in the call example is more verbose:

  (call (ann rest : ((Listof Number) -> (Listof Number))) (list 4))

  However, these are going to be rare and will be mentioned explicitly whenever they’re needed.

Tuesday, September 24th Bindings & Substitution

We now get to an important concept: substitution.

Even in our simple language, we encounter repeated expressions. For example, if we want to compute the square of some expression:

Why would we want to get rid of the repeated sub-expression?

• It introduces a redundant computation. In this example, we want to avoid computing the same sub-expression a second time.

• It makes the computation more complicated than it could be without the repetition. Compare the above with:

  with x = {+ 4 2},
    {* x x}

• This is related to a basic fact in programming that we have already discussed: duplicating information is always a bad thing. Among other bad consequences, it can even lead to bugs that could not happen if we wouldn’t duplicate code. A toy example is “fixing” one of the numbers in one expression and forgetting to fix the corresponding one:

  {* {+ 4 2} {+ 4 1}}

  Real world examples involve much more code, which make such bugs very difficult to find, but they still follow the same principle.

• This gives us more expressive power — we don’t just say that we want to multiply two expressions that both happen to be {+ 4 2}, we say that we multiply the {+ 4 2} expression by itself. It allows us to express identity of two values as well as using two values that happen to be the same.

So, the normal way to avoid redundancy is to introduce an identifier. Even when we speak, we might say: “let x be 4 plus 2, multiply x by x”.

(These are often called “variables”, but we will try to avoid this name: what if the identifier does not change (vary)?)

To get this, we introduce a new form into our language:

We expect to be able to reduce this to:

by substituting 6 for x in the body sub-expression of with.

A little more complicated example:

Tuesday, September 24th WAE: Adding Bindings to AE

PLAI §3

To add this to our language, we start with the BNF. We now call our language “WAE” (With+AE):

Note that we had to introduce two new rules: one for introducing an identifier, and one for using it. This is common in many language specifications, for example define-type introduces a new type, and it comes with cases that allows us to destruct its instances.

For <id> we need to use some form of identifiers, the natural choice in Racket is to use symbols. We can therefore write the corresponding type definition:

The parser is easily extended to produce these syntax objects:

But note that this parser is inconvenient — if any of these expressions:

would result in a “bad syntax” error, which is not very helpful. To make things better, we can add another case for with expressions that are malformed, and give a more specific message in that case:

and finally, to group all of the parsing code that deals with with expressions (both valid and invalid ones), we can use a single case for both of them:

And now we’re done with the syntactic part of the with extension.

Quick note — why would we indent With like a normal function in code like this

(With 'x
      (Num 2)
      (Add (Id 'x) (Num 4)))

instead of an indentation that looks like a let

(With 'x (Num 2)
  (Add (Id 'x) (Num 4)))

?

The reason for this is that the second indentation looks like a binding construct (eg, the indentation used in a let expression), but With is not a binding form — it’s a plain function because it’s at the Racket level. You should therefore keep in mind the huge difference between that With and the with that appears in WAE programs:

{with {x 2}
  {+ x 4}}

Another way to look at it: imagine that we intend for the language to be used by Spanish/Chinese/German/French speakers. In this case we would translate “with”:

{con  {x 2} {+ x 4}}
{he  {x 2} {+ x 4}}
{mit  {x 2} {+ x 4}}
{avec {x 2} {+ x 4}}
{c    {x 2} {+ x 4}}

but we will not do the same for With if we (the language implementors) are English speakers.

Tuesday, September 24th Evaluation of with

Now, to make this work, we will need to do some substitutions.

We basically want to say that to evaluate:

we need to evaluate WAE2 with id substituted by WAE1. Formally:

There is a more common syntax for substitution (quick: what do I mean by this use of “syntax”?):

Side-note: this syntax originates with logicians who used [x/v]e, and later there was a convention that mimicked the more natural order of arguments to a function with e[x->v], and eventually both of these got combined into e[v/x] which is a little confusing in that the left-to-right order of the arguments is not the same as for the subst function.

Now all we need is an exact definition of substitution.

Note that substitution is not the same as evaluation, it’s only a part of the evaluation process. In the previous examples, when we evaluated the expression we did substitutions as well as the usual arithmetic operations that were already part of the AE evaluator. In this last definition there is still a missing evaluation step, see if you can find it.

So let us try to define substitution now:

Substitution (take 1): e[v/i]
To substitute an identifier i in an expression e with an expression v, replace all identifiers in e that have the same name i by the expression v.

This seems to work with simple expressions, for example:

however, we crash with an invalid syntax if we try:

— we got to an invalid expression.

To fix this, we need to distinguish normal occurrences of identifiers, and ones that are used as new bindings. We need a few new terms for this:

1. Binding Instance: a binding instance of an identifier is one that is used to name it in a new binding. In our <WAE> syntax, binding instances are only the <id> position of the with form.

2. Scope: the scope of a binding instance is the region of program text in which instances of the identifier refer to the value bound in the binding instance. (Note that this definition actually relies on a definition of substitution, because that is what is used to specify how identifiers refer to values.)

3. Bound Instance (or Bound Occurrence / Identifier): an instance of an identifier is bound if it is contained within the scope of a binding instance of its name.

4. Free Instance (or Free Occurrence / Identifier): An identifier that is not contained in any binding instance of its name is said to be free.

Using this we can say that the problem with the previous definition of substitution is that it failed to distinguish between bound instances (which should be substituted) and binding instances (which should not). So we try to fix this:

Substitution (take 2): e[v/i]
To substitute an identifier i in an expression e with an expression v, replace all instances of i that are not themselves binding instances with the expression v.

First of all, check the previous examples:

still work, and

also works. However, if we try this:

we get:

but we want that to be 8: the inner x should be bound by the closest with that binds it.

The problem is that the new definition of substitution that we have respects binding instances, but it fails to deal with their scope. In the above example, we want the inner with to shadow the outer with’s binding for x.

Substitution (take 3): e[v/i]
To substitute an identifier i in an expression e with an expression v, replace all instances of i that are not themselves binding instances, and that are not in any nested scope, with the expression v.

This avoids bad substitution above, but it is now doing things too carefully:

becomes

which is an error because x is unbound (and there is reasonable no rule that we can specify to evaluate it).

The problem is that our substitution halts at every new scope, in this case, it stopped at the new y scope, but it shouldn’t have because it uses a different name. In fact, that last definition of substitution cannot handle any nested scope.

Revise again:

Substitution (take 4): e[v/i]
To substitute an identifier i in an expression e with an expression v, replace all instances of i that are not themselves binding instances, and that are not in any nested scope of i, with the expression v.

which, finally, is a good definition. This is just a little too mechanical. Notice that we actually refer to all instances of i that are not in a scope of a binding instance of i, which simply means all free occurrences of i — free in e (why? — remember the definition of “free”?):

Substitution (take 4b): e[v/i]
To substitute an identifier i in an expression e with an expression v, replace all instances of i that are free in e with the expression v.

Based on this we can finally write the code for it:

… and this is just the same as writing a formal “paper version” of the substitution rule.

We still have bugs: but we’ll need some more work to get to them.

---

Before we find the bugs, we need to see when and how substitution is used in the evaluation process.

To modify our evaluator, we will need rules to deal with the new syntax pieces — with expressions and identifiers.

When we see an expression that looks like:

we continue by evaluating E1 to get a value V1, we then substitute the identifier x with the expression V1 in E2, and continue by evaluating this new expression. In other words, we have the following evaluation rule:

So we know what to do with with expressions. How about identifiers? The main feature of subst, as said in the purpose statement, is that it leaves no free instances of the substituted variable around. This means that if the initial expression is valid (did not contain any free variables), then when we go from

to

the result is an expression that has no free instances of x. So we don’t need to handle identifiers in the evaluator — substitutions make them all go away.

We can now extend the formal definition of AE to that of WAE:

If you’re paying close attention, you might catch a potential problem in this definition: we’re substituting eval(E1) for x in E2 — an operation that requires a WAE expression, but eval(E1) is a number. (Look at the type of the eval definition we had for AE, then look at the above definition of subst.) This seems like being overly pedantic, but we it will require some resolution when we get to the code. The above rules are easily coded as follows:

Note the Num expression in the marked line: evaluating the named expression gives us back a number — we need to convert this number into a syntax to be able to use it with subst. The solution is to use Num to convert the resulting number into a numeral (the syntax of a number). It’s not an elegant solution, but it will do for now.

Finally, here are a few test cases. We use a new test special form which is part of the course plugin. The way to use test is with two expressions and an => arrow — DrRacket evaluates both, and nothing will happen if the results are equal. If the results are different, you will get a warning line, but evaluation will continue so you can try additional tests. You can also use an =error> arrow to test an error message — use it with some text from the expected error, ? stands for any single character, and * is a sequence of zero or more characters. (When you use test in your homework, the handin server will abort when tests fail.) We expect these tests to succeed (make sure that you understand why they should succeed).

Putting this all together, we get the following code; trying to run this code will raise an unexpected error…

---

Oops, this program still has problems that were caught by the tests — we encounter unexpected free identifier errors. What’s the problem now? In expressions like:

we forgot to substitute x in the expression that y is bound to. We need to the recursive substitute in both the with’s body expression as well as its named expression:

And still we have a problem… Now it’s

that halts with an error, but we want it to evaluate to 5! Carefully trying out our substitution code reveals the problem: when we substitute 5 for the outer x, we don’t go inside the inner with because it has the same name — but we do need to go into its named expression. We need to substitute in the named expression even if the identifier is the same one we’re substituting:

The complete (and, finally, correct) version of the code is now:

---

Reminder:

• We started doing substitution, with a let-like form: with.

• Reasons for using bindings:

  • Avoid writing expressions twice.
    • More expressive language (can express identity).
    • Duplicating is bad! (“DRY”: Don’t Repeat Yourself.)
    • Avoids static redundancy.
  • Avoid redundant computations.
    • More than just an optimization when it avoids exponential resources.
    • Avoids dynamic redundancy.

• BNF:

  <WAE> ::= <num>
          | { + <WAE> <WAE> }
          | { - <WAE> <WAE> }
          | { * <WAE> <WAE> }
          | { / <WAE> <WAE> }
          | { with { <id> <WAE> } <WAE> }
          | <id>

  Note that we had to introduce two new rules: one for introducing an identifier, and one for using it.

• Type definition:

  (define-type WAE
    [Num  Number]
    [Add  WAE WAE]
    [Sub  WAE WAE]
    [Mul  WAE WAE]
    [Div  WAE WAE]
    [Id  Symbol]
    [With Symbol WAE WAE])

• Parser:

  (: parse-sexpr : Sexpr -> WAE)
  ;; parses s-expressions into WAEs
  (define (parse-sexpr sexpr)
    (match sexpr
      [(number: n)    (Num n)]
      [(symbol: name) (Id name)]
      [(cons 'with more)
      (match sexpr
        [(list 'with (list (symbol: name) named) body)
          (With name (parse-sexpr named) (parse-sexpr body))]
        [else (error 'parse-sexpr "bad `with' syntax in ~s"
                      sexpr)])]
      [(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))]
      [(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))]
      [(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
      [(list '/ lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
      [else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))

• We need to define substitution. Terms:

  1. Binding Instance.
  2. Scope.
  3. Bound Instance.
  4. Free Instance.

• After lots of attempts:

  e[v/i] — To substitute an identifier i in an expression e with an expression v, replace all instances of i that are free in e with the expression v.

• Implemented the code, and again, needed to fix a few bugs:

  (: subst : WAE Symbol WAE -> WAE)
  ;; substitutes the second argument with the third argument in the
  ;; first argument, as per the rules of substitution; the resulting
  ;; expression contains no free instances of the second argument
  (define (subst expr from to)
    (cases expr
      [(Num n) expr]
      [(Add l r) (Add (subst l from to) (subst r from to))]
      [(Sub l r) (Sub (subst l from to) (subst r from to))]
      [(Mul l r) (Mul (subst l from to) (subst r from to))]
      [(Div l r) (Div (subst l from to) (subst r from to))]
      [(Id name) (if (eq? name from) to expr)]
      [(With bound-id named-expr bound-body)
      (With bound-id
            (subst named-expr from to)
            (if (eq? bound-id from)
              bound-body
              (subst bound-body from to)))]))

  (Note that the bugs that we fixed clarify the exact way that our scopes work: in {with {x 2} {with {x {+ x 2}} x}}, the scope of the first x is the {+ x 2} expression.)

• We then extended the AE evaluation rules:

  eval(...) = ... same as the AE rules ...
  eval({with {x E1} E2}) = eval(E2[eval(E1)/x])
  eval(id) = error!

  and noted the possible type problem.

• The above translated into a Racket definition for an eval function (with a hack to avoid the type issue):

  (: eval : WAE -> Number)
  ;; evaluates WAE expressions by reducing them to numbers
  (define (eval expr)
    (cases expr
      [(Num n) n]
      [(Add l r) (+ (eval l) (eval r))]
      [(Sub l r) (- (eval l) (eval r))]
      [(Mul l r) (* (eval l) (eval r))]
      [(Div l r) (/ (eval l) (eval r))]
      [(With bound-id named-expr bound-body)
      (eval (subst bound-body
                    bound-id
                    (Num (eval named-expr))))]
      [(Id name) (error 'eval "free identifier: ~s" name)]))

Tuesday, September 24th Formal Specs

Note the formal definitions that were included in the WAE code. They are ways of describing pieces of our language that are more formal than plain English, but still not as formal (and as verbose) as the actual code.

A formal definition of subst:

(N is a <num>, E1, E2 are <WAE>s, x is some <id>, y is a different <id>)

And a formal definition of eval:

Tuesday, October 1st Lazy vs Eager Evaluation

As we have previously seen, there are two basic approaches for evaluation: either eager or lazy. In lazy evaluation, bindings are used for sort of textual references — it is only for avoiding writing an expression twice, but the associated computation is done twice anyway. In eager evaluation, we eliminate not only the textual redundancy, but also the computation.

Which evaluation method did our evaluator use? The relevant piece of formalism is the treatment of with:

And the matching piece of code is:

How do we make this lazy?

In the formal equation:

and in the code:

We can verify the way this works by tracing eval (compare the trace you get for the two versions):

Ignoring the traces for now, the modified WAE interpreter works as before, specifically, all tests pass. So the question is whether the language we get is actually different than the one we had before. One difference is in execution speed, but we can’t really notice a difference, and we care more about meaning. Is there any program that will run differently in the two languages?

The main feature of the lazy evaluator is that it is not evaluating the named expression until it is actually needed. As we have seen, this leads to duplicating computations if the bound identifier is used more than once — meaning that it does not eliminate the dynamic redundancy. But what if the bound identifier is not used at all? In that case the named expression simply evaporates. This is a good hint at an expression that behaves differently in the two languages — if we add division to both languages, we get a different result when we try running:

The eager evaluator stops with an error when it tries evaluating the division — and the lazy evaluator simply ignores it.

Even without division, we get a similar behavior for

but it is questionable whether the fact that this evaluates to 7 is correct behavior — we really want to forbid program that use free variable.

Furthermore, there is an issue with name capturing — we don’t want to substitute an expression into a context that captures some of its free variables. But our substitution allows just that, which is usually not a problem because by the time we do the substitution, the named expression should not have free variables that need to be replaced. However, consider evaluating this program:

under the two evaluation regimens: the eager version stops with an error, and the lazy version succeed. This points at a bug in our substitution, or rather not dealing with an issue that we do not encounter.

So the summary is: as long as the initial program is correct, both evaluation regimens produce the same results. If a program contains free variables, they might get captured in a naive lazy evaluator implementation (but this is a bug that should be fixed). Also, there are some cases where eager evaluation runs into a run-time problem which does not happen in a lazy evaluator because the expression is not used. It is possible to prove that when you evaluate an expression, if there is an error that can be avoided, lazy evaluation will always avoid it, whereas an eager evaluator will always run into it. On the other hand, lazy evaluators are usually slower than eager evaluator, so it’s a speed vs. robustness trade-off.

Note that with lazy evaluation we say that an identifier is bound to an expression rather than a value. (Again, this is why the eager version needed to wrap eval’s result in a Num and this one doesn’t.)

(It is possible to change things and get a more well behaved substitution, we basically will need to find if a capture might happen, and rename things to avoid it. For example,

With this, we might have gone through this path in evaluating the above:

But you can see that this is much more complicated (more code: requires a free-in predicate, being able to invent new fresh names, etc). And it’s not even the end of that story…)

Tuesday, October 1st de Bruijn Indexes

This whole story revolves around names, specifically, name capture is a problem that should always be avoided (it is one major source of PL headaches).

But are names the only way we can use bindings?

There is a least one alternative way: note that the only thing we used names for are for references. We don’t really care what the name is, which is pretty obvious when we consider the two WAE expressions:

or the two Racket function definitions:

Both of these show a pair of expressions that we should consider as equal in some sense (this is called “alpha-equality”). The only thing we care about is what variable points where: the binding structure is the only thing that matters. In other words, as long as DrRacket produces the same arrows when we use Check Syntax, we consider the program to be the same, regardless of name choices (for argument names and local names, not for global names like foo in the above).

The alternative idea uses this principle: if all we care about is where the arrows go, then simply get rid of the names… Instead of referencing a binding through its name, just specify which of the surrounding scopes we want to refer to. For example, instead of:

we can use a new “reference” syntax — [N] — and use this instead of the above:

So the rules for [N] are — [0] is the value bound in the current scope, [1] is the value from the next one up etc.

Of course, to do this translation, we have to know the precise scope rules. Two more complicated examples:

is translated to:

(note how x appears as a different reference based on where it appeared in the original code.) Even more subtle:

is translated to:

because the inner with does not have its own named expression in its scope, so the named expression is immediately in the scope of the outer with.

This is called “de Bruijn Indexes”: instead of referencing identifiers by their name, we use an index into the surrounding binding context. The major disadvantage, as can be seen in the above examples, is that it is not convenient for humans to work with. Specifically, the same identifier is referenced using different numbers, which makes it hard to understand what some code is doing. After all, abstractions are the main thing we deal with when we write programs, and having labels make the bindings structure much easier to understand than scope counts.

However, practically all compilers use this for compiled code (think about stack pointers). For example, GCC compiles this code:

to:

Tuesday, October 1st Functions & Function Values

PLAI §4

Now that we have a form for local bindings, which forced us to deal with proper substitutions and everything that is related, we can get to functions. The concept of a function is itself very close to substitution, and to our with form. For example, when we write:

then the {* x x} body is itself parametrized over some value for x. If we take this expression and take out the 5, we’re left with something that has all of the necessary ingredients of a function — a bunch of code that is parameterized over some input identifier:

We only need to replace with and use a proper name that indicates that it’s a function:

Now we have a new form in our language, one that should have a function as its meaning. As we have seen in the case of with expressions, we also need a new form to use these functions. We will use call for this, so that

will be the same as the original with expression that we started with — the fun expression is like the with expression with no value, and applying it on 5 is providing that value back:

Of course, this does not help much — all we get is a way to use local bindings that is more verbose from what we started with. What we’re really missing is a way to name these functions. If we get the right evaluation rules, we can evaluate a fun expression to some value — which will allow us to bind it to a variable using with. Something like this:

In this expression, we say that x is the formal parameter (or argument), and the 5 and 6 are actual parameters (sometimes abbreviated as formals and actuals). Note that naming functions often helps, but many times there are small functions that are fine to specify without a name — for example, consider a two-stage addition function, where there is no apparent good name for the returned function:

Tuesday, October 1st Implementing First Class Functions

PLAI §6 (uses some stuff from PLAI §5, which we do later)

This is a simple plan, but it is directly related to how functions are going to be used in our language. We know that {call {fun {x} E1} E2} is equivalent to a with expression, but the new thing here is that we do allow writing just the {fun ...} expression by itself, and therefore we need to have some meaning for it. The meaning, or the value of this expression, should roughly be “an expression that needs a value to be plugged in for x”. In other words, our language will have these new kinds of values that contain an expression to be evaluated later on.

There are three basic approaches that classify programming languages in relation to how the deal with functions:

1. First order: functions are not real values. They cannot be used or returned as values by other functions. This means that they cannot be stored in data structures. This is what most “conventional” languages used to have in the past. (You will be implementing such a language in homework 4.)

   An example of such a language is the Beginner Student language that is used in HtDP, where the language is intentionally first-order to help students write correct code (at the early stages where using a function as a value is usually an error). It’s hard to find practical modern languages that fall in this category.

2. Higher order: functions can receive and return other functions as values. This is what you get with C and modern Fortran.

3. First class: functions are values with all the rights of other values. In particular, they can be supplied to other functions, returned from functions, stored in data structures, and new functions can be created at run-time. (And most modern languages have first class functions.)

The last category is the most interesting one. Back in the old days, complex expressions were not first-class in that they could not be freely composed. This is still the case in machine-code: as we’ve seen earlier, to compute an expression such as

you have to do something like this:

In other words, every intermediate value needs to have its own name. But with proper (“high-level”) programming languages (at least most of them…) you can just write the original expression, with no names for these values.

With first-class functions something similar happens — it is possible to have complex expressions that consume and return functions, and they do not need to be named.

What we get with our fun expression (if we can make it work) is exactly this: it generates a function, and you can choose to either bind it to a name, or not. The important thing is that the value exists independently of a name.

This has a major effect on the “personality” of a programming language as we will see. In fact, just adding this feature will make our language much more advanced than languages with just higher-order or first-order functions.

---

Quick Example: the following is working JavaScript code, that uses first class functions.

Note that the above definition of foo does not use an anonymous “lambda expression” — in Racket terms, it’s translated to

The returned function is not anonymous, but it’s not really named either: the bar name is bound only inside the body of foo, and outside of it that name no longer exists since it’s not its scope. It gets used in the printed form if the function value is displayed, but this is merely a debugging aid. The anonymous lambda version that is common in Racket can be used in JavaScript too:

Side-note: GCC includes extensions that allow internal function definitions, but it still does not have first class functions — trying to do the above is broken:

#include <stdio.h>
typedef int(*int2int)(int);
int2int foo(int x) {
  int bar(int y) { return x + y; }
  return bar;
}
int main() {
  int2int f = foo(1);
  int2int g = foo(10);
  printf(">> %d, %d\n", f(2), g(2));
}

Tuesday, October 1st Side-note: how important is it to have anonymous functions?

You’ll see many places where people refer to the feature of first-class functions as the ability to create anonymous functions, but this is a confusion and it’s not accurate. Whether a function has a name or not is not the important question — instead, the important question is whether functions can exist with no bindings that refers to them.

As a quick example in Racket:

in Javascript:

and in Python:

In all three of these, we have a foo function that returns a function named bar — but the bar name, is only available in the scope of foo. The fact that the name is displayed as part of the textual rendering of the function value is merely a debugging feature.

Tuesday, October 1st The FLANG Language

Now for the implementation — we call this new language FLANG.

First, the BNF:

And the matching type definition:

The parser for this grammar is, as usual, straightforward:

We also need to patch up the substitution function to deal with these things. The scoping rule for the new function form is, unsurprisingly, similar to the rule of with, except that there is no extra expression now, and the scoping rule for call is the same as for the arithmetic operators:

And the matching code:

---

Now, before we start working on an evaluator, we need to decide on what exactly do we use to represent values of this language. Before we had functions, we had only number values and we used Racket numbers to represent them. Now we have two kinds of values — numbers and functions. It seems easy enough to continue using Racket numbers to represent numbers, but what about functions? What should be the result of evaluating

? Well, this is the new toy we have: it should be a function value, which is something that can be used just like numbers, but instead of arithmetic operations, we can call these things. What we need is a way to avoid evaluating the body expression of the function — delay it — and instead use some value that will contain this delayed expression in a way that can be used later.

To accommodate this, we will change our implementation strategy a little: we will use our syntax objects for numbers ((Num n) instead of just n), which will be a little inconvenient when we do the arithmetic operations, but it will simplify life by making it possible to evaluate functions in a similar way: simply return their own syntax object as their values. The syntax object has what we need: the body expression that needs to be evaluated later when the function is called, and it also has the identifier name that should be replaced with the actual input to the function call. This means that evaluating:

now yields

and a number (Num 5) evaluates to (Num 5).

In a similar way, (Fun 'x (Num 2)) evaluates to (Fun 'x (Num 2)).

Why would this work? Well, because call will be very similar to with — the only difference is that its arguments are ordered a little differently, being retrieved from the function that is applied and the argument.

The formal evaluation rules are therefore treating functions like numbers, and use the syntax object to represent both values:

Note that the last rule could be written using a translation to a with expression:

And alternatively, we could specify with using call and fun:

There is a small problem in these rules which is intuitively seen by the fact that the evaluation rule for a call is expected to be very similar to the one for arithmetic operations. We now have two kinds of values, so we need to check the arithmetic operation’s arguments too:

The corresponding code is:

Note that the Call case is doing the same thing we do in the With case. In fact, we could have just generated a With expression and evaluate that instead:

The arith-op function is in charge of checking that the input values are numbers (represented as FLANG numbers), translating them to plain numbers, performing the Racket operation, then re-wrapping the result in a Num. Note how its type indicates that it is a higher-order function.

It uses the following function to convert FLANG numbers to Racket numbers. (Note that else is almost always a bad idea since it can prevent the compiler from showing you places to edit code — but this case is an exception since we never want to deal with anything other than Nums.) The reason that this function is relatively trivial is that we chose the easy way and represented numbers using Racket numbers, but we could have used strings or anything else.

We can also make things a little easier to use if we make run convert the result to a number:

Adding few simple tests we get:

---

There is still a problem with this version. First a question — if call is similar to arithmetic operations (and to with in what it actually does), then how come the code is different enough that it doesn’t even need an auxiliary function?

Second question: what should happen if we evaluate these code snippets:

Third question, what will happen if we do the above?

What we’re missing is an evaluation of the function expression, in case it’s not a literal fun form. The following fixes this:

The complete code is:

Tuesday, October 1st Introducing Racket’s lambda

Quick laundry list of things to go over:

• fun & lambda
• difference between lambda and simple values
• not being able to do recursive functions with let
• let* as a derived form
• let with lambda in Racket –> can be a derived form
• how if can be used to implement and and or as derived forms
• Newtonian syntax vs. a lambda expression.

Almost all modern languages have this capability. For example, this:

Can be written in JavaScript like this:

or in ES6 JavaScript:

In Perl:

In Ruby:

etc. Even Java has lambda expressions, and “recently” C++ added them too.

Tuesday, October 1st Using Functions as Objects

A very important aspect of Racket — using “higher order” functions — functions that get and return functions. Here is a very simple example:

Note: what we get is actually an object that remembers (by the substitution we’re doing) a number. How about:

Take this idea to the next level:

Or, with types:

Even more — why should the internal function expect a boolean and choose what to return? We can simply expect a function that will take the two values and return one:

And a typed version, using our own constructor to make it a little less painful:

Note that the Kons type definition is the same as:

so All is to polymorphic type definitions what lambda is for function definitions.

Finally, in JavaScript:

or with ES6 arrow functions, the function definitions become:

and using Typescript to add types:

Tuesday, October 1st Using define-type for new “type aliases”

As seen in these examples, there is another way to use define-type, using a = to create a new type name “alias” for an existing type. For example:

These uses of define-type do not define any new kind of type, they are essentially a convenience tool for making code shorter and more readable.

Note in particular that this can also be used to define “alias type constructors” too: somewhat similar to creating new “type functions”. For example:

This is something that we will only need in a few rare cases.

Tuesday, October 1st Currying

A curried function is a function that, instead of accepting two (or more) arguments, accepts only one and returns a function that accepts the rest. For example:

It’s easy to write functions for translating between normal and curried versions.

Typed version of that, with examples:

Usages — common with H.O. functions like map, where we want to fix one argument.

When dealing with such higher-order code, the types are very helpful, since every arrow corresponds to a function:

It is common to make the -> function type associate to the right, so you can find this type written as:

or even as

but that can be a little confusing…

Tuesday, October 1st Using Higher-Order & Anonymous Functions

Say that we have a function for estimating derivatives of a function at a specific point:

And say that we want to try out various functions given some plot function that draws graphs of numeric functions, for example:

The problem is that plot expects a single (Number -> Number) function — if we want to try it with a derivative, we can do this:

But this will get very tedious very fast — it is much simpler to use an anonymous function:

we can even verify that our derivative is correct by comparing a known function to its derivative

But it’s still not completely natural to do these things — you need to explicitly combine functions, which is not too convenient. Instead of doing this, we can write H.O. functions that will work with functional inputs and outputs. For example, we can write a function to subtract functions:

and the same for the derivative:

Now we can try the same in a much easier way:

More than that — our fderiv could be created from deriv automatically:

Same principle with fsub: we can write a function that converts a binary arithmetical function into a function that operates on unary numeric function. But to make things more readable we can define new types for unary and binary numeric functions:

We can do this with anything — developing a rich library of functions and functionals (functions over functions) is extremely easy… Here’s a pretty extensive yet very short library of functions:

This is written in the “untyped dialect” of the class language, but it should be easy now to add the types.

Examples:

Tuesday, October 1st Side-note: “Point-Free” combinators

Forming functions without using lambda (or an implicit lambda using a define syntactic sugar) is called point-free style. It’s especially popular in Haskell, where it is easier to form functions this way because of implicit currying and a large number of higher level function combinators. If used too much, it can easily lead to obfuscated code.

Tuesday, October 1st This is not Runtime Code Generation

All of this is similar to run-time code generation, but not really. The only thing that fderiv does is take a function and store it somewhere in the returned function, then when that function receives a number, it uses the stored function and send it to deriv with the number. We could simply write deriv as what fderiv is — which is the real derivative function:

but again, this is not faster or slower than the plain deriv. However, there are some situations where we can do some of the computation on the first-stage argument, saving work from the second stage. Here is a cooked-to-exaggeration example — we want a function that receives two inputs x, y and returns fib(x)*y, but we must use a stupid fib:

The function we want is:

If we currify it as usual (or just use currify), we get:

And try this several times:

But in the definition of bogus, notice that (fib x) does not depend on y — so we can rewrite it a little differently:

and trying the above again is much faster now:

This is therefore not doing any kind of runtime code generation, but it enables doing similar optimizations in our code. A proper RTCG facility would recompile the curried function for a given first input, and (hopefully) automatically achieve the optimization that we did in a manual way.

Tuesday, October 8th Substitution Caches

PLAI §5 (called “deferred substitutions” there)

Evaluating using substitutions is very inefficient — at each scope, we copy a piece of the program AST. This includes all function calls which implies an impractical cost (function calls should be cheap!).

To get over this, we want to use a cache of substitutions.

Basic idea: we begin evaluating with no cached substitutions, then collect them as we encounter bindings.

[Implies another change for our evaluator: we don’t really substitute identifiers until we get to them; when we reach an identifier, it is no longer an error — we must consult the substitution cache.]

Tuesday, October 8th Implementation of Cache Functionality

First, we need a type for a substitution cache. For this we will use a list of lists of two elements each — a name and its value FLANG:

We need to have an empty substitution cache, a way to extend it, and a way to look things up:

Actually, the reason to use such list of lists is that Racket has a built-in function called assq that will do this kind of search (assq is a search in an association list using eq? for the key comparison). This is a version of lookup that uses assq:

Tuesday, October 8th Formal Rules for Cached Substitutions

The formal evaluation rules are now different. Evaluation carries along a substitution cache that begins its life as empty: so eval needs an extra argument. We begin by writing the rules that deal with the cache, and use the above function names for simplicity — the behavior of the three definitions can be summed up in a single rule for lookup:

And now we can write the new rules for eval

Note that there is no mention of subst — the whole point is that we don’t really do substitution, but use the cache instead. The lookup rules, and the places where extend is used replaces subst, and therefore specifies our scoping rules.

Also note that the rule for call is still very similar to the rule for with, but it looks like we have lost something — the interesting bit with substituting into fun expressions.

Tuesday, October 8th Evaluating with Substitution Caches

Implementing the new eval is easy now — it is extended in the same way that the formal eval rule is extended:

Again, note that we don’t need subst anymore, but the rest of the code (the data type definition, parsing, and arith-op) is exactly the same.

Finally, we need to make sure that eval is initially called with an empty cache. This is easy to change in our main run entry point:

The full code (including the same tests, but not including formal rules for now) follows. Note that one test does not pass.

Tuesday, October 8th Dynamic and Lexical Scopes

This seems like it should work, and it even worked on a few examples, except for one which was hard to follow. Seems like we have a bug…

Now we get to a tricky issue that managed to be a problem for lots of language implementors, including the first version of Lisp. Lets try to run the following expression — try to figure out what it will evaluate to:

We expect it to return 7 (at least I do!), but we get 9 instead… The question is — should it return 9?

What we have arrived to is called dynamic scope. Scope is determined by the dynamic run-time environment (which is represented by our substitution cache). This is almost always undesirable, as I hope to convince you.

Before we start, we define two scope options for a programming language:

• Static Scope (also called Lexical Scope): In a language with static scope, each identifier gets its value from the scope of its definition, not its use.

• Dynamic Scope: In a language with dynamic scope, each identifier gets its value from the scope of its use, not its definition.

Racket uses lexical scope, our new evaluator uses dynamic, the old substitution-based evaluator was static etc.

As a side-remark, Lisp began its life as a dynamically-scoped language. The artifacts of this were (sort-of) dismissed as an implementation bug. When Scheme was introduced, it was the first Lisp dialect that used strictly lexical scoping, and Racket is obviously doing the same. (Some Lisp implementations used dynamic scope for interpreted code and lexical scope for compiled code!) In fact, Emacs Lisp is the only live dialects of Lisp that is still dynamically scoped by default. To see this, compare a version of the above code in Racket:

and the Emacs Lisp version (which looks almost the same):

which also happens when we use another function on the way:

and note that renaming identifiers can lead to different code — change that val to x:

and you get 8 because the argument name changed the x that the internal function sees!

Consider also this Emacs Lisp function:

which has no meaning by itself (x is unbound),

but can be given a dynamic meaning using a let:

or a function application:

There is also a dynamically-scoped language in the course languages:

Note how bad the last example gets: you basically cannot call any function and know in advance what it will do.

There are some cases where dynamic scope can be useful in that it allows you to “remotely” customize any piece of code. A good example of where this is taken to an extreme is Emacs: originally, it was based on an ancient Lisp dialect that was still dynamically scoped, but it retained this feature even when practically all Lisp dialects moved on to having lexical scope by default. The reason for this is that the danger of dynamic scope is also a way to make a very open system where almost anything can be customized by changing it “remotely”. Here’s a concrete example for a similar kind of dynamic scope usage that makes a very hackable and open system:

Obviously, this power to customize everything is also the main source of problems with getting no guarantees for code. A common way to get the best of both worlds is to have controllable dynamic scope. For example, Common Lisp also has lexical scope everywhere by default, but some variables can be declared as special, which means that they are dynamically scoped. The main problem with that is that you can’t tell when a variable is special by just looking at the code that uses it, so a more popular approach is the one that is used in Racket: all bindings are always lexically scoped, but there are parameters which are a kind of dynamically scoped value containers — but they are bound to plain (lexically scoped) identifiers. Here’s the same code as above, translated to Racket with parameters:

The main point here is that the points where a dynamically scoped value is used are under the programmer’s control — you cannot “customize” what - is doing, for example. This gives us back the guarantees that we like to have (= that code works), but of course these points are pre-determined, unlike an environment where everything can be customized including things that are unexpectedly useful.

As a side-note, after many decades of debating this, Emacs has finally added lexical scope in its core language, but this is still determined by a flag — a global lexical-binding variable.

Tuesday, October 8th Dynamic versus Lexical Scope

And back to the discussion of whether we should use dynamic or lexical scope:

• The most important fact is that we want to view programs as executed by the normal substituting evaluator. Our original motivation was to optimize evaluation only — not to change the semantics! It follows that we want the result of this optimization to behave in the same way. All we need is to evaluate:

  (run "{with {x 3}
          {with {f {fun {y} {+ x y}}}
            {with {x 5}
              {call f 4}}}}")

  in the original evaluator to get convinced that 7 should be the correct result (note also that the same code, when translated into Racket, evaluates to 7).

  (Yet, this is a very important optimization, which without it lots of programs become too slow to be feasible, so you might claim that you’re fine with the modified semantics…)

• It does not allow using functions as objects, for example, we have seen that we have a functional representation for pairs:

  (define (kons x y)
    (lambda (n)
      (match n
        ['first  x]
        ['second y]
        [else (error ...)])))
  
  (define my-pair (kons 1 2))

  If this is evaluated in a dynamically-scoped language, we do get a function as a result, but the values bound to x and y are now gone! Using the substitution model we substituted these values in, but now they were only held in a cache which no has no entries for them…

  In the same way, currying would not work, our nice deriv function would not work etc etc etc.

• Makes reasoning impossible, because any piece of code behaves in a way that cannot be predicted until run-time. For example, if dynamic scoping was used in Racket, then you wouldn’t be able to know what this function is doing:

  (define (foo)
    x)

  As it is, it will cause a run-time error, but if you call it like this:

  (let ([x 1])
    (foo))

  then it will return 1, and if you later do this:

  (define (bar x)
    (foo))
  
  (let ([x 1])
    (bar 2))

  then you would get 2!

  These problems can be demonstrated in Emacs Lisp too, but Racket goes one step further — it uses the same rule for evaluating a function as well as its values (Lisp uses a different name-space for functions). Because of this, you cannot even rely on the following function:

  (define (add x y)
    (+ x y))

  to always add x and y! — A similar example to the above:

  (let ([+ -])
    (add 1 2))

  would return -1!

• Many so-called “scripting” languages begin their lives with dynamic scoping. The main reason, as we’ve seen, is that implementing it is extremely simple (no, nobody does substitution in the real world! (Well, almost nobody…)).

  Another reason is that these problems make life impossible if you want to use functions as object like you do in Racket, so you notice them very fast — but in a normal language without first-class functions, problems are not as obvious.

• For example, bash has local variables, but they have dynamic scope:

  x="the global x"
  print_x() { echo "The current value of x is \"$x\""; }
  foo() { local x="x from foo"; print_x; }
  print_x; foo; print_x

  Perl began its life with dynamic scope for variables that are declared local:

  $x="the global x";
  sub print_x { print "The current value of x is \"$x\"\n"; }
  sub foo { local($x); $x="x from foo"; print_x; }
  print_x; foo; print_x;

  When faced with this problem, “the Perl way” was, obviously, not to remove or fix features, but to pile them up — so local still behaves in this way, and now there is a my declaration which achieves proper lexical scope (and every serious Perl programmer knows that you should always use my)…

  There are other examples of languages that changed, and languages that want to change (e.g, nobody likes dynamic scope in Emacs Lisp, but there’s just too much code now).

• This is still a tricky issue, like any other issue with bindings. For example, googling got me quickly to a Python blog post which is confused about what “dynamic scoping” is… It claims that Python uses dynamic scope (Search for “Python uses dynamic as opposed to lexical scoping”), yet python always used lexical scope rules, as can be seen by translating their code to Racket (ignore side-effects in this computation):

  (define (orange-juice)
    (* x 2))
  (define x 3)
  (define y (orange-juice)) ; y is now 6
  (define x 1)
  (define y (orange-juice)) ; y is now 2

  or by trying this in Python:

  def orange_juice():
    return x*2
  def foo(x):
    return orange_juice()
  foo(2)

  The real problem of python (pre 2.1, and pre 2.2 without the funny

  from __future__ import nested_scope

  line) is that it didn’t create closures, which we will talk about shortly.

• Another example, which is an indicator of how easy it is to mess up your scope is the following Ruby bug — running in irb:

  % irb
  irb(main):001:0> x = 0
  => 0
  irb(main):002:0> lambda{|x| x}.call(5)
  => 5
  irb(main):003:0> x
  => 5

  (This is a bug due to weird scoping rules for variables, which was fixed in newer versions of Ruby. See this Ruby rant for details, or read about Ruby and the principle of unwelcome surprise for additional gems (the latter is gone, so you’ll need the web archive to read it).)

• Another thing to consider is the fact that compilation is something that you do based only on the lexical structure of programs, since compilers never actually run code. This means that dynamic scope makes compilation close to impossible.

• There are some advantages for dynamic scope too. Two notable ones are:

  • Dynamic scope makes it easy to have a “configuration variable” easily change for the extent of a calling piece of code (this is used extensively in Emacs, for example). The thing is that usually we want to control which variables are “configurable” in this way, statically scoped languages like Racket often choose a separate facility for these. To rephrase the problem of dynamic scoping, it’s that all variables are modifiable.

    The same can be said about functions: it is sometimes desirable to change a function dynamically (for example, see “Aspect Oriented Programming”), but if there is no control and all functions can change, we get a world where no code can every be reliable.

  • It makes recursion immediately available — for example,

    {with {f {fun {x} {call f x}}}
      {call f 0}}

    is an infinite loop with a dynamically scoped language. But in a lexically scoped language we will need to do some more work to get recursion going.

Tuesday, October 8th Implementing Lexical Scope: Closures and Environments

So how do we fix this?

Lets go back to the root of the problem: the new evaluator does not behave in the same way as the substituting evaluator. In the old evaluator, it was easy to see how functions can behave as objects that remember values. For example, when we do this:

the result was a function value, which actually was the syntax object for this:

Now if we call this function from someplace else like:

it is clear what the result will be: f is bound to a function that adds 1 to its input, so in the above the later binding for x has no effect at all.

But with the caching evaluator, the value of

is simply:

and there is no place where we save the 1 — that’s the root of our problem. (That’s also what makes people suspect that using lambda in Racket and any other functional language involves some inefficient code-recompiling magic.) In fact, we can verify that by inspecting the returned value, and see that it does contain a free identifier.

Clearly, we need to create an object that contains the body and the argument list, like the function syntax object — but we don’t do any substitution, so in addition to the body an argument name(s) we need to remember that we still need to substitute x by 1 . This means that the pieces of information we need to know are:

and that last bit has the missing 1. The resulting object is called a closure because it closes the function body over the substitutions that are still pending (its environment).

So, the first change is in the value of functions which now need all these pieces, unlike the Fun case for the syntax object.

A second place that needs changing is the when functions are called. When we’re done evaluating the call arguments (the function value and the argument value) but before we apply the function we have two values — there is no more use for the current substitution cache at this point: we have finished dealing with all substitutions that were necessary over the current expression — we now continue with evaluating the body of the function, with the new substitutions for the formal arguments and actual values given. But the body itself is the same one we had before — which is the previous body with its suspended substitutions that we still did not do.

Rewrite the evaluation rules — all are the same except for evaluating a fun form and a call form:

As a side note, these substitution caches are a little more than “just a cache” now — they actually hold an environment of substitutions in which expression should be evaluated. So we will switch to the common environment name now.:

In case you find this easier to follow, the “flat algorithm” for evaluating a call is:

Note how the scoping rules that are implied by this definition match the scoping rules that were implied by the substitution-based rules. (It should be possible to prove that they are the same.)

The changes to the code are almost trivial, except that we need a way to represent <{fun {x} B}, env> pairs.

---

The implication of this change is that we now cannot use the same type for function syntax and function values since function values have more than just syntax. There is a simple solution to this — we never do any substitutions now, so we don’t need to translate values into expressions — we can come up with a new type for values, separate from the type of abstract syntax trees.

When we do this, we will also fix our hack of using FLANG as the type of values: this was merely a convenience since the AST type had cases for all kinds of values that we needed. (In fact, you should have noticed that Racket does this too: numbers, strings, booleans, etc are all used by both programs and syntax representation (s-expressions) — but note that function values are not used in syntax.) We will now implement a separate VAL type for runtime values.

First, we need now a type for such environments — we can use Listof for this:

but we can just as well define a new type for environment values:

Reimplementing lookup is now simple:

… we don’t need extend because we get Extend from the type definition, and we also get (EmptyEnv) instead of empty-subst.

We now use this with the new type for values — two variants of these:

And now the new implementation of eval which uses the new type and implements lexical scope:

We also need to update arith-op to use VAL objects. The full code follows — it now passes all tests, including the example that we used to find the problem.

Tuesday, October 8th Fixing an Overlooked Bug

Incidentally, this version fixes a bug we had previously in the substitution version of FLANG:

This bug was due to our naive subst, which doesn’t avoid capturing renames. But note that since that version of the evaluator makes its way from the outside in, there is no difference in semantics for valid programs — ones that don’t have free identifiers.

(Reminder: This was not a dynamically scoped language, just a bug that happened when x wasn’t substituted away before f was replaced with something that refers to x.)

Tuesday, October 15th Lexical Scope using Racket Closures

PLAI §11 (without the last part about recursion)

An alternative representation for an environment.

We’ve already seen how first-class functions can be used to implement “objects” that contain some information. We can use the same idea to represent an environment. The basic intuition is — an environment is a mapping (a function) between an identifier and some value. For example, we can represent the environment that maps 'a to 1 and 'b to 2 (using just numbers for simplicity) using this function:

An empty mapping that is implemented in this way has the same type:

We can use this idea to implement our environments: we only need to define three things — EmptyEnv, Extend, and lookup. If we manage to keep the contract to these functions intact, we will be able to simply plug it into the same evaluator code with no other changes. It will also be more convenient to define ENV as the appropriate function type for use in the VAL type definition instead of using the actual type:

Now we get to EmptyEnv — this is expected to be a function that expects no arguments and creates an empty environment, one that behaves like the empty-mapping function defined above. We could define it like this (changing the empty-mapping type to return a VAL):

but we can skip the need for an extra definition and simply return an empty mapping function:

(The un-Rackety name is to avoid replacing previous code that used the EmptyEnv name for the constructor that was created by the type definition.)

The next thing we tackle is lookup. The previous definition that was used is:

How should it be modified now? Easy — an environment is a mapping: a Racket function that will do the searching job itself. We don’t need to modify the contract since we’re still using ENV, except a different implementation for it. The new definition is:

Note that lookup does almost nothing — it simply delegates the real work to the env argument. This is a good hint for the error message that empty mappings should throw —

Finally, Extend — this was previously created by the variant case of the ENV type definition:

keeping the same type that is implied by this variant means that the new Extend should look like this:

The question is — how do we extend a given environment? Well, first, we know that the result should be mapping — a symbol -> VAL function that expects an identifier to look for:

Next, we know that in the generated mapping, if we look for id then the result should be val:

If the name that we’re looking for is not the same as id, then we need to search through the previous environment:

But we know what lookup does — it simply delegates back to the mapping function (which is our rest argument), so we can take a direct route instead:

To see how all this works, try out extending an empty environment a few times and examine the result. For example, the environment that we began with:

behaves in the same way (if the type of values is numbers) as

The new code is now the same, except for the environment code:

Tuesday, October 15th More Closures (on both levels)

Racket closures (= functions) can be used in other places too, and as we have seen, they can do more than encapsulate various values — they can also hold the behavior that is expected of these values.

To demonstrate this we will deal with closures in our language. We currently use a variant that holds the three pieces of relevant information:

We can replace this by a functional object, which will hold the three values. First, change the VAL type to hold functions for FunV values:

And note that the function should somehow encapsulate the same information that was there previously, the question is how this information is going to be done, and this will determine the actual type. This information plays a role in two places in our evaluator — generating a closure in the Fun case, and using it in the Call case:

we can simply fold the marked functionality bit of Call into a Racket function that will be stored in a FunV object — this piece of functionality takes an argument value, extends the closure’s environment with its value and the function’s name, and continues to evaluate the function body. Folding all of this into a function gives us:

where the values of bound-body, bound-id, and val are known at the time that the FunV is constructed. Doing this gives us the following code for the two cases:

And now the type of the function is clear:

And again, the rest of the code is unmodified:

Tuesday, October 15th Types of Evaluators

What we did just now is implement lexical environments and closures in the language we implement using lexical environments and closures in our own language (Racket)!

This is another example of embedding a feature of the host language in the implemented language, an issue that we have already discussed.

There are many examples of this, even when the two languages involved are different. For example, if we have this bit in the C implementation of Racket:

then the special semantics of evaluating a Racket and form is being inherited from C’s special treatment of &&. You can see this by the fact that if there is a bug in the C compiler, then it will propagate to the resulting Racket implementation too. A different solution is to not use && at all:

and we can say that this is even better since it evaluates the second expression in tail position. But in this case we don’t really get that benefit, since C itself is not doing tail-call optimization as a standard feature (though some compilers do so under some circumstances).

We have seen a few different implementations of evaluators that are quite different in flavor. They suggest the following taxonomy.

• A syntactic evaluator is one that uses its own language to represent expressions and semantic runtime values of the evaluated language, implementing all the corresponding behavior explicitly.

• A meta evaluator is an evaluator that uses language features of its own language to directly implement behavior of the evaluated language.

While our substitution-based FLANG evaluator was close to being a syntactic evaluator, we haven’t written any purely syntactic evaluators so far: we still relied on things like Racket arithmetics etc. The most recent evaluator that we have studied, is even more of a meta evaluator than the preceding ones: it doesn’t even implement closures and lexical scope, and instead, it uses the fact that Racket itself has them.

With a good match between the evaluated language and the implementation language, writing a meta evaluator can be very easy. With a bad match, though, it can be very hard. With a syntactic evaluator, implementing each semantic feature will be somewhat hard, but in return you don’t have to worry as much about how well the implementation and the evaluated languages match up. In particular, if there is a particularly strong mismatch between the implementation and the evaluated language, it may take less effort to write a syntactic evaluator than a meta evaluator.

As an exercise, we can build upon our latest evaluator to remove the encapsulation of the evaluator’s response in the VAL type. The resulting evaluator is shown below. This is a true meta evaluator: it uses Racket closures to implement FLANG closures, Racket function application for FLANG function application, Racket numbers for FLANG numbers, and Racket arithmetic for FLANG arithmetic. In fact, ignoring some small syntactic differences between Racket and FLANG, this latest evaluator can be classified as something more specific than a meta evaluator:

• A meta-circular evaluator is a meta evaluator in which the implementation and the evaluated languages are the same.

  This is essentially the concept of a “universal” evaluator, as in a “universal turing machine”.

(Put differently, the trivial nature of the evaluator clues us in to the deep connection between the two languages, whatever their syntactic differences may be.)

Tuesday, October 15th Feature Embedding

We saw that the difference between lazy evaluation and eager evaluation is in the evaluation rules for with forms, function applications, etc:

is eager, and

is lazy. But is the first rule really eager? The fact is that the only thing that makes it eager is the fact that our understanding of the mathematical notation is eager — if we were to take math as lazy, then the description of the rule becomes a description of lazy evaluation.

Another way to look at this is — take the piece of code that implements this evaluation:

and the same question applies: is this really implementing eager evaluation? We know that this is indeed eager — we can simply try it and check that it is, but it is only eager because we are using an eager language for the implementation! If our own language was lazy, then the evaluator’s implementation would run lazily, which means that the above applications of the eval and the subst functions would also be lazy, making our evaluator lazy as well.

This is a general phenomena where some of the semantic features of the language we use (math in the formal description, Racket in our code) gets embedded into the language we implement.

Here’s another example — consider the code that implements arithmetics:

what if it was written like this:

Would it still implement unlimited integers and exact fractions? That depends on the language that was used to implement it: the above syntax suggests C, C++, Java, or some other relative, which usually come with limited integers and no exact fractions. But this really depends on the language — even our own code has unlimited integers and exact rationals only because Racket has them. If we were using a language that didn’t have such features (there are such Scheme implementations), then our implemented language would absorb these (lack of) features too, and its own numbers would be limited in just the same way. (And this includes the syntax for numbers, which we embedded intentionally, like the syntax for identifiers).

The bottom line is that we should be aware of such issues, and be very careful when we talk about semantics. Even the language that we use to communicate (semi-formal logic) can mean different things.

---

Aside: read “Reflections on Trusting Trust” by Ken Thompson (You can skip to the “Stage II” part to get to the interesting stuff.)

(And when you’re done, look for “XcodeGhost” to see a relevant example, and don’t miss the leaked document on the wikipedia page…)

---

Here is yet another variation of our evaluator that is even closer to a meta-circular evaluator. It uses Racket values directly to implement values, so arithmetic operations become straightforward. Note especially how the case for function application is similar to arithmetics: a FLANG function application translates to a Racket function application. In both cases (applications and arithmetics) we don’t even check the objects since they are simple Racket objects — if our language happens to have some meaning for arithmetics with functions, or for applying numbers, then we will inherit the same semantics in our language. This means that we now specify less behavior and fall back more often on what Racket does.

We use Racket values with this type definition:

And the evaluation function can now be:

Note how the arithmetics implementation is simple — it’s a direct translation of the FLANG syntax to Racket operations, and since we don’t check the inputs to the Racket operations, we let Racket throw type errors for us. Note also how function application is just like the arithmetic operations: a FLANG application is directly translated to a Racket application.

However, this does not work quite as simply in Typed Racket. The whole point of typechecking is that we never run into type errors — so we cannot throw back on Racket errors since code that might produce them is forbidden! A way around this is to perform explicit checks that guarantee that Racket cannot run into type errors. We do this with the following two helpers that are defined inside eval:

Note that Typed Racket is “smart enough” to figure out that in evalF the result of the recursive evaluation has to be either Number or (VAL -> VAL); and since the if throws out on numbers, we’re left with (VAL -> VAL) functions, not just any function.

Tuesday, October 15th Recursion, Recursion, Recursion

PLAI §9

There is one major feature that is still missing from our language: we have no way to perform recursion (therefore no kind of loops). So far, we could only use recursion when we had names. In FLANG, the only way we can have names is through with which not good enough for recursion.

To discuss the issue of recursion, we switch to a “broken” version of (untyped) Racket — one where a define has a different scoping rules: the scope of the defined name does not cover the defined expression. Specifically, in this language, this doesn’t work:

In our language, this translation would also not work (assuming we have if etc):

And similarly, in plain Racket this won’t work if let is the only tool you use to create bindings:

In the broken-scope language, the define form is more similar to a mathematical definition. For example, when we write:

it is actually shorthand for

we can then replace defined names with their definitions:

and this can go on, until we get to the actual code that we wrote:

This means that the above fact definition is similar to writing:

which is not a well-formed definition — it is meaningless (this is a formal use of the word “meaningless”). What we’d really want, is to take the equation (using = instead of :=)

and find a solution which will be a value for fact that makes this true.

If you look at the Racket evaluation rules handout on the web page, you will see that this problem is related to the way that we introduced the Racket define: there is a hand-wavy explanation that talks about knowing things.

The big question is: can we define recursive functions without Racket’s magical define form?

Note: This question is a little different than the question of implementing recursion in our language — in the Racket case we have no control over the implementation of the language. As it will eventually turn out, implementing recursion in our own language will be quite easy when we use mutation in a specific way. So the question that we’re now facing can be phrased as either “can we get recursion in Racket without Racket’s magical definition forms?” or “can we get recursion in our interpreter without mutation?”.

Tuesday, October 15th Recursion without the Magic

PLAI §22.4 (we go much deeper)

Note: This explanation is similar to the one you can find in “The Why of Y”, by Richard Gabriel.

To implement recursion without the define magic, we first make an observation: this problem does not come up in a dynamically-scoped language. Consider the let-version of the problem:

This works fine — because by the time we get to evaluate the body of the function, fact is already bound to itself in the current dynamic scope. (This is another reason why dynamic scope is perceived as a convenient approach in new languages.)

Regardless, the problem that we have with lexical scope is still there, but the way things work in a dynamic scope suggest a solution that we can use now. Just like in the dynamic scope case, when fact is called, it does have a value — the only problem is that this value is inaccessible in the lexical scope of its body.

Instead of trying to get the value in via lexical scope, we can imitate what happens in the dynamically scoped language by passing the fact value to itself so it can call itself (going back to the original code in the broken-scope language):

except that now the recursive call should still send itself along:

The problem is that this required rewriting calls to fact — both outside and recursive calls inside. To make this an acceptable solution, calls from both places should not change. Eventually, we should be able to get a working fact definition that uses just

The first step in resolving this problem is to curry the fact definition.

Now fact is no longer our factorial function — it’s a function that constructs it. So call it make-fact, and bind fact to the actual factorial function.

We can try to do the same thing in the body of the factorial function: instead of calling (self self), just bind fact to it:

This works fine, but if we consider our original goal, we need to get that local fact binding outside of the (lambda (n) ...) — so we’re left with a definition that uses the factorial expression as is. So, swap the two lines:

But the problem is that this gets us into an infinite loop because we’re trying to evaluate (self self) too ea(ge)rly. In fact, if we ignore the body of the let and other details, we basically do this:

And this expression has an interesting property: it reduces to itself, so evaluating it gets stuck in an infinite loop.

So how do we solve this? Well, we know that (self self) should be the same value that is the factorial function itself — so it must be a one-argument function. If it’s such a function, we can use a value that is equivalent, except that it will not get evaluated until it is needed, when the function is called. The trick here is the observation that (lambda (n) (add1 n)) is really the same as add1 (provided that add1 is a one-argument function), except that the add1 part doesn’t get evaluated until the function is called. Applying this trick to our code produces a version that does not get stuck in the same infinite loop:

Continuing from here — we know that

(remember how we derived fun from a with), so we can turn that let into the equivalent function application form:

And note now that the (lambda (fact) …) expression is everything that we need for a recursive definition of fact — it has the proper factorial body with a plain recursive call. It’s almost like the usual value that we’d want to define fact as, except that we still have to abstract on the recursive value itself. So lets move this code into a separate definition for fact-step:

We can now proceed by moving the (make-fact make-fact) self application into its own function which is what creates the real factorial:

Rewrite the make-fact definition using an explicit lambda:

and fold the functionality of make-fact and make-real-fact into a single make-fact function by just using the value of make-fact explicitly instead of through a definition:

We can now observe that make-real-fact has nothing that is specific to factorial — we can make it take a “core function” as an argument:

and call it make-recursive:

We’re almost done now — there’s no real need for a separate fact-step definition, just use the value for the definition of fact:

turn the let into a function form:

do some renamings to make things simpler — make and self turn to x, and core to f:

or we can manually expand that first (lambda (x) (x x)) application to make the symmetry more obvious (not really surprising because it started with a let whose purpose was to do a self-application):

And we finally got what we were looking for: a general way to define any recursive function without any magical define tricks. This also work for other recursive functions:

A convenient tool that people often use on paper is to perform a kind of a syntactic abstraction: “assume that whenever I write (twice foo) I really meant to write (foo foo)”. This can often be done as plain abstractions (that is, using functions), but in some cases — for example, if we want to abstract over definitions — we just want such a rewrite rule. (More on this towards the end of the course.) The broken-scope language does provide such a tool — rewrite extends the language with a rewrite rule. Using this, and our make-recursive, we can make up a recursive definition form:

In other words, we’ve created our own “magical definition” form. The above code can now be written in almost the same way it is written in plain Racket:

Finally, note that make-recursive is limited to 1-argument functions only because of the protection from eager evaluation. In any case, it can be used in any way you want, for example,

is a function that returns itself rather than calling itself. Using the rewrite rule, this would be:

which is the same as:

in plain Racket.

Tuesday, October 15th The Core of make-recursive

As in Racket, being able to express recursive functions is a fundamental property of the language. It means that we can have loops in our language, and that’s the essence of making a language powerful enough to be TM-equivalent — able to express undecidable problems, where we don’t know whether there is an answer or not.

The core of what makes this possible is the expression that we have seen in our derivation:

which reduces to itself, and therefore has no value: trying to evaluate it gets stuck in an infinite loop. (This expression is often called “Omega”.)

This is the key for creating a loop — we use it to make recursion possible. Looking at our final make-recursive definition and ignoring for a moment the “protection” that we need against being stuck prematurely in an infinite loop:

we can see that this is almost the same as the Omega expression — the only difference is that application of f. Indeed, this expression (the result of (make-recursive F) for some F) reduces in a similar way to Omega:

which means that the actual value of this expression is:

This definition would be sufficient if we had a lazy language, but to get things working in a strict one we need to bring back the protection. This makes things a little different — if we use (protect f) to be a shorthand for the protection trick,

then we have:

which makes the (make-recursive F) evaluation reduce to

and this is still the same result (as long as F is a single-argument function).

(Note that protect cannot be implemented as a plain function!)

Tuesday, October 15th Denotational Explanation of Recursion

Note: This explanation is similar to the one you can find in “The Little Schemer” called “(Y Y) Works!”, by Dan Friedman and Matthias Felleisen.

The explanation that we have now for how to derive the make-recursive definition is fine — after all, we did manage to get it working. But this explanation was done from a kind of an operational point of view: we knew a certain trick that can make things work and we pushed things around until we got it working like we wanted. Instead of doing this, we can re-approach the problem from a more declarative point of view.

So, start again from the same broken code that we had (using the broken-scope language):

This is as broken as it was when we started: the occurrence of fact in the body of the function is free, which means that this code is meaningless. To avoid the compilation error that we get when we run this code, we can substitute anything for that fact — it’s even better to use a replacement that will lead to a runtime error:

This function will not work in a similar way to the original one — but there is one case where it does work: when the input value is 0 (since then we do not reach the bogus application). We note this by calling this function fact0:

Now that we have this function defined, we can use it to write fact1 which is the factorial function for arguments of 0 or 1:

And remember that this is actually just shorthand for:

We can continue in this way and write fact2 that will work for n<=2:

or, in full form:

If we continue this way, we will get the true factorial function, but the problem is that to handle any possible integer argument, it will have to be an infinite definition! Here is what it is supposed to look like:

The true factorial function is fact-infinity, with an infinite size. So, we’re back at the original problem…

To help make things more concise, we can observe the repeated pattern in the above, and extract a function that abstracts this pattern. This function is the same as the fact-step that we have seen previously:

which is actually:

Do this a little differently — rewrite fact0 as:

Similarly, fact1 is written as:

and so on, until the real factorial, which is still infinite at this stage:

Now, look at that (lambda (mk) ...) — it is an infinite expression, but for every actual application of the resulting factorial function we only need a finite number of mk applications. We can guess how many, and as soon as we hit an application of 777 we know that our guess is too small. So instead of 777, we can try to use the maker function to create and use the next.

To make things more explicit, here is the expression that is our fact0, without the definition form:

This function has a very low guess — it works for 0, but with 1 it will run into the 777 application. At this point, we want to somehow invoke mk again to get the next level — and since 777 does get applied, we can just replace it with mk:

The resulting function works just the same for an input of 0 because it does not attempt a recursive call — but if we give it 1, then instead of running into the error of applying 777:

we get to apply fact-step there:

and this is still wrong, because fact-step expects a function as an input. To see what happens more clearly, write fact-step explicitly:

The problem is in what we’re going to pass into fact-step — its fact argument will not be the factorial function, but the mk function constructor. Renaming the fact argument as mk will make this more obvious (but not change the meaning):

It should now be obvious that this application of mk will not work, instead, we need to apply it on some function and then apply the result on (- n 1). To get what we had before, we can use 777 as a bogus function:

This will allow one recursive call — so the definition works for both inputs of 0 and 1 — but not more. But that 777 is used as a maker function now, so instead, we can just use mk itself again:

And this is a working version of the real factorial function, so make it into a (non-magical) definition:

But we’re not done — we “broke” into the factorial code to insert that (mk mk) application — that’s why we dragged in the actual value of fact-step. We now need to fix this. The expression on that last line

is close enough — it is (fact-step (mk mk)). So we can now try to rewrite our fact as:

… and would fail in a familiar way! If it’s not familiar enough, just rename all those mks as xs:

We’ve run into the eagerness of our language again, as we did before. The solution is the same — the (x x) is the factorial function, so protect it as we did before, and we have a working version:

The rest should not be surprising now… Abstract the recursive making bit in a new make-recursive function:

and now we can do the first reduction inside make-recursive and write the fact-step expression explicitly:

and this is the same code we had before.

Tuesday, October 15th The Y Combinator

Our make-recursive function is usually called the fixpoint operator or the Y combinator.

It looks really simple when using the lazy version (remember: our version is the eager one):

Note that if we do allow a recursive definition for Y itself, then the definition can follow the definition that we’ve seen:

(define (Y f) (f (Y f)))

And this all comes from the loop generated by:

This expression, which is also called Omega (the (lambda (x) (x x)) part by itself is usually called omega and then (omega omega) is Omega), is also the idea behind many deep mathematical facts. As an example for what it does, follow the next rule:

(Note the usage of colon for the first and quotes for the second — what is the equivalent of that in the lambda expression?)

By itself, this just gets you stuck in an infinite loop, as Omega does, and the Y combinator adds F to that to get an infinite chain of applications — which is similar to:

Sidenote: see this SO question and my answer, which came from the PLQ implementation.

Tuesday, October 15th The main property of Y

fact-step is a function that given any limited factorial, will generate a factorial that is good for one more integer input. Start with 777, which is a factorial that is good for nothing (because it’s not a function), and you can get fact0 as

and that’s a good factorial function only for an input of 0. Use that with fact-step again, and you get

which is the factorial function when you only look at input values of 0 or 1. In a similar way

is good for 0…2 — and we can continue as much as we want, except that we need to have an infinite number of applications — in the general case, we have:

which is good for 0…n. The real factorial would be the result of running fact-step on itself infinitely, it is fact-infinity. In other words (here fact is the real factorial):

but note that since this is really infinity, then

so we get an equation:

and a solution for this is going to be the real factorial. The solution is the fixed-point of the fact-step function, in the same sense that 0 is the fixed point of the sin function because

And the Y combinator does just that — it has this property:

or, using the more common name:

This property encapsulates the real magical power of Y. You can see how it works: since (Y f) = (f (Y f)), we can add an f application to both sides, giving us (f (Y f)) = (f (f (Y f))), so we get:

and we can conclude that

Tuesday, October 15th Yet another explanation for Y

Here’s another explanation of how the Y combinator works. Remember that our fact-step function was actually a function that generates a factorial function based on some input, which is supposed to be the factorial function:

As we’ve seen, you can apply this function on a version of factorial that is good for inputs up to some n, and the result will be a factorial that is good for those values up to n+1. The question is what is the fixpoint of fact-step? And the answer is that if it maps factₙ factorial to factₙ₊₁, then the input will be equal to the output on the infinitieth fact, which is the actual factorial. Since Y is a fixpoint combinator, it gives us exactly that answer:

Tuesday, October 15th Typing the Y Combinator

Typing the Y combinator is a tricky issue. For example, in standard ML you must write a new type definition to do this:

Can you find a pattern in the places where T is used? — Roughly speaking, that type definition is

;; `t' is the type name, `T' is the constructor (aka the variant)
(define-type (RecTypeOf t)
  [T ((RecTypeOf t) -> t)])

First note that the two fn a => ... parts are the same as our protection, so ignoring that we get:

val y = fn f => (fn (T x) => (f (x (T x))))
                  (T (fn (T x) => (f (x (T x)))))

if you now replace T with Quote, things make more sense:

val y = fn f => (fn (Quote x) => (f (x (Quote x))))
                  (Quote (fn (Quote x) => (f (x (Quote x)))))

and with our syntax, this would be:

(define (Y f)
  ((lambda (qx)
    (cases qx
      [(Quote x) (f (x qx))]))
  (Quote
    (lambda (qx)
      (cases qx
        [(Quote x) (f (x qx))])))))

it’s not really quotation — but the analogy should help: it uses Quote to distinguish functions as values that are applied (the xs) from functions that are passed as arguments.

In OCaml, this looks a little different:

but OCaml has also a -rectypes command line argument, which will make it infer the type by itself:

The translation of this to #lang pl is a little verbose because we don’t have auto-currying, and because we need to declare input types to functions, but it’s essentially a direct translation of the above:

It is also possible to write this expression in “plain” Typed Racket, without a user-defined type — and we need to start with a proper type definition. First of all, the type of Y should be straightforward: it is a fixpoint operation, so it takes a T -> T function and produces its fixpoint. The fixpoint itself is some T (such that applying the function on it results in itself). So this gives us:

However, in our case make-recursive computes a functional fixpoint, for unary S -> T functions, so we should narrow down the type

Now, in the body of make-recursive we need to add a type for the x argument which is behaving in a weird way: it is used both as a function and as its own argument. (Remember — I will say the next sentence twice: “I will say the next sentence twice”.) We need a recursive type definition helper (not a new type) for that:

This type is tailored for our use of x: it is a type for a function that will consume itself (hence the Rec) and spit out the value that the f argument consumes — an S -> T function.

The resulting full version of the code:

Tuesday, October 22nd Lambda Calculus — Schlac

PLAI §22 (we do much more)

We know that many constructs that are usually thought of as primitives are not really needed — we can implement them ourselves given enough tools. The question is how far can we go?

The answer: as far as we want. For example:

We begin with a very minimal language, which is based on the Lambda Calculus. In this language we get a very minimal set of constructs and values.

In DrRacket, this we will use the Schlac language level (stands for “SchemeRacket as Lambda Calculus”). This language has a Racket-like syntax, but don’t be confused — it is very different from Racket. The only constructs that are available in this language are: lambda expressions of at least one argument, function application (again, at least one argument), and simple definition forms which are similar to the ones in the “Broken define” language — definitions are used as shorthand, and cannot be used for recursive function definition. They’re also only allowed at the toplevel — no local helpers, and a definition is not an expression that can appear anywhere. The BNF is therefore:

Since this language has no primitive values (other than functions), Racket numbers and booleans are also considered identifiers, and have no built-in value that come with the language. In addition, all functions and function calls are curried, so

is actually shorthand for

The rules for evaluation are simple, there is one very important rule for evaluation which is called “beta reduction”:

where substitution in this context requires being careful so you won’t capture names. This requires you to be able to do another kind of transformation which is called “alpha conversion”, which basically says that you can rename identifiers as long as you keep the same binding structure (eg, a valid renaming does not change the de-Bruijn form of the expression). There is one more rule that can be used, eta conversion which says that (lambda (x) (f x)) is the same as f (we used this rule above when deriving the Y combinator).

One last difference between Schlac and Racket is that Schlac is a lazy language. This will be important since we do not have any built-in special forms like if.

Here is a Schlac definition for the identity function:

and there is not much that we can do with this now:

(In the last expression, note that (id id id) is shorthand for ((id id) id), and since (id id) is the identity, applying that on id returns it again.)

Something to think about: are we losing anything because we have no no-argument functions?

Tuesday, October 22nd Church Numerals

So far, it seems like it is impossible to do anything useful in this language, since all we have are functions and applications. We know how to write the identity function, but what about other values? For example, can you write code that evaluates to zero?

What’s zero? I only know how to write functions!

(Turing Machine / Assembly programmer: “What’s a function? — I only know how to write 0s and 1s!”)

The first thing we therefore need is to be able to encode numbers as functions. For zero, we will use a function of two arguments that simply returns its second value:

or, more concisely

This is the first step in an encoding that is known as Church Numerals: an encoding of natural numbers as functions. The number zero is encoded as a function that takes in a function and a second value, and applies the function zero times on the argument (which is really what the above definition is doing). Following this view, the number one is going to be a function of two arguments, that applies the first on the second one time:

and note that 1 is just like the identity function (as long as you give it a function as its first input, but this is always the case in Schlac). The next number on the list is two — which applies the first argument on the second one twice:

We can go on doing this, but what we really want is a way to perform arbitrary arithmetic. The first requirement for that is an add1 function that increments its input (an encoded natural number) by one. To do this, we write a function that expects an encoded number:

and this function is expected to return an encoded number, which is always a function of f and x:

Now, in the body, we need to apply f on x n+1 times — but remember that n is a function that will do n applications of its first argument on its second:

and all we have left to do now is to apply f one more time, yielding this definition for add1:

Using this, we can define a few useful numbers:

This is all nice theoretically, but how can we make sure that it is correct? Well, Schlac has a few additional built-in functions that translate Church numerals into Racket numbers. To try our definitions we use the ->nat (read: to natural number):

You can now verify that the identity function is really the same as the number 1:

We can even write a test case, since Schlac contains the test special form, but we have to be careful in that — first of all, we cannot test whether functions are equal (why?) so we must use ->nat, but

will not work since 7 is undefined. To overcome this, Schlac has a back-door for primitive Racket values — just use a quote:

We can now define natural number addition — one simple idea is to get two encoded numbers m and n, then start with x, apply f on it n times by using it as a function, then apply f m more times on the result in the same way:

or equivalently:

Another idea is to use add1 and increment n by m using add1:

We can also define multiplication of m and n quite easily — begin with addition — (lambda (x) (+ n x)) is a function that expects an x and returns (+ x n) — it’s an increment-by-n function. But since all functions and applications are curried, this is actually the same as (lambda (x) ((+ n) x)) which is the same as (+ n). Now, what we want to do is repeat this operation m times over zero, which will add n to zero m times, resulting in m * n. The definition is therefore:

An alternative approach is to consider

for some encoded number n and a function f — this function is like f^n (f composed n times with itself). But remember that this is shorthand for

and we know that (lambda (x) (foo x)) is just like foo (if it is a function), so this is equivalent to just

So (n f) is f^n, and in the same way (m g) is g^m — if we use (n f) for g, we get (m (n f)) which is n self-compositions of f, self-composed m times. In other words, (m (n f)) is a function that is like m*n applications of f, so we can define multiplication as:

which is the same as

The same principle can be used to define exponentiation (but now we have to be careful with the order since exponentiation is not commutative):

And there is a similar alternative here too —

• a Church numeral m is the m-self-composition function,

• and (1 m) is just like m^1 which is the same as m (1=identity)

• and (2 m) is just like m^2 — it takes a function f, self composes it m times, and self composes the result m times — for a total of f^(m*m)

• and (3 m) is similarly f^(m*m*m)

• so (n m) is f^(m^n) (note that the first ^ is self-compositions, and the second one is a mathematical exponent)

• so (n m) is a function that returns m^n self-compositions of an input function, Which means that (n m) is the Church numeral for m^n, so we get:

  (define ^ (lambda (m n) (n m)))

which basically says that any number encoding n is also the ?^n operation.

All of this is was not too complicated — but all so far all we did is write functions that increment their inputs in various ways. What about sub1? For that, we need to do some more work — we will need to encode booleans.

Tuesday, October 22nd More Encodings

Our choice of encoding numbers makes sense — the idea is that the main feature of a natural number is repeating something a number of times. For booleans, the main property we’re looking for is choosing between two values. So we can encode true and false by functions of two arguments that return either the first or the second argument:

Note that this encoding of #f is really the same as the encoding of 0, so we have to know what type to expect an use the proper operations (this is similar to C, where everything is just integers). Now that we have these two, we can define if:

it expects a boolean which is a function of two arguments, and passes it the two expressions. The #t boolean will simply return the first, and the #f boolean will return the second. Strictly speaking, we don’t really need this definition, since instead of writing (if c t e), we can simply write (c t e). In any case, we need the language to be lazy for this to work. To demonstrate this, we’ll intentionally use the quote back-door to use a non-functional value, using this will normally result in an error:

But testing our if definition, things work just fine:

and we see that DrRacket leaves the second addition expression in red, which indicates that it was not executed. We can also make sure that even when it is defined as a function, it is still working fine because the language is lazy:

What about and and or? Simple, or takes two arguments, and returns either true or false if one of the inputs is true:

but (if b #t #f) is really the same as just b because it must be a boolean (we cannot use more than one “truty” or “falsy” values):

also, if a is true, we want to return #t, but that is exactly the value of a, so:

and finally, we can get rid of the if (which is actually breaking the if abstraction, if we encode booleans in some other way):

Similarly, you can convince yourself that the definition of and is:

Schlac has to-Racket conversion functions for booleans too:

and

A not function is quite simple — one alternative is to choose from true and false in the usual way:

and another is to return a function that switches the inputs to an input boolean:

which is the same as

We can now put numbers and booleans together: we define a zero? function.

(Good question: is this fast?)

(Note that it is better to test that the value is explicitly #t, if we just use (test (->bool ...)) then the test will work even if the expression in question evaluated to some bogus value.)

The idea is simple — if n is the encoding of zero, it will return it’s second argument which is #t:

if n is an encoding of a bigger number, then it is a self-composition, and the function that we give it is one that always returns #f, no matter how many times it is self-composed. Try 2 for example:

Now, how about an encoding for compound values? A minimal approach is what we use in Racket — a way to generate pairs (cons), and encode lists as chains of pairs with a special value at the end (null). There is a natural encoding for pairs that we have previously seen — a pair is a function that expects a selector, and will apply that on the two values:

Or, equivalently:

To extract the two values from a pair, we need to pass a selector that consumes two values and returns one of them. In our framework, this is exactly what the two boolean values do, so we get:

We can even do this:

or write a list-ref function:

Note that we don’t need a recursive function for this: our encoding of natural numbers makes it easy to “iterate N times”. What we get with this encoding is essentially free natural-number recursion.

We now need a special null value to mark list ends. This value should have the same number of arguments as a cons value (one: a selector/boolean function), and it should be possible to distinguish it from other values. We choose

Testing the list encoding:

And as with natural numbers and booleans, Schlac has built-in facility to convert encoded lists to Racket values, except that this requires specifying the type of values in a list so it’s a higher-order function:

which (“as usual”) can be written as

We can even do this:

Defining null? is now relatively easy (and it’s actually already used by the above ->listof conversion). The following definition

works because if x is null, then it simply ignores its argument and returns #t, and if it’s a pair, then it uses the input selector, which always returns #f in its turn. Using some arbitrary A and B:

We can use the Y combinator to create recursive functions — we can even use the rewrite rules facility that Schlac contains (the same one that we have previously seen):

and using it:

And to complete this, um, journey — we’re still missing subtraction. There are many ways to solve the problem of subtraction, and for a challenge try to come up with a solution yourself. One of the clearer solutions uses a simple idea — begin with a pair of two zeroes <0,0>, and repeat this transformation n times: <a,b> -> <b,b+1>. After n steps, we will have <n-1,n> — so we get:

And from this the road is short to general subtraction, m-n is simply n applications of sub1 on m:

We now have a normal-looking language, and we’re ready to do anything we want. Here are two popular examples:

To get generalized arithmetic capability, Schlac has yet another built-in facility for translating Racket natural numbers into Church numerals:

… and to get to that frightening expression in the beginning, all you need to do is replace all definitions in the fib definition over and over again until you’re left with nothing but lambda expressions and applications, then reformat the result into some cute shape. For extra fun, you can look for immediate applications of lambda expressions and reduce them manually.

All of this is in the following code:

Tuesday, October 22nd Alternative Church Encoding

Finally, note that this is just one way to encode things — other encodings are possible. One alternative encoding is in the following code — it uses a list of N falses as the encoding for N. This encoding makes it easier to add1 (just cons another #f), and to sub1 (simply cdr). The tradeoff is that some arithmetics operations becomes more complicated, for example, the definition of + requires the fixpoint combinator. (As expected, some people want to see what can we do with a language without recursion, so they don’t like jumping to Y too fast.)

Tuesday, October 22nd Implementing define-type & cases in Schlac

(The following explanation originates from Chris Okasaki.)

Another interesting way to implement lists follows the pattern matching approach, where both pairs and the null value are represented by a function that serves as a kind of a match dispatcher. This function takes in two inputs — if it is the representation of null then it will return the first input, and if it is a pair, then it will apply the second input on the two parts of the pair. This is implemented as follows (with type comments to make it clear):

This might seem awkward, but it follows the intended use of pairs and null as a match-like construct. Here is an example, with the equivalent Racket code on the side:

In fact, it’s easy to implement our selectors and predicate using this:

The same approach can be used to define any kind of new data type in a way that looks like our own define-type definitions. For example, consider a much-simplified definition of the AE type we’ve seen early in the semester, and a matching eval definition as an example for using cases:

We can follow the above approach now to write Schlac code that more than being equivalent, is also very similar in nature. Note that the type definition is replaced by two definitions for the two constructors:

We can even take this further: the translations from define-type and cases are mechanical enough that we could implement them almost exactly via rewrites (there are a subtle change in that we’re list field names rather than types):

And using that, an evluator is simple:

Tuesday, October 22nd Recursive Environments

PLAI §11.5

What we really need for recursion, is a special kind of an environment, one that can refer to itself. So instead of doing (note: calls removed for readability):

which does not work for the usual reasons, we want to use some

that will do the necessary magic.

One way to achieve this is using the Y combinator as we have seen — a kind of a “constructor” for recursive functions. We can do that in a similar way to the rewrite rule that we have seen in Schlac — translate the above expression to:

or even:

Now, we will see how it can be used in our code to implement a recursive environment.

If we look at what with does in

then we can say that to evaluate this expression, we evaluate the body expression in an extended environment that contains fact, even if a bogus one that is good for 0 only — the new environment is created with something like this:

so we can take this whole thing as an operation over env

This gives us the first-level fact. But fact itself is still undefined in env, so it cannot call itself. We can try this:

but that still doesn’t work, and it will never work no matter how far we go:

What we really want is infinity: a place where add-fact works and the result is the same as what we’ve started with — we want to create a “magical” environment that makes this possible:

which basically gives us the illusion of being at the infinity point. This magic-env thing is exactly the fixed-point of the add-fact operation. We can use:

and following the main property of the Y combinator, we know that:

What does all this mean? It means that if we have a fixed-point operator at the level of the implementation of our environments, then we can use it to implement a recursive binder. In our case, this means that a fixpoint in Racket can be used to implement a recursive language. But we have that — Racket does have recursive functions, so we should be able to use that to implement our recursive binder.

There are two ways that make it possible to write recursive functions in Racket. One is to define a function, and use its name to do a recursive call — using the Racket formal rules, we can see that we said that we mark that we now know that a variable is bound to a value. This is essentially a side-effect — we modify what we know, which corresponds to modifying the global environment. The second way is a new form: letrec. This form is similar to let, except that the scope that is established includes the named expressions — it is exactly what we want rec to do. A third way is using recursive local definitions, but that is equivalent to using letrec, more on this soon.

Tuesday, October 22nd Recursion: Racket’s letrec

So we want to add recursion to our language, practically. We already know that Racket makes it possible to write recursive functions, which is possible because of the way it implements its “global environment”: our evaluator can only extend an environment, while Racket modifies its global environment. This means that whenever a function is defined in the global environment, the resulting closure will have it as its environment “pointer”, but the global environment was not extended — it stays the same, and was just modified with one additional binding.

But Racket has another, a bit more organized way of using recursion: there is a special local-binding construct that is similar to let, but allows a function to refer to itself. It is called letrec:

Some people may remember that there was a third way for creating recursive functions: using local definition in function bodies. For example, we have seen things like:

This looks like the same kind of environment magic that happens with a global define — but actually, Racket defines the meaning of internal definitions using letrec — so the above code is exactly the same as:

The scoping rules for a letrec is that the scope of the bound name covers both the body and the named expression. Furthermore, multiple names can be bound to multiple expressions, and the scope of each name covers all named expression as well as the body. This makes it easy to define mutually recursive functions, such as:

But it is not a required functionality — it could be done with a single recursive binding that contains several functions:

This is basically the same problem we face if we want to use the Y combinator for mutually recursive bindings. The above solution is inconvenient, but it can be improved using more lets to have easier name access. For example:

Tuesday, October 22nd Implementing Recursion using letrec

We will see how to add a similar construct to our language — for simplicity, we will add a rec form that handles a single binding:

Using this, things can get a little tricky. What should we get if we do:

? Currently, it seems like there is no point in using any expression except for a function expression in a rec expression, so we will handle only these cases.

(BTW, under what circumstances would non-function values be useful in a letrec?)

---

One way to achieve this is to use the same trick that we have recently seen: instead of re-implementing language features, we can use existing features in our own language, which hopefully has the right functionality in a form that can be re-used to in our evaluator.

Previously, we have seen a way to implement environments using Racket closures:

We can use this implementation, and create circular environments using Racket’s letrec. The code for handling a with expressions is:

It looks like we should be able to handle rec in a similar way (the AST constructor name is WRec (“with-rec”) so it doesn’t collide with TR’s Rec constructor for recursive types):

but this won’t work because the named expression is evaluated prematurely, in the previous environment. Instead, we will move everything that needs to be done, including evaluation, to a separate extend-rec function:

Now, the extend-rec function needs to provide the new, “magically circular” environment. Following what we know about the arguments to extend-rec, and the fact that it returns a new environment (= a lookup function), we can sketch a rough definition:

What should the missing expression be? It can simply evaluate the object given itself:

But how do we get this environment, before it is defined? Well, the environment is itself a Racket function, so we can use Racket’s letrec to make the function refer to itself recursively:

It’s a little more convenient to use an internal definition, and add a type for clarity:

This works, but there are several problems:

1. First, we no longer do a simple lookup in the new environment. Instead, we evaluate the expression on every such lookup. This seems like a technical point, because we do not have side-effects in our language (also because we said that we want to handle only function expressions). Still, it wastes space since each evaluation will allocate a new closure.

2. Second, a related problem — what happens if we try to run this:

   {rec {x x} x}

   ? Well, we do that stuff to extend the current environment, then evaluate the body in the new environment, this body is a single variable reference:

   (eval (Id 'x) the-new-env)

   so we look up the value:

   (lookup 'x the-new-env)

   which is:

   (the-new-env 'x)

   which goes into the function which implements this environment, there we see that name is the same as name1, so we return:

   (eval expr rec-env)

   but the expr here is the original named-expression which is itself (Id 'x), and we’re in an infinite loop.

We can try to get over these problems using another binding. Racket allows several bindings in a single letrec expression or multiple internal function definitions, so we change extend-rec to use the newly-created environment:

This runs into an interesting type error, which complains about possibly getting some Undefined value. It does work if we switch to the untyped language for now (using #lang pl untyped) — and it seems to run fine too. But it raises more questions, beginning with: what is the meaning of:

or equivalently, an internal block of

? Well, DrRacket seems to do the “right thing” in this case, but what about:

? As a hint, see what happens when we now try to evaluate the problematic

expression, and compare that with the result that you’d get from Racket. This also clarifies the type error that we received.

It should be clear now why we want to restrict usage to just binding recursive functions. There are no problems with such definitions because when we evaluate a fun expression, there is no evaluation of the body, which is the only place where there are potential references to the same function that is defined — a function’s body is delayed, and executed only when the function is applied later.

But the biggest question that is still open: we just implemented a circular environment using Racket’s own circular environment implementation, and that does not explain how they are actually implemented. The cycle of pointers that we’ve implemented depends on the cycle of pointers that Racket uses, and that is a black box we want to open up.

For reference, the complete code is below.

Tuesday, October 29th Implementing rec Using Cyclic Structures

PLAI §10

Looking at the arrows in the environment diagrams, what we’re really looking for is a closure that has an environment pointer which is the same environment in which it was defined. This will make it possible for fact to be bound to a closure that can refer to itself since its environment is the same one in which it is defined. However, so far we have no tools that makes it possible to do this.

What we need is to create a “cycle of pointers”, and so far we do not have a way of achieving that: when we create a closure, we begin with an environment which is saved in the slot’s environment slot, but we want that closure to be the value of a binding in that same environment.

Tuesday, October 29th Boxes and Mutation

To actually implement a circular structure, we will now use side-effects, using a new kind of Racket value which supports mutation: a box. A box value is built with the box constructor:

the value is retrieved with the `unbox’ function,

and finally, the value can be changed with the set-box! function.

An important thing to note is that set-box! is much like display etc, it returns a value that is not printed in the Racket REPL, because there is no point in using the result of a set-box!, it is called for the side-effect it generates. (Languages like C blur this distinction between returning a value and a side-effect with its assignment statement.)

As a side note, we now have side effects of two kinds: mutation of state, and I/O (at least the O part). (Actually, there is also infinite looping that can be viewed as another form of a side effect.) This means that we’re now in a completely different world, and lots of new things can make sense now. A few things that you should know about:

• We never used more than one expression in a function body because there was no point in it, but now there is. To evaluate a sequence of Racket expressions, you wrap them in a begin expression.

• In most places you don’t actually need to use begin — these are places that are said to have an implicit begin: the body of a function (or any lambda expression), the body of a let (and let-relatives), the consequence positions in cond, match, and cases clauses and more. One of the common places where a begin is used is in an if expression (and some people prefer using cond instead when there is more than a single expression).

• cond without an else in the end can make sense, if all you’re using it it for is side-effects.

• if could get a single expression which is executed when the condition is true (and an unspecified value is used otherwise), but our language (as well as the default Racket language) always forbids this — there are convenient special forms for a one-sided ifs: when & unless, and they can have any number of expressions (they have an implicit begin). They have an advantage of saying “this code does some side-effects here” more explicit.

• There is a function called for-each which is just like map, except that it doesn’t collect the list of results, it is used only for performing side effects.

• Aliasing and the concept of “object equality”: equal? vs eq?. For example:

  (: foo : (Boxof ...) (Boxof ...) -> ...)
  (define (foo a b)
    (set-box! a 1)) ;*** this might change b, can check `eq?`

When any one of these things is used (in Racket or other languages), you can tell that side-effects are involved, because there is no point in any of them otherwise. In addition, any name that ends with a ! (“bang”) is used to mark a function that changes state (usually a function that only changes state).

So how do we create a cycle? Simple, boxes can have any value, and they can be put in other values like lists, so we can do this:

and we get a circular value. (Note how it is printed.) And with types:

Tuesday, October 29th Types for Boxes

Obviously, Any is not too great — it is the most generic type, so it provides the least information. For example, notice that

returns the right list, which is equal to foo itself — but if we try to grab some part of the resulting list:

we get a type error, because the result of the unbox is Any, so Typed Racket knows nothing about it, and won’t allow you to treat it as a list. It is not too surprising that the type constructor that can help in this case is Rec which we have already seen — it allows a type that can refer to itself:

Note that either foo or the value in the box are both printed with a Rec type — the value in the box can’t just have a (U #f this) type, since this doesn’t mean anything in there, so the whole type needs to still be present.

There is another issue to be aware of with Boxof types. For most type constructors (like Listof), if T1 is a subtype of T2, then we also know that(Listof T1) is a subtype of (Listof T2). This makes the following code typecheck:

Since the (Listof Integer) is a subtype of the (Listof Number) input for foo, the application typechecks. But this is not the same for the output type, for example — if we change the bar type to:

we get a type error since Number is not a subtype of Integer. So subtypes are required to “go higher” on the input side and “lower” on the other. So, in a sense, the fact that boxes are mutable means that their contents can be considered to be on the other side of the arrow, which is why for such T1 subtype of T2, it is (Boxof T2) that is a subtype of (Boxof T1), instead of the usual. For example, this doesn’t work:

And you can see why this is the case — the marked line is fine given a Number contents, so if the type checker allows passing in a box holding an integer, then that expression would mutate the contents and make it an invalid value.

However, boxes are not only mutable, they hold a value that can be read too, which means that they’re on both sides of the arrow, and this means that (Boxof T1) is a subtype of (Boxof T2) if T2 is a subtype of T1 and T1 is a subtype of T2 — in other words, this happens only when T1 and T2 are the same type. (See below for an extended demonstration of all of this.)

Note also that this demonstration requires that extra b definition, if it’s skipped:

then this will typecheck again — Typed Racket will just consider the context that requires a box holding a Number, and it is still fine to initialize such a box with an Integer value.

As a side comment, this didn’t always work. Earlier in its existence, Typed Racket would always choose a specific type for values, which would lead to confusing errors with boxes. For example, the above would need to be written as

(define (bar x)
  (foo (box (ann x : Number))))

to prevent Typed Racket from inferring a specific type. This is no longer the case, but there can still be some surprises. A similar annotation was needed in the case of a list holding a self-referential box, to avoid the initial #f from getting a specific-but-wrong type.

Another way to see the problem these days is to enter the following expressions and see what types Typed Racket guesses for them:

> (define a 0)
> (define b (box 0))
> a
- : Integer [more precisely: Zero]        ;***
0
> b
- : (Boxof Integer)                        ;***
'#&0

Note that for a, the assigned type is very specific, because Typed Racket assumes that it will not change. But with a boxed value, using a type of (Boxof Zero) would lead to a useless box, since it’ll only allow using set-box! with 0, and therefore can never change. This shows that this is exactly that: a guess given the lack or explicit user-specified type, so there’s no canonical guess that can be inferred here.

Tuesday, October 29th Boxof’s Lack of Subtyping

The lack of any subtype relations between (Boxof T) and (Boxof S) regardless of S and T can roughly be explained as follows.

First, a box is a container that you can pull a value out of — which makes it similar to lists. In the case of lists, we have:

This is true for all such containers that you can pull a value out of: if you expect to pull a T but you’re given a container of a subtype S, then things are still fine (you’ll get an S which is also a T). Such “containers” include functions that produce a value — for example:

However, functions also have the other side, where things are different — instead of a side of some produced value, it’s the side of the consumed value. We get the opposite rule there:

To see why this is right, use Number and Integer for S and T:

so — if you expect a function that takes an integer, a valid subtype value that I can give you is a function that takes a number. In other words, every function that takes a number is also a function that takes an integer, but not the other way.

To summarize all of this, when you make the output type of a function “smaller” (more constrained), the resulting type is smaller (a subset), but on the input side things are flipped — a bigger input type means a more constrained function.

The technical names for these properties are: a “covariant” type is one that preserves the subtype relationship, and a “contravairant” type is one that reverses it. (Which is similar to how these terms are used in math.)

(Side note: this is related to the fact that in logic, P => Q is roughly equivalent to not(P) or Q — the left side, P, is inside negation. It also explains why in ((S -> T) -> Q) the S obeys the first rule, as if it was on the right side — because it’s negated twice.)

Now, a (Boxof T) is a producer of T when you pull a value out of the box, but it’s also a consumer of T when you put such a value in it. This means that — using the above analogy — the T is on both sides of the arrow. This means that

which is actually:

A different way to look at this conclusion is to consider the function type of (A -> A): when is it a subtype of some other (B -> B)? Only when A is a subtype of B and B is a subtype of A, which means that this happens only when A and B are the same type.

The term for this is “nonvariant” (or “invariant”): (A -> A) is unrelated to (B -> B) regardless of how A and B are related. The only exception is, of course, when they are the same type. The Wikipedia entry about these puts the terms together nicely in the face of mutation:

Read-only data types (sources) can be covariant; write-only data types (sinks) can be contravariant. Mutable data types which act as both sources and sinks should be invariant.

The following piece of code makes the analogy to function types more formally. Boxes behave as if their contents is on both sides of a function arrow — on the right because they’re readable, and on the left because they’re writable, which the conclusion that a (Boxof A) type is a subtype of itself and no other (Boxof B).

Tuesday, October 29th Implementing a Circular Environment

We now use this to implement rec in the following way:

1. Change environments so that instead of values they hold boxes of values: (Boxof VAL) instead of VAL, and whenever lookup is used, the resulting boxed value is unboxed,

2. In the WRec case, create the new environment with some temporary binding for the identifier — any value will do since it should not be used (when named expressions are always fun expressions),

3. Evaluate the expression in the new environment,

4. Change the binding of the identifier (the box) to the result of this evaluation.

The resulting definition is:

Racket has another let relative for such cases of multiple-nested lets — let*. This form is a derived form — it is defined as a shorthand for using nested lets. The above is therefore exactly the same as this code:

This let* form can be read almost as a C/Java-ish kind of code:

The code can be simpler if we fold the evaluation into the set-box! (since value is used just there), and if use lookup to do the mutation — since this way there is no need to hold onto the box. This is a bit more expensive, but since the binding is guaranteed to be the first one in the environment, the addition is just one quick step. The only binding that we need is the one for the new environment, which we can do as an internal definition, leaving us with:

A complete rehacked version of FLANG with a rec binding follows. We can’t test rec easily since we have no conditionals, but you can at least verify that

is an infinite loop.

Tuesday, October 29th Variable Mutation

PLAI §12 and PLAI §13 (different: adds boxes to the language)

PLAI §14 (that’s what we do)

The code that we now have implements recursion by changing bindings, and to make that possible we made environments hold boxes for all bindings, therefore bindings are all mutable now. We can use this to add more functionality to our evaluator, by allowing changing any variable — we can add a set! form:

to the evaluator that will modify the value of a variable. To implement this functionality, all we need to do is to use lookup to retrieve some box, then evaluate the expression and put the result in that box. The actual implementation is left as a homework exercise.

One thing that should be considered here is — all of the expressions in our language evaluate to some value, the question is what should be the value of a set! expression? There are three obvious choices:

1. return some bogus value,

2. return the value that was assigned,

3. return the value that was previously in the box.

Each one of these has its own advantage — for example, C uses the second option to chain assignments (eg, x = y = 0) and to allow side effects where an expression is expected (eg, while (x = x-1) ...).

The third one is useful in cases where you might use the old value that is overwritten — for example, if C had this behavior, you could pop a value from a linked list using something like:

because the argument to first will be the old value of stack, before it changed to be its rest. You could also swap two variables in a single expression: x = y = x.

(Note that the expression x = x + 1 has the meaning of C’s ++x when option (2) is used, and x++ when option (3) is used.)

Racket chooses the first option, and we will do the same in our language. The advantage here is that you get no discounts, therefore you must be explicit about what values you want to return in situations where there is no obvious choice. This leads to more robust programs since you do not get other programmers that will rely on a feature of your code that you did not plan on.

In any case, the modification that introduces mutation is small, but it has a tremendous effect on our language: it was true for Racket, and it is true for FLANG. We have seen how mutation affects the language subset that we use, and in the extension of our FLANG the effect is even stronger: since any variable can change (no need for explicit boxes). In other words, a binding is not always the same — in can change as a result of a set! expression. Of course, we could extend our language with boxes (using Racket boxes to implement FLANG boxes), but that will be a little more verbose.

Note that Racket does have a set! form, and in addition, fields in structs can be made modifiable. However, we do not use any of these. At least not for now.

Tuesday, October 29th State and Environments

A quick example of how mutation can be used:

and compare that to:

It is a good idea if you follow the exact evaluation of

and see how both bindings have separate environment so each one gets its own private state. The equivalent code in the homework interpreter extended with set! doesn’t need boxes:

To see multiple values from a single expression you can extend the language with a list binding. As a temporary hack, we can use dummy function inputs to cover for our lack of nullary functions, and use with (with dummy bound ids) to sequence multiple expressions:

Note that we cannot describe this behavior with substitution rules! We now use the environments to make it possible to change bindings — so finally an environment is actually an environment rather than a substitution cache.

When you look at the above, note that we still use lexical scope — in fact, the local binding is actually a private state that nobody can access. For example, if we write this:

then the resulting function that us bound to counter keeps a local integer state which no other code can access — you cannot modify it, reset it, or even know if it is really an integer that is used in there.

Tuesday, October 29th Implementing Objects with State

We have already seen how several pieces of information can be encapsulate in a Racket closure that keeps them all; now we can do a little more — we can actually have mutable state, which leads to a natural way to implement objects. For example:

implements a constructor for point objects which keep two values and can move one of them. Note that the messages act as a form of methods, and that the values themselves are hidden and are accessible only through the interface that these messages make. For example, if these points correspond to some graphic object on the screen, we can easily incorporate a necessary screen update:

and be sure that this is always done when the value changes — since there is no way to change the value except through this interface.

A more complete example would define functions that actually send these messages — here is a better implementation of a point object and the corresponding accessors and mutators:

And a quick imitation of inheritance can be achieved using delegation to an instance of the super-class:

You can see how all of these could come from some preprocessing of a more normal-looking class definition form, like:

Tuesday, October 29th The Toy Language

Not in PLAI

A quick note: from now on, we will work with a variation of our language — it will change the syntax to look a little more like Racket, and we will use Racket values for values in our language and Racket functions for built-ins in our language.

Main highlights:

• There can be multiple bindings in function arguments and local bind forms — the names are required to be distinct.

• There are now a few keywords like bind that are parsed in a special way. Other forms are taken as function application, which means that there are no special parse rules (and AST entries) for arithmetic functions. They’re now bindings in the global environment, and treated in the same way as all bindings. For example, * is an expression that evaluates to the primitive multiplication function, and {bind {{+ *}} {+ 2 3}} evaluates to 6.

• Since function applications are now the same for primitive functions and user-bound functions, there is no need for a call keyword. Note that the function call part of the parser must be last, since it should apply only if the input is not some other known form.

• Note the use of make-untyped-list-function: it’s a library function (included in the course language) that can convert a few known Racket functions to a function that consumes a list of any Racket values, and returns the result of applying the given Racket function on these values. For example:

  (define add (make-untyped-list-function +))
  (add (list 1 2 3 4))

  evaluates to 10.

• Another important aspect of this is its type — the type of add in the previous example is (List -> Any), so the resulting function can consume any input values. If it gets a bad value, it will throw an appropriate error. This is a hack: it basically means that the resulting add function has a very generic type (requiring just a list), so errors can be thrown at run-time. However, in this case, a better solution is not going to make these run-time errors go away because the language that we’re implementing is not statically typed.

• The benefit of this is that we can avoid the hassle of more verbose code by letting these functions dynamically check the input values, so we can use a single RktV variant in VAL which wraps any Racket value. (Otherwise we’d need different wrappers for different types, and implement these dynamic checks.)

The following is the complete implementation.

Tuesday, November 5th Compilation

Instead of interpreting an expression, which is performing a full evaluation, we can think about compiling it: translating it to a different language which we can later run more easily, more efficiently, on more platforms, etc. Another feature that is usually associated with compilation is that a lot more work was done at the compilation stage, making the actual running of the code faster.

For example, translating an AST into one that has de-Bruijn indexes instead of identifier names is a form of compilation — not only is it translating one language into another, it does the work involved in name lookup before the program starts running.

This is something that we can experiment with now. An easy way to achieve this is to start with our evaluation function:

and change it so it compiles a given expression to a Racket function. (This is, of course, just to demonstrate a conceptual point, it is only the tip of what compilers actually do…) This means that we need to turn it into a function that receives a TOY expression and compiles it. In other words, eval no longer consumes and environment argument which makes sense because the environment is a place to hold run-time values, so it is a data structure that is not part of the compiler (it is usually represented as the call stack).

So we split the two arguments into a compile-time and run-time, which can be done by simply currying the eval function — here this is done, and all calls to eval are also curried:

We also need to change the eval call in the main run function:

Not much has changed so far.

Note that in the general case of a compiler we need to run a program several times, so we’d want to avoid parsing it over and over again. We can do that by keeping a single parsed AST of the input. Now we went one step further by making it possible to do more work ahead and keep the result of the first “stage” of eval around (except that “more work” is really not saying much at the moment):

At this point, even though our “compiler” is not much more than a slightly different representation of the same functionality, we rename eval to compile which is a more appropriate description of what we intend it to do (so we change the purpose statement too):

---

Not much changed, still. We curried the eval function and renamed it to compile. But when we actually call compile almost nothing happens — all it does is create a Racket closure which will do the rest of the work. (And this closure closes over the given expression.)

Running this “compiled” code is going to be very much like the previous usage of eval, except a little slower, because now every recursive call involves calling compile to generate a closure, which is then immediately used — so we just added some allocations at the recursive call points! (Actually, the extra cost is minimal because the Racket compiler will optimize away such immediate closure applications.)

Another way to see how this is not really a compiler yet is to consider when compile gets called. A proper compiler is something that does all of its work before running the code, which means that once it spits out the compiled code it shouldn’t be used again (except for compiling other code, of course). Our current code is not really a compiler since it breaks this feature. (For example, if GCC behaved this way, then it would “compile” files by producing code that invokes GCC to compile the next step, which, when run, invokes GCC again, etc.)

However, the conceptual change is substantial — we now have a function that does its work in two stages — the first part gets an expression and can do some compile-time work, and the second part does the run-time work, and includes anything inside the (lambda (env) …). The thing is that so far, the code does nothing at the compilation stage (remember: only creates a closure). But because we have two stages, we can now shift work from the second stage (the run-time) to the first (the compile-time).

For example, consider the following simple example:

We can do the same thing here — separate foo it into two stages using currying, and modify bar appropriately:

Now instead of a simple multiplication, lets expand it a little, for example, do a case split on common cases where x is 0, 1, or 2:

This is not much faster, since Racket already optimizes multiplication in a similar way.

Now comes the real magic: deciding what branch of the cond to take depends only on x, so we can push the lambda inside:

We just made an improvement — the comparisons for the common cases are now done as soon as (foo x) is called, they’re not delayed to when the resulting function is used. Now go back to the way this is used in bar and make it call foo once for the given c:

Now foo-c is generated once, and if c happens to be one of the three common cases (as in the last expression), we can avoid doing any multiplication. (And if we hit the default case, then we’re doing the same thing we did before.)

[However, the result runs at roughly the same speed! This heavily depends on what kind of optimizations the compiler can do, in this case, optimizing multiplications (which are essentially a single machine-code instruction) vs optimizing multiple-stage function calls.]

Here is another useful example that demonstrates this:

(Question: when can you do that?)

This is not unique to Racket, it can happen in any language. Racket (or any language with first class function values) only makes it easy to create a local function that is specialized for the flag.

---

Getting our thing closer to a compiler is done in a similar way — we push the (lambda (env) ...) inside the various cases. (Note that compile* depends on the env argument, so it also needs to move inside — this is done for all cases that use it, and will eventually go away.) We actually need to use (lambda ([env : ENV]) ...) though, to avoid upsetting the type checker:

and with this we shifted a bit of actual work to compile time — the code that checks what structure we have, and extracts its different slots. But this is still not good enough — it’s only the first top-level cases that is moved to compile-time — recursive calls to compile are still there in the resulting closures. This can be seen by the fact that we have those calls to compile in the Racket closures that are the results of our compiler, which, as discussed above, mean that it’s not an actual compiler yet.

For example, consider the Bind case:

At compile-time we identify and deconstruct the Bind structure, then create a the runtime closure that will access these parts when the code runs. But this closure will itself call compile on bound-body and each of the expressions. Both of these calls can be done at compile time, since they only need the expressions — they don’t depend on the environment. Note that compile* turns to run here, since all it does is run a compiled expression on the current environment.

We can move it back up, out of the resulting functions, by making it a function that consumes an environment and returns a “caller” function:

Once this is done, we have a bunch of work that can happen at compile time: we pre-scan the main “bind spine” of the code.

We can deal in a similar way with other occurrences of compile calls in compiled code. The two branches that need to be fixed are:

1. In the If branch, there is not much to do. After we make it pre-compile the cond-expr, we also need to make it pre-compile both the then-expr and the else-expr. This might seem like doing more work (since before changing it only one would get compiled), but since this is compile-time work, then it’s not as important. Also, if expressions are evaluated many times (being part of a loop, for example), so overall we still win.

2. The Call branch is a little trickier: the problem here is that the expressions that are compiled are coming from the closure that is being applied. The solution for this is obvious: we need to change the closure type so that it closes over compiled expressions instead of over plain ones. This makes sense because closures are run-time values, so they need to close over the compiled expressions since this is what we use as “code” at run-time.

Again, the goal is to have no compile calls that happen at runtime: they should all happen before that. This would allow, for example, to obliterate the compiler once it has done its work, similar to how you don’t need GCC to run a C application. Yet another way to look at this is that we shouldn’t look at the AST at runtime — again, the analogy to GCC is that the AST is a data structure that the compiler uses, and it does not exist at runtime. Any runtime reference to the TOY AST is, therefore, as bad as any runtime reference to compile.

When we’re done with this process we’ll have something that is an actual compiler: translating TOY programs into Racket closures. To see how this is an actual compiler consider the fact that Racket uses a JIT to translate bytecode into machine code when it’s running functions. This means that the compiled version of our TOY programs are, in fact, translated all the way down to machine code.

Tuesday, November 5th Is this really a compiler?

Yes, it is, though it’s hard to see it when we’re compiling TOY code directly to Racket closures.

Another way to see this in a more obvious way is to change the compiler code so instead of producing a Racket closure it spits out the Racket code that makes up these closures when evaluated.

The basic idea is to switch from a function that has code that “does stuff”, to a function that emits that code indtead. For example, consider a function that computes the average of two numbers

to one that instead returns the actual code

It is, however, inconvenient to use strings to represent code: S-expressions are a much better fit for representing code:

This is still tedious though, since the clutter of lists and quotes makes it hard to see the actual code that is produced. It would be nice if we could quote the whole thing instead:

but that’s wrong since we don’t want to include the x and y symbols in the result, but rather their values. Racket (and all other lisp dialects) have a tool for that: quasiquote. In code, you just use a backquote ` instead of a ', and then you can unquote parts of the quasi-quoted code using , (which is called unquote). (Later in the course we’ll talk about these "`"s and ","s more.)

So the above becomes:

Note that this would work fine if x and y are numbers, but they’re now essentially arguments that hold expression values (as S-expressions). For example, see what you get with:

Back to the compiler, we change the closure-generating compiler code

into

Doing this systematically would result in something that is more clearly a compiler: the result of compile would be an S-expression that you can then paste in the Racket file to run it.

An example of this idea taken seriously is the graal + truffle combination for implementing fast JIT compiled languages:

• https://medium.com/graalvm/writing-truly-memory-safe-jit-compilers-f79ad44558dd

Tuesday, November 5th Lazy Evaluation: Using a Lazy Racket

PLAI §7 (done with Haskell)

For this part, we will use a new language, Lazy Racket.

As the name suggests, this is a version of the normal (untyped) Racket language that is lazy.

First of all, let’s verify that this is indeed a lazy language:

That went without a problem — the argument expression was indeed not evaluated. In this language, you can treat all expressions as future promises to evaluate. There are certain points where such promises are actually forced, all of these stem from some need to print a resulting value, in our case, it’s the REPL that prints such values:

The expression by itself only generates a promise, but when we want to print it, this promise is forced to evaluate — this forces the addition, which forces its arguments (plain values rather than computation promises), and at this stage we get an error. (If we never want to see any results, then the language will never do anything at all.) So a promise is forced either when a value printout is needed, or if it is needed to recursively compute a value to print:

Note that the error was raised by the internal expression: the outer expression uses *, and + requires actual values not promises.

Another example, which is now obvious, is that we can now define an if function:

Actually, in this language if, and, and or are all function values instead of special forms:

(By now, you should know that these have no value in Racket — using them like this in plain will lead to syntax errors.) There are some primitives that do not force their arguments. Constructors fall in this category, for example cons and list:

Nothing — the definition simply worked, but that’s expected, since nothing is printed. If we try to inspect this value, we can get some of its parts, provided we do not force the bogus one:

The same holds for cons:

Now if this is the case, then how about this:

Everything is fine, as expected — but what is the value of ones now? Clearly, it is a list that has 1 as its first element:

But what do we have in the tail of this list? We have ones which we already know is a list that has 1 in its first place — so following Racket’s usual rules, it means that the second element of ones is, again, 1. If we continue this, we can see that ones is, in fact, an infinite list of 1s:

In this sense, the way define behaves is that it defines a true equation: if ones is defined as (cons 1 ones), then the real value does satisfy

which means that the value is the fixpoint of the defined expression.

We can use append in a similar way:

This looks like it has some common theme with the discussion of implementing recursive environments — it actually demonstrates that in this language, letrec can be used for simple values too. First of all, a side note — here an expression that indicated a bug in our substituting evaluator:

When our evaluator returned 1 for this, we noticed that this was a bug: it does not obey the lexical scoping rules. As seen above, Lazy Racket is correctly using lexical scope. Now we can go back to the use of letrec — what do we get by this definition:

we get an error about xs being undefined.

xs is unbound because of the usual scope that let uses. How can we make this work? — We simply use letrec:

As expected, if we try to print an infinite list will cause an infinite loop, which DrRacket catches and prints in that weird way:

How would we inspect an infinite list? We write a function that returns part of it:

Dealing with infinite lists can lead to lots of interesting things, for example:

To see how it works, see what you know about fibs[n] which will be our notation for the nth element of fibs (starting from 1):

and for all n>2:

so it follows the exact definition of Fibonacci numbers.

Note that the list examples demonstrate that laziness applies to nested values (actually, nested computations) too: a value that is not needed is not computed, even if it is contained in a value that is needed. For example, in:

the if needs to know only whether its first argument (note: it is an argument, since this if is a function) is #f or not. Once it is determined that it is a pair (a cons cell), there is no need to actually look at the values inside the pair, and therefore (+ 1 x) (and more specifically, x) is never evaluated and we see no error.

Tuesday, November 5th Lazy Evaluation: Some Issues

There are a few issues that we need to be aware of when we’re dealing with a lazy language. First of all, remember that our previous attempt at lazy evaluation has made

evaluate to 1, which does not follow the rules of lexical scope. This is not a problem with lazy evaluation, but rather a problem with our naive implementation. We will shortly see a way to resolve this problem. In the meanwhile, remember that when we try the same in Lazy Racket we do get the expected error:

A second issue is a subtle point that you might have noticed when we played with Lazy Racket: for some of the list values we have see a “.” printed. This is part of the usual way Racket displays an improper list — any list that does not terminate with a null value. For example, in plain Racket:

In the dialect that we’re using in this course, this is not possible. The secret is that the cons that we use first checks that its second argument is a proper list, and it will raise an error if not. So how come Lazy Racket’s cons is not doing the same thing? The problem is that to know that something is a proper list, we will have to force it, which will make it not behave like a constructor.

As a side note, we can achieve some of this protection if we don’t insist on immediately checking the second argument completely, and instead we do the check when needed — lazily:

(define (safe-cons x l)
  (cons x (if (pair? l) l (error "poof"))))

Finally, there are two consequences of using a lazy language that make it more difficult to debug (or at lease take some time to get used to). First of all, control tends to flow in surprising ways. For example, enter the following into DrRacket, and run it in the normal language level for the course:

In the normal language level, we get an error, and red arrows that show us how where in the computation the error was raised. The arrows are all expected, except that foo2 is not in the path — why is that? Remember the discussion about tail-calls and how they are important in Racket since they are the only tool to generate loops? This is what we’re seeing here: foo2 calls foo3 in a tail position, so there is no need to keep the foo2 context anymore — it is simply replaced by foo3. (Incidentally, there is also no arrow that goes through foo1: Racket does some smart inlining, and it figures out that foo0+foo1 are simply returning the same value, so it skips foo1.)

Now switch to Lazy Racket and re-run — you’ll see no arrows at all. What’s the problem? The call of foo0 creates a promise that is forced in the top-level expression, that simply returns the first of the list that foo1 created — and all of that can be done without forcing the foo2 call. Going this way, the computation is finally running into an error after the calls to foo0, foo1, and foo2 are done — so we get the seemingly out-of-context error.

To follow what’s happening here, we need to follow how promise are forced: when we have code like

then the foo call is a strict point, since we need an actual value to display on the REPL. Since it’s in a strict position, we do the call, but when we’re in the function there is no need to compute the division result — so it is returned as a lazy promise value back to the toplevel. It is only then that we continue the process of getting an actual value, which leads to trying to compute the division and get the error.

Finally, there are also potential problems when you’re not careful about memory use. A common technique when using a lazy language is to generate an infinite list and pull out its Nth element. For example, to compute the Nth Fibonacci number, we’ve seen how we can do this:

and we can also do this (reminder: letrec is the same as an internal definition):

but the problem here is that when list-ref is making its way down the list, it might still hold a reference to fibs, which means that as the list is forced, all intermediate values are held in memory. In the first of these two, this is guaranteed to happen since we have a binding that points at the head of the fibs list. With the second form things can be confusing: it might be that our language implementation is smart enough to see that fibs is not really needed anymore and release the offending reference. If it isn’t, then we’d have to do something like

to eliminate it. But even if the implementation does know that there is no need for that reference, there are other tricky situations that are hard to avoid.

Side note: Racket didn’t use to do this optimization, but now it does, and the lazy language helped in clarifying more cases where references should be released. To see that, consider these two variants:

If we try to use them with a big input:

then nat1 would work fine, whereas nat2 will likely run into DrRacket’s memory limit and the computation will be terminated. The problem is that nat2 uses the nats value after the list-ref call, which will make a reference to the head of the list, preventing it from being garbage-collected while list-ref is cdr-ing down the list and making more cons cells materialize.

It’s still possible to show the extra information though — just save it:

It looks like it’s spending a redundant runtime cycle in the extra computation, but it’s a lazy language so this is not a problem.

Tuesday, November 5th Lazy Evaluation: Shell Examples

There is a very simple and elegant principle in shell programming — we get a single data type, a character stream, and many small functions, each doing a single simple job. With these small building blocks, we can construct more sequences that achieve more complex tasks, for example — a sorted frequency table of lines in a file:

This is very much like a programming language — we get small blocks, and build stuff out of them. Of course there are swiss army knives like awk that try to do a whole bunch of stuff, (the same attitude that brought Perl to the world…) and even these respect the “stream” data type. For example, a simple { print $1 } statement will work over all lines, one by one, making it a program over an infinite input stream, which is what happens in reality in something like:

But there is something else in shell programming that makes so effective: it is implementing a sort of a lazy evaluation. For example, compare this:

to:

Each element in the pipe is doing its own small job, and it is always doing just enough to feed its output. Each basic block is designed to work even on infinite inputs! (Even sort works on unlimited inputs…) (Soon we will see a stronger connection with lazy evaluation.)

Side note: (Alan Perlis) “It is better to have 100 functions operate on one data structure than 10 functions on 10 data structures”… But the uniformity comes at a price: the biggest problem shells have is in their lack of a recursive structure, contaminating the world with way too many hacked up solutions. More than that, it is extremely inefficient and usually leads to data being re-parsed over and over and over — each small Unix command needs to always output stuff that is human readable, but the next command in the pipe will need to re-parse that, eg, rereading decimal numbers. If you look at pipelines as composing functions, then a pipe of numeric commands translates to something like:

itoa(baz(atoi(itoa(bar(atoi(itoa(foo(atoi(inp)))))))))

and it is impossible to get rid of the redundant atoi(itoa(...))s.

Tuesday, November 5th Lazy Evaluation: Programming Examples

We already know that when we use lazy evaluation, we are guaranteed to have more robust programs. For example, a function like:

is completely useless in Racket because all functions are eager, but in a lazy language, it would behave exactly like the real if. Note that we still need some primitive conditional, but this primitive can be a function (and it is, in Lazy Racket).

But we get more than that. If we have a lazy language, then computations are pushed around as if they were values (computations, because these are expressions that are yet to be evaluated). In fact, there is no distinction between computations and values, it just happens that some values contain “computational promises”, things that will do something in the future.

To see how this happens, we write a simple program to compute the (infinite) list of prime numbers using the sieve of Eratosthenes. To do this, we begin by defining the list of all natural numbers:

And now define a sift function: it receives an integer n and an infinite list of integers l, and returns a list without the numbers that can be divided by n. This is simple to write using filter:

and it requires a definition for divides? — we use Racket’s modulo for this:

Now, a sieve is a function that consumes a list that begins with a prime number, and returns the prime numbers from this list. To do this, it returns a list that has the same first number, and for its tail it sifts out numbers that are divisible by the first from the original list’s tail, and calls itself recursively on the result:

Finally, the list of prime numbers is the result of applying sieve on the list of numbers from 2. The whole program is now:

To see how this runs, we trace modulo to see which tests are being used. The effect of this is that each time divides? is actually required to return a value, we will see a line with its inputs, and its output. This output looks quite tricky — things are computed only on a “need to know” basis, meaning that debugging lazy programs can be difficult, since things happen when they are needed which takes time to get used to. However, note that the program actually performs the same tests that you’d do using any eager-language implementation of the sieve of Eratosthenes, and the advantage is that we don’t need to decide in advance how many values we want to compute — all values will be computed when you want to see the corresponding result. Implementing this behavior in an eager language is more difficult than a simple program, yet we don’t need such complex code when we use lazy evaluation.

Note that if we trace divides? we see results that are some promise struct — these are unevaluated expressions, and they point at the fact that when divides? is used, it doesn’t really force its arguments — this happens later when these results are forced.

The analogy with shell programming using pipes should be clear now — for example, we have seen this:

The last head -5 means that no computation is done on parts of the original file that are not needed. It is similar to a (take 5 l) expression in Lazy Racket.

Tuesday, November 5th Side Note: Similarity to Generators and Channels

Using infinite lists is similar to using channels — a tool for synchronizing threads and (see a Rob Pike’s talk), and generators (as they exist in Python). Here are examples of both, note how similar they both are, and how similar they are to the above definition of primes. (But note that there is an important difference, can you see it? It has to be with whether a stream is reusable or not.)

First, the threads & channels version:

And here is the generator version:

Tuesday, November 5th Call by Need vs Call by Name

Finally, note that on requiring different parts of the primes, the same calls are not repeated. This indicates that our language implements “call by need” rather than “call by name”: once an expression is forced, its value is remembered, so subsequent usages of this value do not require further computations.

Using “call by name” means that we actually use expressions which can lead to confusing code. An old programming language that used this is Algol. A confusing example that demonstrates this evaluation strategy is:

x and i are arguments that are passed by name, which means that they can use the same memory location. This is called aliasing, a problem that happens when pointers are involved (eg, pointers in C and reference arguments in C++). The code, BTW, is called “Jensen’s device”.

Tuesday, November 5th Example of Feature Embedding

Another interesting behavior that we can now observe, is that the TOY evaluation rule for with:

is specifying an eager evaluator only if the language that this rule is written in is itself eager. Indeed, if we run the TOY interpreter in Lazy Racket (or other interpreters we have implemented), we can verify that running:

is perfectly fine — the call to Racket’s division is done in the evaluation of the TOY division expression, but since Lazy Racket is lazy, then if this value is never used then we never get to do this division! On the other hand, if we evaluate

we do get an error when DrRacket tries to display the result, which forces strictness. Note how the arrows in DrRacket that show where the computation is are quite confusing: the computation seem to go directly to the point of the arithmetic operations (arith-op) since the rest of the evaluation that the evaluator performed was already done, and succeeded. The actual failure happens when we try to force the resulting promise which contains only the strict points in our code.

Tuesday, November 12th Implementing Laziness (in plain Racket)

PLAI §8

Generally, we know how lazy evaluation works when we use the substitution model. We even know that if we have:

then the result should be an error because we cannot substitute the binding of x into the body expression because it will capture the y — changing the binding structure. As an indication, the original expression contains a free reference to y, which is exactly why we shouldn’t substitute it. But what about:

Evaluating this eagerly returns 18, we therefore expect any other evaluation (eager or lazy, using substitutions or environments) to return 18 too, because any of these options should not change the meaning of numbers, of addition, or of the scoping rules. (And we know that no matter what evaluation strategy we choose, if we get to a value (no infinite loop or exception) then it’ll always be the same value.) For example, try using lazy evaluation with substitutions:

And what about lazy evaluation using environments:

We have a problem! This problem should be familiar now, it is very similar to the problem that led us down the mistaken path of dynamic scoping when we tried to have first-class functions. In both cases, substitution always worked, and it looks like in both cases the problem is that we don’t remember the environment of an expression: in the case of functions, it is the environment at the time of creating the closure that we want to capture and use when we go back later to evaluate the body of the function. Here we have a similar situation, except that we don’t need a function to defer computation: most expressions get evaluated at some time in the future, so every time we defer such a computation we need to remember the lexical environment of the expression.

This is the major point that will make things work again: every expression creates something like a closure — an object that closes over an expression and an environment at the (lexical) place where that expression was used, and when we actually want to evaluate it later, we need to do it in the right lexical context. So it is like a closure except it doesn’t need to be applied, and there are no arguments. In fact it is also a form of a closure — instead of closing over a function body and an environment, it closes over any expression and an environment. (As we shall see, lazy evaluation is tightly related to using nullary functions: thunks.)

Tuesday, November 12th Sloth: A Lazy Evaluator

So we implement this by creating such closure values for all expressions that are not evaluated right now. We begin with the Toy language, and rename it to “Sloth”. We then add one more case to the data type of values which implements the new kind of expression closures, which contains the expression and its environment:

(Intuition#1: ExprV is a delayed evaluation and therefore it has the two values that are ultimately passed to eval. Intuition#2: laziness can be implemented with thunks, so we hold the same information as a FunV does, only there’s no need for the argument names.)

Where should we use the new ExprV? — At any place where we want to be lazy and defer evaluating an expression for later. The two places in the interpreter where we want to delay evaluation are the named expressions in a bind form and the argument expressions in a function application. Both of these cases use the helper eval* function to do their evaluations, for example:

To delay these evaluations, we need to change eval* so it returns an expression closure instead of actually doing the evaluation — change:

to:

Note how simple this change is — instead of an eval function call, we create a value that contains the parts that would have been used in the eval function call. This value serves as a promise to do this evaluation (the eval call) later, if needed. (This is exactly why a Lazy Racket would make this a lazy evaluator: in it, all function calls are promises.)

---

Side note: this can be used in any case when you’re using an eager language, and you want to delay some function call — all you need to do is replace (using a C-ish syntax)

with

now all calls to foo return a delayed_foo instance instead of an integer. Whenever we want to force the delayed promise, we can use this function:

You might even want to make sure that each such promise is evaluated exactly once — this is simple to achieve by adding a cache field to the struct:

As we will see shortly, this corresponds to switching from a call-by-name lazy language to a call-by-need one.

---

Back to our Sloth interpreter — given the eval* change, we expect that eval-uating:

will return:

and the same goes for eval-uating

Similarly, evaluating

should return

But what about evaluating an expression like this one:

?

Using what we have so far, we will get to evaluate the body, which is a (Call …) expression, but when we evaluate the arguments for this function call, we will get ExprV values — so we will not be able to perform the addition. Instead, we will get an error from the function that racket-func->prim-val creates, due to the value being an ExprV instead of a RktV.

What we really want is to actually add two values, not promises. So maybe distinguish the two applications — treat PrimV differently from FunV closures?

This still doesn’t work — the problem is that the function now gets a bunch of values, where some of these can still be ExprVs because the evaluation itself can return such values… Another way to see this problem is to consider the code for evaluating an If conditional expression:

…we need to take care of a possible ExprV here. What should we do? The obvious solution is to use eval if we get an ExprV value:

Note how this translates back the data structure that represents a delayed eval promise back into a real eval call…

Going back to our code for Call, there is a problem with it — the

will indeed evaluate the expression instead of lazily deferring this to the future, but this evaluation might itself return such lazy values. So we need to inspect the resulting value again, forcing the promise if needed:

But we still have a problem — programs can get an arbitrarily long nested chains of ExprVs that get forced to other ExprVs.

What we really need is to write a loop that keeps forcing promises over and over until it gets a proper non-ExprV value.

Note that it’s close to real-eval*, but there’s no need to mix it with eval. The recursive call is important: we can never be sure that eval didn’t return an ExprV promise, so we have to keep looping until we get a “real” value.

Now we can change the evaluation of function calls to something more manageable:

The code is fairly similar to what we had previously — the only difference is that we wrap a strict call where a proper value is needed — the function value itself, and arguments to primitive functions.

The If case is similar (note that it doesn’t matter if strict is used with the result of eval or eval* (which returns an ExprV)):

Note that, like before, we always return #t for non-RktV values — this is because we know that the value there is never an ExprV. All we need now to get a working evaluator, is one more strictness point: the outermost point that starts our evaluation — run — needs to use strict to get a proper result value.

With this, all of the tests that we took from the Toy evaluator run successfully. To make sure that the interpreter is lazy, we can add a test that will fail if the language is strict:

[In fact, we can continue and replace all eval calls with ExprV, leaving only the one call in strict. This doesn’t make any difference, because the resulting promises will eventually be forced by strict anyway.]

Tuesday, November 12th Getting more from Sloth

As we’ve seen, using strict in places where we need an actual value rather than a delayed promise is enough to get a working lazy evaluator. Our current implementation assumes that all primitive functions need strict values, therefore the argument values are all passed through the strict function — but this is not always the case. Specifically, if we have constructor functions, then we don’t need (and usually don’t want) to force the promises. This is basically what allows us to use infinite lists in Lazy Racket: the fact that list and cons do not require forcing their arguments.

To allow some primitive functions to consume strict values and some to leave them as is, we’re going to change racket-func->prim-val and add a flag that indicates whether the primitive function is strict or not. Obviously, we also need to move the strict call around arguments to a primitive function application into the racket-func->prim-val generated function — which simplifies the Call case in eval (we go from (proc (map strict arg-vals)) back to (proc arg-vals)). The new code for racket-func->prim-val and its helper is:

We now need to annotate the primitives in the global environment, as well as add a few constructors:

Note that this last change raises a subtle type issue: we’re actually abusing the Racket list and cons constructors to hold Sloth values. One way in which this becomes a problem is the current assumption that a primitive function always returns a Racket value (it is always wrapped in a RktV) — but this is no longer the case for first and rest: when we use

in Sloth, the resulting value will be

This leads to two problems: first, if we use Racket’s first and rest, they will complain (throw a runtime error) since the input value is not a proper list (it’s a pair that has a non-list value in its tail). To resolve that, we use the more primitive car and cdr functions to implement Sloth’s first and rest.

The second problem happens when we try and grab the first value of this

we will eventually get back the ExprV and wrap it in a RktV:

and finally run will strip off the RktV and return the ExprV. A solution to this is to make our first and rest functions return a value without wrapping it in a RktV — we can identify this situation by the fact that the returned value is already a VAL instead of some other Racket value. We can identify such values with the VAL? predicate that gets defined by our define-type, implemented by a new wrap-in-val helper:

Note that we don’t need to worry about the result being an ExprV — that will eventually be taken care of by strict.

Tuesday, November 12th The Sloth Implementation

The complete Sloth code follows. It can be used to do the same fun things we did with Lazy Racket.

Tuesday, November 12th Shouldn’t there be more ExprV promises?

You might notice that there are some apparently missing promises. For example, consider our evaluation of Bind forms:

The named expressions are turned into expression promises via eval*, but shouldn’t we change the first eval (the one that evaluates the body) into a promise too? This is a confusing point, and the bottom line is that there is no need to create a promise there. The main idea is that the eval function is actually called from contexts that actually need to be evaluated. One example is when we force a promise via strict, and another one is when run calls eval. Note that in both of these cases, we actuallly need a (forced) value, so creating a promise in there doesn’t make any difference.

To see this differently, consider how bind might be used within the language. The first case is when bind is the topmost expression, or part of a bind “spine”:

In these cases we evaluate the bind expression when we need to return a result for the whole run, so adding an ExprV is not going to make a difference. The second case is when bind is used in an expression line a function argument:

Here there is also no point in adding an ExprV to the Bind case, since the evaluation of the whole argument (the Bind value) will be wrapped in an ExprV, so it is already delayed. (And when it get forced, we will need to do the bind evaluation anyway, so again, it adds no value.)

A generalization of this is that when we actually call eval (either directly or via strict), there is never any point in making the result that it returns a promise.

(And if you’ll follow this carefully and look at all of the eval calls, you will see that this means that neither of the eval*s in the If case are needed!)

Tuesday, November 12th Implementing Call by Need

As we have seen, there are a number of advantages for lazy evaluation, but its main disadvantage is the fact that it is extremely inefficient, to the point of rendering lots of programs impractical, for example, in:

we end up adding 4 and 5 twice. In other words, we don’t suffer from textual redundancy (each expression is written once), but we don’t avoid dynamic redundancy. We can get it back by simply caching evaluation results, using a box that will be used to remember the results. The box will initially hold #f, and it will change to hold the VAL that results from evaluation:

We need a utility function to create an evaluation promise, because when an ExprV is created, its initial cache box needs to be initialized.

(And note that Typed Racket needs to figure out that the #f in this definition has a type of (U #f VAL) and not just #f.)

This eval-promise is used instead of ExprV in eval. Finally, whenever we force such an ExprV promise, we need to check if it was already evaluated, otherwise force it and cache the result. This is simple to do since there is a single field that is used both as a flag and a cached value:

But note that this makes using side-effects in our interpreter even more confusing. (It was true with call-by-name too.)

The resulting code follows.

Tuesday, November 12th Side Effects in a Lazy Language

We’ve seen that a lazy language without the call-by-need optimization is too slow to be practical, but the optimization makes using side-effects extremely confusing. Specifically, when we deal with side-effects (I/O, mutation, errors, etc) the order of evaluation matters, but in our interpreter expressions are getting evaluated as needed. (Remember tracing the prime-numbers code in Lazy Racket — numbers are tested as needed, not in order.) If we can’t do these things, the question is whether there is any point in using a purely functional lazy language at all — since computer programs often interact with an imperative world.

There is a solution for this: the lazy language does not have any (sane) facilities for doing things (like printf that prints something in plain Racket), but it can use a data structure that describes such operations. For example, in Lazy Racket we cannot print stuff sanely using printf, but we can construct a string using format (which is just like printf, except that it returns the formatted string instead of printing it). So (assuming Racket syntax for simplicity), instead of:

we will write:

and get back a string. We can now change the way that our interpreter deals with the output value that it receives after evaluating a lazy expression: if it receives a string, then it can take that string as denoting a request for printout, and simply print it. Such an evaluator will do the printout when the lazy evaluation is done, and everything works fine because we don’t try to use any side-effects in the lazy language — we just describe the desired side-effects, and constructing such a description does not require performing side-effects.

But this only solves printing a single string, and nothing else. If we want to print two strings, then the only thing we can do is concatenate the two strings — but that is not only inefficient, it cannot describe infinite output (since we will not be able to construct the infinite string in memory). So we need a better way to chain several printout representations. One way to do so is to use a list of strings, but to make things a little easier to manage, we will create a type for I/O descriptions — and populate it with one variant holding a string (for plain printout) and one for holding a chain of two descriptions (which can be used to construct an arbitrarily long sequence of descriptions):

Now we can use this to chain any number of printout representations by turning them into a single Begin2 request, which is very similar to simply using a loop to print the list. For example, the eager printout code:

turns to the following code:

This will basically scan an input list like the eager version, but instead of printing the list, it will convert it into a single output request that forms a recipe for this printout. Note that within the lazy world, the result of print-list is just a value, there are no side effects involved. Turning this value into the actual printout is something that needs to be done on the eager side, which must be part of the implementation. In the case of Lazy Racket, we have no access to the implementation, but we can do so in our Sloth implementation: again, run will inspect the result and either print a given string (if it gets a Print value), or print two things recursively (if it gets a Begin2 value). (To implement this, we will add an IOV variant to the VAL type definition, and have it contain an IO description of the above type.)

Because the sequence is constructed in the lazy world, it will not require allocating the whole sequence in memory — it can be forced bits by bits (using strict) as the imperative back-end (the run part of the implementation) follows the instructions in the resulting IO description. More concretely, it will also work on an infinite list: the translation of an infinite-loop printout function will be one that returns an infinite IO description tree of Begin2 values. This loop will also force only what it needs to print and will go on recursively printing the whole sequence (possibly not terminating). For example (again, using Racket syntax), the infinite printout loop

is translated into a function that returns an infinite tree of print operations:

When this tree is converted to actions, it will result in an infinite loop that produces the same output — it is essentially the same infinite loop, only now it’s derived by an infinite description rather than an infinite process.

Finally, how should we deal with inputs? We can add another variant to our type definition that represents a read-line operation, assuming that like read-line it does not require any arguments:

Now the eager implementation can invoke read-line when it encounters a ReadLine value — but what should it do with the resulting string? Even worse, naively binding a value to ReadLine

doesn’t get us the string that is read — instead, the value is a description of a read operation, which is very different from the actual string value that we want in the binding.

The solution is to take the “code that acts on the string value” and make it be the argument to ReadLine. In the above example, that could would be the let expression without the (ReadLine) part — and as you rememebr from the time we introduced fun into WAE, taking away a named expression from a binding expression leads to a function. With this in mind, it makes sense to make ReadLine take a function value that represents what to do in the future, once the reading is actually done.

This receiver value is a kind of a continuation of the computation, provided as a callback value — it will get the string that was read on the terminal, and will return a new description of side-effects that represents the rest of the process:

Now, when the eager side sees a ReadLine value, it will read a line, and invoke the callback function with the string that it has read. By doing this, the control goes back to the lazy world to process the value and get back another IO value to continue the processing. This results in a process where the lazy code generates some IO descriptions, then the imperative side will execute it and control goes back to the lazy code, then back to the imperative side, etc.

As a more verbose example of all of the above, this silly loop:

is now translated to:

Using this strategy to implement side-effects is possible, and you will do that in the homework — some technical details are going to be different but the principle is the same as discussed above. The last problem is that the above code is difficult to work with — in the homework you will see how to use syntactic abstractions to make things much simpler.

Tuesday, November 19th Designing Domain Specific Languages (DSLs)

PLAI §35

Programming languages differ in numerous ways:

1. Each uses different notations for writing down programs. As we’ve observed, however, syntax is only partially interesting. (This is, however, less true of languages that are trying to mirror the notation of a particular domain.)

2. Control constructs: for instance, early languages didn’t even support recursion, while most modern languages still don’t have continuations.

3. The kinds of data they support. Indeed, sophisticated languages like Racket blur the distinction between control and data by making fragments of control into data values (such as first-class functions and continuations).

4. The means of organizing programs: do they have functions, modules, classes, namespaces, …?

5. Automation such as memory management, run-time safety checks, and so on.

Each of these items suggests natural questions to ask when you design your own languages in particular domains.

Obviously, there are a lot of domain specific languages these days — and that’s not new. For example, four of the oldest languages were conceived as domain specific languages:

• Fortran — Formula Translator
• Algol — Algorithmic Language
• Cobol — Common Business Oriented Language
• Lisp — List Processing

Only in the late 60s / early 70s languages began to get free from their special purpose domain and become general purpose languages (GPLs). These days, we usually use some GPL for our programs and often come up with small domain specific languages (DSLs) for specific jobs. The problem is designing such a specific language. There are lots of decisions to make, and as should be clear now, many ways of shooting your self in the foot. You need to know:

• What is your domain?

• What are the common notations in this domain (need to be convenient both for the machine and for humans)?

• What do you expect to get from your DSL? (eg, performance gains when you know that you’re dealing with a certain limited kind of functionality like arithmetics.)

• Do you have any semantic reason for a new language? (For example, using special scoping rules, or a mixture of lazy and eager evaluation, maybe a completely different way of evaluation (eg, makefiles).)

• Is your language expected to envelope other functionality (eg, shell scripts, TCL), perhaps throwing some functionality on a different language (makefiles and shell scripts), or is it going to be embedded in a bigger application (eg, PHP), or embedded in a way that exposes parts of an application to user automation (Emacs Lisp, Word Basic, Visual Basic for Office Application or Some Other Long List of Buzzwords).

• If you have one language embedded in another enveloping language — how do you handle syntax? How can they communicate (eg, share variables)?

And very important:

• Is there a benefit for implementing a DSL over using a GPL — how much will your DSL grow (usually more than you think)? Will it get to a point where it will need the power of a full GPL? Do you want to risk doing this just to end up admitting that you need a “Real Language” and dump your solution for “Visual Basic for Applications”? (It might be useful to think ahead about things that you know you don’t need, rather than things you need.)

To clarify why this can be applicable in more situations than you think, consider what programming languages are used for. One example that should not be ignored is using a programming language to implement a programming language — for example, what we did so far (or any other interpreter or compiler). In the same way that some piece of code in a PL represent functions about the “real world”, there are other programs that represent things in a language — possibly even the same one. To make a side-effect-full example, the meaning of one-brick might abstract over laying a brick when making a wall — it abstracts all the little details into a function:

and we can now write

instead of all of the above. We might use that in a loop:

This is a common piece of looping code that we’ve seen in many forms, and a common complaint of newcomers to functional languages is the lack of some kind of a loop. But once you know the template, writing such loops is easy — and in fact, you can write code that would take something like:

and produce the previous code. Note the main point here: we switch from code that deals with bricks to code that deals with code.

Now, a viable option for implementing a new DSL is to do so by transforming it into an existing language. Such a process is usually tedious and error prone — tedious because you need to deal with the boring parts of a language (making a parser etc), and error prone because it’s easy to generate bad code (especially when you’re dealing with strings) and you get bad errors in terms of the translated code instead of the actual code, resorting to debugging the intermediate generated programs. Lisp languages traditionally have taken this idea one level further than other languages: instead of writing a new transformer for your language, you use the host language, but you extend and customize it by adding you own forms.

Tuesday, November 19th Syntax Transformations: Macros

PLAI §36

Macros are one of the biggest advantages of all Lisps, and specifically even more so an advantage of Scheme implementations, and yet more specifically, it is a major Racket feature: this section is therefore specific to Racket (which has this unique feature), although most of this is the same in most Schemes.

As we have previously seen, it is possible to implement one language construct using another. What we did could be described as bits of a compiler, since they translate one language to another.

We will see how this can be done in Racket: implementing some new linguistic forms in terms of ones that are already known. In essence, we will be translating Racket constructs to other Racket constructs — and all that is done in Racket, no need to go back to the language Racket was implemented in (C).

This is possible with a simple “trick”: the Racket implementation uses some syntax objects. These objects are implemented somehow inside Racket’s own source code. But these objects are also directly available for our use — part of the implementation is exposed to our level. This is quite similar to the way we have implemented pairs in our language — a TOY or a SLOTH pair is implemented using a Racket pair, so the same data object is available at both levels.

This is the big idea in Lisp, which Scheme (and therefore Racket) inherited from (to some extent): programs are made of numbers, strings, symbols and lists of these — and these are all used both at the meta-level as well as the user level. This means that instead of having no meta-language at all (locking away a lot of useful stuff), and instead of having some crippled half-baked meta language (CPP being both the most obvious (as well as the most popular) example for such a meta language), instead of all this we get exactly the same language at both levels.

How is this used? Well, the principle is simple. For example, say we want to write a macro that will evaluate two forms in sequence, but if the first one returns a result that is not false then it returns it instead of evaluating the second one too. This is exactly how or behaves, so pretend we don’t have it — call our version orelse:

in effect, we add a new special form to our language, with its own evaluation rule, only we know how to express this evaluation rule by translating it to things that are already part of our language.

We could do this as a simple function — only if we’re willing to explicitly delay the arguments with a lambda, and use the thunks in the function:

or:

and using it like this:

But this is clearly not the right way to do this: whoever uses this code must be aware of the need to send us thunks, and it’s verbose and inconvenient.

Note that this could be a feasible solution if there was a uniform way to have an easy syntactic way to say “a chunk of code” instead of immediately execute it — this is exactly what (lambda () ...) does. So we could, for example, make {...} be a shorthand for that, which is what Perl-6 is doing. However, we will soon see examples where we want more than just delay the evaluation of some code.

We want to translate

If we look at the code as an s-expression, then we can write the following function:

We can now try it with a simple list:

and note that the result is correct.

How is this used? Well, all we need is to hook our function into our implementation’s evaluator. In Lisp, we get a defmacro form for this, and many Schemes inherited it or something similar. In Racket, we need to

but it requires the transformation to be a little different in a way that makes life easier: the above contains a lot of boilerplate code. Usually, we will require the input to be a list of some known length, the first element to be a symbol that specifies our form, and then do something with the other arguments. So we’d want to always follow a template that looks like:

But this looks very similar to making sure that a function call is a specific function call (and for a good reason — macro usages look just like function calls). So make the translation function get a number of arguments one each for each part of the input, an s-expression. For example, the above translation and test become:

The number of arguments is used to check the input (turning an arity error for the macro to an arity error for the translator function call), and we don’t need to “caddr our way” to arguments.

This gives us the simple definition — but what about the promised hook? — All we need is to use define-macro instead of define, and change the name to the name that will trigger this translation (providing the last missing test of the input):

and test it:

Note that this is basically a (usually purely functional) lazy language of transformations which is built on top of Racket. It is possible for macros to generate pieces of code that contain references to these same macros, and they will be used to expand those instances again.

Now we start with the problems, one by one.

Tuesday, November 19th Macro Problems

There is an inherent problem when macros are being used, in any form and any language (even in CPP): you must remember that you are playing with expressions, not with values — which is why this is problematic:

And the reason for this should be clear. The standard solution for this is to save the value as a binding — so back to the drawing board, we want this transformation instead:

(Note that we would have the same problem in the version that used simple functions and thunks.)

And to write the new code:

and this works like we want it to.

Tuesday, November 19th Complexity of S-expression transformations

As can be seen, writing a simple macro doesn’t look too good — what if we want to write a more complicated macro? A solution to this is to look at the above macro and realize that it almost looks like the code we want — we basically want to return a list of a certain fixed shape, we just want some parts to be filled in by the given arguments. Something like:

if we had a way to make the <...>s not be a fixed part of the result, but we actually want the values that the transformation function received. (Remember that the < and > are just a part of the name, no magic, just something to make these names stand out.) This is related to notational problems that logicians and philosophers had problems with for centuries. One solution that Lisp uses for this is: instead of a quote, use backquote (called quasiquote in Racket) which is almost like quote, except that you can unquote parts of the value inside. This is done with a “,” comma. Using this, the above code can be written like this:

Tuesday, November 19th Scoping problems

You should be able to guess what’s this problem about. The basic problem of these macros is that they cannot be used reliably — the name that is produced by the macro can shadow a name that is in a completely different place, therefore destroying lexical scope. For example, in:

the val in the macro shadows the use of this name in the above. One way to solve this is to write macros that look like this:

or:

or (this is actually similar to using UUIDs):

Which is really not too good because such obscure variables tend to clobber each other too, in all kinds of unexpected ways.

Another way is to have a function that gives you a different variable name every time you call it:

but this is not safe either since there might still be clashes of these names (eg, if they’re using a counter that is specific to the current process, and you start a new process and load code that was generated earlier). The Lisp solution for this (which Racket’s gensym function implements as well) is to use uninterned symbols — symbols that have their own identity, much like strings, and even if two have the same name, they are not eq?.

Note also that there is the mirror side of this problem — what happens if we try this:

? This leads to capture in the other direction — the code above shadows the if binding that the macro produces.

Some Schemes will allow something like

but this is a hack since the macro outputs something that is not a pure s-expression (and it cannot work for a syntactic keyword like if). Specifically, it is not possible to write the resulting expression (to a compiled file, for example).

We will ignore this for a moment.

---

Another problem — manageability of these transformations.

Quasiquotes gets us a long way, but it is still insufficient.

For example, lets write a Racket bind that uses lambda for binding. The transformation we now want is:

The code for this looks like this:

This already has a lot more pitfalls. There are lists and conses that you should be careful of, there are maps and there are cadrs that would be catastrophic if you use cars instead. The quasiquote syntax is a little more capable — you can write this:

where “,@” is similar to “,” but the unquoted expression should evaluate to a list that is spliced into its surrounding list (that is, its own parens are removed and it’s made into elements in the containing list). But this is still not as readable as the transformation you actually want, and worse, it is not checking that the input syntax is valid, which can lead to very confusing errors.

This is yet another problem — if there is an error in the resulting syntax, the error will be reported in terms of this result rather than the syntax of the code. There is no easy way to tell where these errors are coming from. For example, say that we make a common mistake: forget the “@” character in the above macro:

Now, someone else (the client of this macro), tries to use it:

Yes? Now what? Debugging this is difficult, since in most cases it is not even clear that you were using a macro, and in any case the macro comes from code that you have no knowledge of and no control over. [The problem in this specific case is that the macro expands the code to:

so Racket will to use 1 as a function and throw a runtime error.]

Adding error checking to the macro results in this code:

Such checks are very important, yet writing this is extremely tedious.

Tuesday, November 19th Scheme (and Racket) Macros

Scheme, Racket included (and much extended), has a solution that is better than defmacro: it has define-syntax and syntax-rules. First of all, define-syntax is used to create the “magical connection” between user code and some macro transformation code that does some rewriting. This definition:

makes foo be a special syntax that, when used in the head of an expression, will lead to transforming the expression itself, where the result of this transformation is what gets used instead of the original expression. The “...something...” in this code fragment should be a transformation function — one that consumes the expression that is to be transformed, and returns the new expression to run.

Next, syntax-rules is used to create such a transformation in an easy way. The idea is that what we thought to be an informal specification of such rewrites, for example:

• let can be defined as the following transformation:

  (let ((x v) ...) body ...)
  --> ((lambda (x ...) body ...) v ...)

• let* is defined with two transformation rules:

  1. (let* () body …) –> (let () body …)
  2. (let* ((x1 v1) (x2 v2) …) body …) –> (let ((x1 v1)) (let* ((x2 v2) …) body …))

can actually be formalized by automatically creating a syntax transformation function from these rule specifications. (Note that this example has round parentheses so we don’t fall into the illusion that square brackets are different: the resulting transformation would be the same.) The main point is to view the left hand side as a pattern that can match some forms of syntax, and the right hand side as producing an output that can use some matched patterns.

syntax-rules is used with such rewrite specifications, and it produces the corresponding transformation function. For example, this:

evaluates to a function that is somewhat similar to:

but match is a little closer, since it uses similar input patterns:

Such transformations are used in a define-syntax expression to tie the transformer back into the compiler by hooking it on a specific keyword. You can now appreciate how all this work when you see how easy it is to define macros that are very tedious with define-macro. For example, the above bind:

and let* with its two rules:

These transformations are so convenient to follow, that Scheme specifications (and reference manuals) describe forms by specifying their definition. For example, the Scheme report, specifies let* as a “derived form”, and explains its semantics via this transformation.

The input patterns in these rules are similar to match patterns, and the output patterns assemble an s-expression using the matched parts in the input. For example:

does the thing you expect it to do — matches a parenthesized form with two sub-forms, and produces a form with the two sub-forms swapped. The rules for “...” on the left side are similar to match, as we have seen many times, and on the right side it is used to splice a matched sequence into the resulting expression and it is required to use the ... for sequence-matched pattern variables. For example, here is a list of some patterns, and a description of how they match an input when used on the left side of a transformation rule and how they produce an output expression when they appear on the right side:

• (x ...)

  LHS: matches a parenthesized sequence of zero or more expressions, and the x pattern variable is bound to this whole sequence; match analogy: (match ? [(list x ...) ?])

  RHS: when x is bound to a sequence, this will produce a parenthesized expression containing this sequence; match analogy: (match ? [(list x ...) x])

• (x1 x2 ...)

  LHS: matches a parenthesized sequence of one or more expressions, the first is bound to x1 and the rest of the sequence is bound to x2;

  match analogy: (match ? [(list x1 x2 ...) ?])

  RHS: produces a parenthesized expression that contains the expression bound to x1 first, then all of the expressions in the sequence that x2 is bound to;

  match analogy: (match ? [(list x1 x2 ...) (cons x1 x2)])

• ((x y) ...)

  LHS: matches a parenthesized sequence of 2-form parenthesized sequences, binding x to all the first forms of these, and y to all the seconds of these (so they will both have the same number of items);

  match analogy: (match ? [(list (list x y) ...) ?])

  RHS: produces a list of forms where each one is made of consecutive forms in the x sequence and consecutive forms in the y sequence (both sequences should have the same number of elements);

  match analogy:

  (match ? [(list (list x y) ...)
            (map (lambda (x y) (list x y)) x y)])

Some examples of transformations that would be very tedious to write code manually for:

• ((x y) ...) --> ((y x) ...)

  Matches a sequence of 2-item sequences, produces a similar sequence with all of the nested 2-item sequences flipped.

• ((x y) ...) --> ((x ...) (y ...))

  Matches a similar sequence, and produces a sequence of two sequences, one of all the first items, and one of the second ones.

• ((x y ...) ...) --> ((y ... x) ...)

  Similar to the first example, but the nested sequences can have 1 or more items in them, and the nested sequences in the result have the first element moved to the end. Note how the ... are nested: the rule is that for each pattern variable you count how many ...s apply to it, and that tells you what it holds — and you have to use the same ... nestedness for it in the output template.

---

This is solving the problems of easy code — no need for list, cons etc, not even for quasiquotes and tedious syntax massaging. But there were other problems. First, there was a problem of bad scope, one that was previously solved with a gensym:

Translating this to define-syntax and syntax-rules we get something simpler:

Even simpler, Racket has a macro called define-syntax-rule that expands to a define-syntax combined with a syntax-rules — using it, we can write:

This looks like like a function — but you must remember that it is a transformation rule specification which is a very different beast, as we’ll see.

The main thing here is that Racket takes care of making bindings follow the lexical scope rules:

works fine. In fact, it fully respects the scoping rules: there is no confusion between bindings that the macro introduces and bindings that are introduced where the macro is used. (Think about different colors for bindings introduced by the macro and other bindings.) It’s also fine with many cases that are much harder to cope with otherwise (eg, cases where there is no gensym magic solution):

or combining both:

(You can try DrRacket’s macro debugger to see how the various bindings get colored differently.)

define-macro advocates will claim that it is difficult to make a macro that intentionally plants a known identifier. Think about a loop macro that has an abort that can be used inside its body. Or an if-it form that is like if, but makes it possible to use the condition’s value in the “then” branch as an it binding. It is possible with all Scheme macro systems to “break hygiene” in such ways, and we will later see how to do this in Racket. However, Racket also provides a better way to deal with such problems (think about it being always “bound to a syntax error”, but locally rebound in an if-it form).

Scheme macros are said to be hygienic — a term used to specify that they respect lexical scope. There are several implementations of hygienic macro systems across Scheme implementations, Racket uses the one known as “syntax-case system”, named after the syntax-case construct that we discuss below.

All of this can get much more important in the presence of a module system, since you can write a module that provides transformations rules, not just values and functions. This means that the concept of “a library” in Racket is something that doesn’t exist in other languages: it’s a library that has values, functions, as well as macros (or, “compiler plugins”).

---

The way that Scheme implementations achieve hygiene in a macro system is by making it deal with more than just raw S-expressions. Roughly speaking, it deals with syntax objects that are sort of a wrapper structure around S-expression, carrying additional information. The important part of this information when it gets to dealing with hygiene is the “lexical scope” — which can roughly be described as having identifiers be represented as symbols plus a “color” which represents the scope. This way such systems can properly avoid confusing identifiers with the same name that come from different scopes.

There was also the problem of making debugging difficult, because a macro can introduce errors that are “coming out of nowhere”. In the implementation that we work with, this is solved by adding yet more information to these syntax objects — in addition to the underlying S-expression and the lexical scope, they also contain source location information. This allows Racket (and DrRacket) to locate the source of a specific syntax error, so locating the offending code is easy. DrRacket’s macro debugger heavily relies on this information to provide a very useful tool — since writing macros can easily become a hard job.

Finally, there was the problem of writing bad macros. For example, it is easy to forget that you’re dealing with a macro definition and write:

just because you want to inline the addition — but in this case you end up duplicating the input expression which can have a disastrous effect. For example:

expands to a lot of code to compile.

Another example is:

the problem here is that (* foo 2) will be used as an identifier to be bound by the let expression — which can lead to a confusing syntax error.

Racket provides many tools to help macro programmers — in addition to a user-interface tool like the macro debugger there are also programmer-level tools where you can reject an input if it doesn’t contain an identifier at a certain place etc. Still, writing macros is much harder than writing functions — some of these problems are inherent to the problem that macros solve; for example, you may want a twice macro that replicates an expression. By specifying a transformation to the core language, a macro writer has full control over which expressions get evaluated and how, which identifiers are binding instances, and how is the scope of the given expression is shaped.

Tuesday, November 19th Meta Macros

One of the nice results of syntax-rules dealing with the subtle points of identifiers and scope is that things works fine even when we “go up a level”. For example, the short define-syntax-rule form that we’ve seen is itself a defined as a simple macro:

In fact, this is very similar to something that we have already seen: the rewrite form that we have used in Schlac is implemented in just this way. The only difference is that rewrite requires an actual => token to separate the input pattern from the output template. If we just use it in a syntax rule:

it won’t work. Racket treats the above => just like any identifier, which in this case acts as a pattern variable which matches anything. The solution to this is to list the => as a keyword which is expected to appear in the macro use as-is — and that’s what the mysterious () of syntax-rules is used for: any identifier listed there is taken to be such a keyword. This makes the following version

do what we want and throw a syntax error unless rewrite is used with an actual => in the proper place.

Tuesday, November 19th Lazy Constructions in an Eager Language

PLAI §37 (has some examples)

This is not really lazy evaluation, but it gets close, and provides the core useful property of easy-to-use infinite lists.

Using it:

Actually, all Scheme implementations come with a generalized tool for (local) laziness: a delay form that delays computation of its body expression, and a force function that forces such promises. Here is a naive implementation of this:

Proper definitions of delay/force cache the result — and practical ones can get pretty complex, for example, in order to allow tail calls via promises.

Tuesday, November 19th Recursive Macros

Syntax transformations can be recursive. For example, we have seen how let* can be implemented by a transformation that uses two rules, one of which expands to another use of let*:

When Racket expands a let* expression, the result may contain a new let* which needs extending as well. An important implication of this is that recursive macros are fine, as long as the recursive case is using a smaller expression. This is just like any form of recursion (or loop), where you need to be looping over a well-founded set of values — where each iteration uses a new value that is closer to some base case.

For example, consider the following macro:

It seems like this is a good implementation of a while loop — after all, if you were to implement it as a function using thunks, you’d write very similar code:

But if you look at the nested while form in the transformation rule, you’ll see that it is exactly the same as the input form. This means that this macro can never be completely expanded — it specifies infinite code! In practice, this makes the (Racket) compiler loop forever, consuming more and more memory. This is unlike, for example, the recursive let* rule which uses one less binding-value pair than specified as its input.

The reason that the function version of while is fine is that it iterates using the same code, and the condition thunk will depend on some state that converges to a base case (usually the body thunk will perform some side-effects that makes the loop converge). But in the macro case there is no evaluation happening, if the transformed syntax contains the same input pattern, we end up having a macro that expands infinitely.

The correct solution for a while macro is therefore to use plain recursion using a local recursive function:

A popular way to deal with macros like this that revolve around a specific control flow is to separate them into a function that uses thunks, and a macro that does nothing except wrap input expressions as thunks. In this case, we get this solution:

Tuesday, November 19th Another example: a simple loop

Here is an implementation of a macro that does a simple arithmetic loop:

(Note that this is not complete code: it suffers from the usual problem of multiple evaluations of the n expression. We’ll deal with it soon.)

This macro combines both control flow and lexical scope. Control flow is specified by the loop (done, as usual in Racket, as a tail-recursive function) — for example, it determines how code is iterated, and it also determines what the for form will evaluate to (it evaluates to whatever when evaluates to, the void value in this case). Scope is also specified here, by translating the code to a function — this code makes x have a scope that covers the body so this is valid:

but it also makes the boundary expression n be in this scope, making this:

valid. In addition, while evaluating the condition on each iteration might be desirable, in most cases it’s not — consider this example:

This is easily solved by using a let to make the expression evaluate just once:

which makes the previous use result in a “reference to undefined identifier: i” error.

Furthermore, the fact that we have a hygienic macro system means that it is perfectly fine to use nested for expressions:

The transformation is, therefore, completely specifying the semantics of this new form.

Extending this syntax is easy using multiple transformation rules — for example, say that we want to extend it to have a step optional keyword. The standard idiom is to have the step-less pattern translated into one that uses step 1:

Usually, you should remember that syntax-rules tries the patterns one by one until a match is found, but in this case there is no problems because the keywords make the choice unambiguous:

We can even extend it to do a different kind of iteration, for example, iterate over list:

Tuesday, November 19th Yet Another: List Comprehension

At this point it’s clear that macros are a powerful language feature that makes it relatively easy to implement new features, making it a language that is easy to use as a tool for quick experimentation with new language features. As an example of a practical feature rather than a toy, let’s see how we can implement Python’s list comprehenions. These are expressions that conveniently combine map, filter, and nested uses of both.

First, a simple implementation that uses only the map feature:

It is a good exercise to see how everything that we’ve seen above plays a role here. For example, how we get the ID to be bound in EXPR.

Next, add a condition expression with an if keyword, and implemented using a filter:

Again, go over it and see how the binding structure makes the identifier available in both expressions. Note that since we’re just playing around we’re not paying too much attention to performance etc. (For example, if we cared, we could have implemented the if-less case by not using filter at all, or we could implement a filter that accepts #t as a predicate and in that case just returns the list, or even implementing it as a macro that identifies a (lambda (_) #t) pattern and expands to just the list (a bad idea in general).)

The last step: Python’s comprehension accepts multiple for-ins for nested loops, possibly with if filters at each level:

A collection of examples that I found in the Python docs and elsewhere, demonstrating all of these:

Tuesday, November 19th Problems of syntax-rules Macros

As we’ve seen, using syntax-rules solves many of the problems of macros, but it comes with a high price tag: the macros are “just” rewrite rules. As rewrite rules they’re pretty sophisticated, but it still loses a huge advantage of what we had with define-macro — the macro code is no longer Racket code but a simple language of rewrite rules.

There are two big problems with this which we will look into now. (DrRacket’s macro stepper tool can be very useful in clarifying these examples.) The first problem is that in some cases we want to perform computations at the macro level — for example, consider a repeat macro that needs to expand like this:

With a syntax-rules macro we can match over specific integers, but we just cannot do this with any integer. Note that this specific case can be done better via a function — better by not replicating the expression:

or even better, assuming the above for is already implemented:

But still, we want to have the ability to do such computation. A similar, and perhaps better example, is better error reporting. For example, the above for implementation blindly expands its input, so:

we get a bad error message in terms of lambda, which is breaking abstraction (it comes from the expansion of for, which is an implementation detail), and worse — it is an error about something that the user didn’t write.

Yet another aspect of this problem is that sometimes we need to get creative solutions where it would be very simple to write the corresponding Racket code. For example, consider the problem of writing a rev-app macro — (rev-app F E …) should evaluate to a function similar to (F E …), except that we want the evaluation to go from right to left instead of the usual left-to-right that Racket does. This code is obviously very broken:

because it generates a malformed let form — there is no way for the macro expander to somehow know that the reverse should happen at the transformation level. In this case, we can actually solve this using a helper macro to do the reversing:

There are still problems with this — it complains about x ... because there is a single x there rather than a sequence of them; and even if it did somehow work, we also need the xs in that last line in the original order rather than the reversed one. So the solution is complicated by collecting new xs while reversing — and since we need them in both orders, we’re going to collect both orders:

So, this worked, but in this case the simplicity of the syntax-rules rewrite language worked against us, and made a very inconvenient solution. This could have been much easier if we could just write a “meta-level” reverse, and a use of map to generate the names.

… And all of that was just the first problem. The second one is even harder: syntax-rules is designed to avoid all name captures, but what if we want to break hygiene? There are some cases where you want a macro that “injects” a user-visible identifier into its result. The most common (and therefore the classic) example of this is an anaphoric if macro, that binds it to the result of the test (which can be any value, not just a boolean):

which we want to turn to:

The obvious definition of `if-it’ doesn’t work:

The reason it doesn’t work should be obvious now — it is designed to avoid the it that the macro introduced from interfering with the it that the user code uses.

Next, we’ll see Racket’s “low level” macro system, which can later be used to solve these problems.

Tuesday, November 19th Racket’s “Low-Level” Macros

As we’ve previously seen, syntax-rules creates transformation functions — but there are other more direct ways to write these functions. These involve writing a function directly rather than creating one with syntax-rules — and because this is a more low-level approach than using syntax-rules to generate a transformer, it is called a “low level macro system”. All Scheme implementations have such low-level systems, and these systems vary from one to the other. They all involve some particular type that is used as “syntax” — this type is always related to S-expressions, but it cannot be the simple define-macro tool that we’ve seen earlier if we want to avoid the problems of capturing identifiers.

Historical note: For a very long time the Scheme standard had avoided a concrete specification of this low-level system, leaving syntax-rules as the only way to write portable Scheme code. This had lead some people to explore more thoroughly the things that can be done with just syntax-rules rewrites, even beyond the examples we’ve seen. As it turns out, there’s a lot that can be done with it — in fact, it is possible to write rewrite rules that implement a lambda calculus, making it possible to write things that look kind of like “real code”. This is, however, awkward to say the least, and redundant with a macro system that can use the full language for arbitrary computations. It has also became less popular recently, since R6RS dictates something that is known as a “syntax-case macro system” (not really a good name, since syntax-case is just a tool in this system).

Racket uses an extended version of this syntax-case system, which is what we will discuss now. In the Racket macro system, “syntax” is a new type, not just S-expressions as is the case with define-macro. The way to think about this type is as a wrapper around S-expressions, where the S-expression is the “raw” symbolic form of the syntax, and a bunch of “stuff” is added. Two important bits of this “stuff” are the source location information for the syntax, and its lexical scope. The source location is what you’d expect: the file name for the syntax (if it was read from a file), its position in the file, and its line and column numbers; this information is mostly useful for reporting errors. The lexical scope information is used in a somewhat peculiar way: there is no direct way to access it, since usually you don’t need to do so — instead, for the rare cases where you do need to manipulate it, you copy the lexical scope of an existing syntax to create another. This allows the macro interface to be usable without specification of a specific representation for the scope.

The main piece of functionality in this system is syntax-case (which has lead to its common name) — a form that is used to deconstruct the input via pattern-matching similar to syntax-rules. In fact, the syntax of syntax-case looks very similar to the syntax of syntax-rules — there are zero or more parenthesized keywords, and then clauses that begin with the same kind of patterns to match the syntax against. The first obvious difference is that the syntax to be matched is given explicitly:

A macro is written as a plain function, usually used as the value in a define-syntax form (but it could also be used in plain helper functions). For example, here’s how the orelse macro is written using this:

Racket’s define-syntax can also use the same syntactic sugar for functions as define:

The second significant change from syntax-rules is that the right-hand-side expressions in the branches are not patterns. Instead, they’re plain Racket expressions. In this example (as in most uses of syntax-case) the result of the syntax-case form is going to be the result of the macro, and therefore it should return a piece of syntax. So far, the only piece of syntax that we see in this code is just the input stx — and returning that will get the macro expander in an infinite loop (because we’re essentially making a transformer for orelse expressions that expands the syntax to itself).

To return a new piece of syntax, we need a way to write new syntax values. The common way to do this is using a new special form: syntax. This form is similar to quote — except that instead of an S-expression, it results in a syntax. For example, in this code:

the first printout happens during macro expansion, and the second is part of the generated code. Like quote, syntax has a convenient abbreviation — “#'”:

Now the question is how we can get the actual macro working. The thing is that syntax is not completely quoting its contents as a syntax — there could be some meta identifiers that are bound as “pattern variables” in the syntax-case pattern that was matched for the current clause — in this case, we have x and y as such pattern variables. (Actually, orelse is a pattern variable too, but this doesn’t matter for our purpose.) Using these inside a syntax will have them replaced by the syntax that they matched against. The complete orelse definition is therefore very easy:

The same treatment of ... holds here too — in the matching pattern they specify 0 or more occurrences of the preceding pattern, and in the output template they mean that the matching sequence is “spliced” in. Note that syntax-rules is now easy to define as a macro that expands to a function that uses syntax-case to do the actual rewrite work:

extra] [Tuesday, November 19th Solving the syntax-rules problems

So far it looks like we didn’t do anything new, but the important change is already in: the fact that the results of a macro is a plain Racket expression mean that we can now add more API functionality for dealing with syntax values. There is no longer a problem with running “meta-level” code vs generated runtime code: anything that is inside a syntax (anything that is quoted with a “#'”) is generated code, and the rest is code that is executed when the macro expands. We will now introduce some of the Racket macro API by demonstrating the solutions to the syntax-rules problem that were mentioned earlier.

First of all, we’ve talked about the problem of reporting good errors. For example, make this:

throw a proper error instead of leaving it for lambda to complain about. To make it easier to play with, we’ll use a simpler macro:

and using an explicit function:

One of the basic API functions is syntax-e — it takes in a syntax value and returns the S-expression that it wraps. In this case, we can pull out the identifier from this, and check that it is a valid identifier using symbol? on what it wraps:

The error is awkward though — it doesn’t look like the usual kind of syntax errors that Racket throws: it’s shown in an ugly way, and its source is not properly highlighted. A better way to do this is to use `raise-syntax-error’ — it takes an error message, the offending syntax, and the offending part of this syntax:

Another inconvenient issue is with pulling out the identifier. Consider that #'(lambda (id) E) is a new piece of syntax that has the supposed identifier in it — we pull it from that instead of from stx, but it would be even easier with #'(id), and even easier than that with just #'id which will be just the identifier:

Also, checking that something is an identifier is common enough that there is another predicate for this (the combination of syntax-e and symbol?) — identifier?:

As a side note, checking the input pattern for validity is very common, and in some cases might be needed to discriminate patterns (eg, one result when id is an identifier, another when it’s not). For this, syntax-cases clauses have “guard expressions” — so we can write the above more simply as:

This, however, produces a less informative “bad syntax” error, since there is no way to tell what the error message should be. (There is a relatively new Racket tool called syntax-parse where such requirements can be specified and a proper error message is generated on bad inputs.)

We can now resolve the repeat problem — create a (repeat N E) macro:

(Note that we can define an internal helper function, just like we do with plain functions.) But this doesn’t quite work (and if you try it, you’ll see an interesting error message) — the problem is that we’re generating code with a call to n-copies in it, instead of actually calling it. The problem is that we need to take the list that n-copies generates, and somehow “plant” it in the resulting syntax. So far the only things that were planted in it are pattern variables — and we can actually use another syntax-case to do just that: match the result of n-copies against a pattern variable, and then use that variable in the final syntax:

This works — but one thing to note here is that n-copies returns a list, not a syntax. The thing is that syntax-case will automatically “coerce” S-expressions into a syntax in some way, easy to do in this case since we only care about the elements of the list, and those are all syntaxes.

However, this use of syntax-case as a pattern variable binder is rather indirect, enough that it’s hard to read the code. Since this is a common use case, there is a shorthand for that too: with-syntax. It looks as a kind of a let-like form, but instead of binding plain identifiers, it binds pattern identifiers — and in fact, the things to be bound are themselves patterns:

Note that there is no need to implement with-syntax as a primitive form — it is not too hard to implement it as a macro that expands to the actual use of syntax-case. (In fact, you can probably guess now that the Racket core language is much smaller than it seems, with large parts that are implemented as layers of macros.)

There is one more related group of shorthands that is relevant here: quasisyntax, unsyntax, and unsyntax-splicing. These are analogous to the quoting forms by the same names, and they have similar shorthands: “#`”, “#,” and “#,@”. They could be used to implement this macro:

[As you might suspect now, these new forms are also implemented as macros, which expand to the corresponding uses of with-syntax, which in turn expand into syntax-case forms.]

We now have almost enough machinery to implement the rev-app macro, and compare it to the original (complex) version that used syntax-rules. The only thing that is missing is a way to generate a number of new identifiers — which we achieved earlier by a number of macro expansion (each expansion of a macro that has a new identifier x will have this identifier different from other expansions, which is why it worked). Racket has a function for this: generate-temporaries. Since it is common to generate temporaries for input syntaxes, the function expects an input syntax that has a list as its S-expression form (or a plain list).

Note that this is not shorter than the syntax-rules version, but it is easier to read since reverse and generate-temporaries have an obvious direct intention, eliminating the need to wonder through rewrite rules and inferring how they do their work. In addition, this macro expands in one step (use the macro stepper to compare it with the previous version), which makes it much more efficient.

extra] [Tuesday, November 19th Breaking Hygiene, How Bad is it?

We finally get to address the second deficiency of syntax-rules — its inability to intentionally capture an identifier so it is visible in user code. Let’s start with the simple version, the one that didn’t work:

and translate it to syntax-case:

The only problem here is that the it identifier is introduced by the macro, or more specifically, by the syntax form that makes up the return syntax. What we need now is a programmatic way to create an identifier with a lexical context that is different than the default. As mentioned above, Racket’s syntax system (and all other syntax-case systems) doesn’t provide a direct way to manipulate the lexical context. Instead, it provides a way to create a piece of syntax by copying the lexical scope of another one — and this is done with the datum->syntax function. The function consumes a syntax value to get the lexical scope from, and a “datum” which is an S-expression that can contain syntax values. The result will have these syntax values as given on the input, but raw S-expressions will be converted to syntaxes, using the given lexical context. In the above case, we need to convert an it symbol into the same-named identifier, and we can do that using the lexical scope of the input syntax. As we’ve seen before, we use with-syntax to inject the new identifier into the result:

We can even control the scope of the user binding — for example, it doesn’t make much sense to have it in the else branch. We can do this by first binding a plain (hygienic) identifier to the result, and only bind it to that when needed:

[A relevant note: Racket provides something that is known as “The Macro Writer’s Bill of Rights” — in this case, it guarantees that the extra let does not imply a runtime or a memory overhead.]

This works — and it’s a popular way for creating such user-visible bindings. However, breaking hygiene this way can lead to some confusing problems. Such problems are usually visible when we try to compose macros — for example, say that we want to create a cond-it macro, the anaphoric analogue of cond, which binds it in each branch. It seems that an obvious way of doing this is by layering it on top of if-it — it should even be simple enough to be defined with syntax-rules:

Surprisingly, this does not work! Can you see what went wrong?

The problem lies in how the it identifier is generated — it used the lexical context of the whole if-it expression, which seemed like exactly what we wanted. But in this case, the if-it expression is coming from the cond-it macro, not from the user input. Or to be more accurate: it’s the cond-it macro which is the user of if-it, so it is visible to cond-it, but not to its own users…

Note that these anaphoric macros are a popular example, but these problems do pop up elsewhere too. For example, imagine a loop macro that wants to bind break unhygienically, a class macro that binds this, and many others.

How can we solve this? There are several ways for this:

• Don’t break hygiene. For example, instead of if-it and cond-it forms that have an implicit it, use forms with an explicit identifiers. For example: (if* it <test> <then> <else>). This might be a little more verbose at times, but it makes everything behave very well, since the identifiers always have the right scope.

• Try to patch things up with a little more unhygienic in your macros. In this case, try to make cond-it introduce if-it unhygienically, so when it introduces it in its own turn, it will be the right one. This is bad, since we started trying to get hygienic macros, and there are no easy discounts. (For example, what if there’s a different if-it that is used at the place where cond-it is used?) In fact, the unhygienic define-macro that we’ve seen is an extreme example of this: there is no lexical scope anywhere; so it is the same identifier no matter where it’s introduced. But as we’ve seen, this means that hygiene is always broken when possible.

• Try to make cond-it come up with its own unhygienic it, then bind this it to the it that if-it creates. This can work but on one hand it’s difficult and fragile to write such code, and on the other hand it defeats the simplicity of macros.

• Finally, Racket provides an elegant solution in the form of syntax parameters. The idea is to avoid the unhygienic binding: have a single global binding for it, and change the meaning of this binding on uses of if-it. (If you’re interested, see “Keeping it Clean with Syntax Parameters” for details.)

extra] [Tuesday, November 19th Macros in Racket’s Module System

Not in PLAI

One of the main things that Racket pioneered is integrating its syntax system with its module system. In plain Racket (#lang racket, not the course languages), every file is a module that can provide some functionality, for when you put this code in a file:

You get a library that gives you a plus function. This is just the usual thing that you’d expect from a library facility in a language — but Racket allows you to do the same with syntax definitions. For example, if we add the following to this file:

we — the users of this library — also get to have a with binding, which is a “FLANG-compatibility” macro that expands into a let. Now, on a brief look, this doesn’t seem all too impressive, but consider the fact that with is actually a translation function that lives at the syntax level, as a kind of a compiler plugin, and you’ll see that this is not as trivial as it seems. Racket arranges to do this with a concept of instantiating code at the compiler level, so libraries are used in two ways: either the usual thing as a runtime instantiation, or at compile time.

extra] [Tuesday, November 19th Defining Languages in Racket

But then Racket takes this concept even further. So far, we treated the thing that follows a #lang as a kind of a language specification — but the more complete story is that this specification is actually just a module. The only difference between such modules like racket or pl and “library modules” as our above file is that language modules provide a bunch of functionality that is specific to a language implementation. However, you don’t need to know about these things up front: instead, there’s a few tools that allow you to provide everything that some other module provides — if we add this to the above:

then we get a library that provides the same two bindings as above (plus and with) — in addition to everything from the racket library (which it got from its own #lang racket line).

To use this file as a language, the last bit that we need to know about is the actual concrete level syntax. Racket provides an s-exp language which is a kind of a meta language for reading source code in the usual S-expression syntax. Assuming that the above is in a file called mylang.rkt, we can use it (from a different file in the same directory) as follows:

which makes the language of this file be (a) read using the S-expression syntax, and (b) get its bindings from our module, so

will show a result of 40.

So far this seems like just some awkward way to get to the same functionality as a simple library — but now we can use more tools to make things more interesting. First, we can provide everything from racket except for let — change the last provide to:

Next, we can provide our with but make it have the name let instead — by replacing that (provide with) with:

The result is a language that is the same as Racket, except that it has an additional plus “built-in” function, and its let syntax is different, as specified by our macro:

To top things off, there are a few “special” implicit macros that Racket uses. One of them, #%app, is a macro that is used implicitly whenever there’s an expression that looks like a function application. In our terms, that’s the Call AST node that gets used whenever a braced-form isn’t one of the known forms. If we override this macro in a similar way that we did for let, we’re essentially changing the semantics of application syntax. For example, here’s a definition that makes it possible to use a @ keyword to get a list of results of applying a function on several arguments:

This makes the (my-app add1 @ 1 2) application evaluate to '(2 3), but if @ is not used (as the second subexpression), we get the usual function application. (Note that this is because the last clause expands to (x ...) which implicitly has the usual Racket function application.) We can now make our language replace Racket’s implicit #%app macro with this, in the same way as we did before: first, drop Racket’s version from what we provide:

and then provide our definition instead

Users of our language get this as the regular function application:

Since #%app is a macro, it can evaluate to anything, even to things that are not function applications at all. For example, here’s an extended definition that adds an arrow syntax that expands to a lambda expression not to an actual application:

And an example of using it

Another such special macro is #%module-begin: this is a macro that is wrapped around the whole module body. Changing it makes it possible to change the semantics of a sequence of toplevel expressions in our language. The following is our complete language, with an example of redefining #%module-begin to create a “verbose” language that prints out expressions and what they evaluate to (note the verbose helper macro that is completely internal):

And for reference, try that language with the above example:

Tuesday, November 19th Macro Conclusions

PLAI §37.5

Macros are extremely powerful, but this also means that their usage should be restricted only to situations where they are really needed. You can view any function as extending the current collection of tools that you provide — where these tools are much more difficult for your users to swallow than plain functions: evaluation can happen in any way, with any scope, unlike the uniform rules of function application. An analogy is that every function (or value) that you provide is equivalent to adding nouns to a vocabulary, but macros can add completely new rules for reading, since using them might result in a completely different evaluation. Because of this, adding macros carelessly can make code harder to read and debug — and using them should be done in a way that is as clear as possible for users.

When should a macro be used?

• Providing cosmetics: eliminating some annoying repetitiveness and/or inconvenient verbosity. This is usually macros that are intended to beautify code, for example, we could use a macro to make this bit of the Sloth source:

  (list '+ (box (racket-func->prim-val + #t)))
  (list '- (box (racket-func->prim-val - #t)))
  (list '* (box (racket-func->prim-val + #t)))

  look much better, by using a macro instead of the above. We can try to use a function, but we still need two inputs for each call — the name and the function:

  (rfpv '+ + #t)
  (rfpv '- - #t)
  (rfpv '* + #t)

  and a macro can eliminate this (small, but potentially dangerous) redundancy. For example:

  (define-syntax-rule (rfpv fun flag)
    (list 'fun (box (racket-func->prim-val fun flag))))

  and then:

  (rfpv + #t)
  (rfpv - #t)
  (rfpv * #t)

  eliminates the typo that was in the previous examples (did you catch that?).

• Altering the order of evaluation: as seen with the orelse macro, we can control evaluation order in our macro. This is achieved by translating the macro into Racket code with a known evaluation order. We even choose not to evaluate some parts, or evaluate some parts multiple times (eg, the for macro).

  Note that by itself, we could get this if only we had a more light-weight notation for thunks, since then we could simply use functions. For example, a while function could easily be used with thunks:

  (define (while cond body)
    (when (cond)
      (body)
      (while cond body)))

  if the syntax for a thunk would be as easy as, for example, using curly braces:

  (let ([i 0])
    (while { (< i 10) }
      { (printf "i = ~s\n" i) (set! i (+ i 1)) }))

• Introducing binding constructs: macros that have a different binding structure from Racket built-ins. These kind of macros are ones that makes a powerful language — for example, we’ve seen how we can survive without basic built-ins like let. For example, the for macro has its own binding structure.

  Note that with a sufficiently concise syntax for functions such as the arrow functions in JavaScript, we can get away with plain functions here too. For example, instead of a with macro, we could do it with a function:

  function with(val,fun) { return fun(val); }
  with( 123, x => x*x );

  (The obvious inconvenience is that the order can be weird.)

• Defining data languages: macros can be used for expressions that are not Racket expressions themselves. For example, the parens that wrap binding pairs in a let form are not function applications. Some times it is possible to use quotes for that, but then we get run-time values rather than being able to translate them into Racket code. Another usage of this category is to hide representation details that might involve implicit lambda’s (for example, delay) — if we define a macro, then there is a single point where we control whether an expression is used within some lambda — but it it was a function, we’d have to change every usage of it to add an explicit lambda.

It is also important to note that macros should not be used too frequently. As said above, every macro adds a completely different way of reading your code — a way that doesn’t use the usual “nouns” and “verbs”, but there are other reasons not to use a macro.

One common usage case is as an optimization — trying to avoid an extra function call. For example, this:

might seem wasteful if you don’t want a full function call on every usage of min. So you might be tempted to use this instead:

you even know the pitfalls of C macros so you make it more robust:

But small functions like the above are things that any decent compiler should know how to optimize, and even if your compiler doesn’t, it’s still not worth doing this optimization because programmer time is the most expensive factor in any computer system. In addition, a compiler is committed to doing these optimizations only when possible (eg, it is not possible to in-line a recursive function) and to do proper in-lining — unlike the min CPP macro above which is erroneous in case x or y are expressions that have side-effects.

Tuesday, November 19th Side-note: macros in mainstream languages

Macros are an extremely powerful tool in Racket (and other languages in the Lisp family) — how come nobody else uses them?

Well, people have tried to use them in many contexts. The problem is that you cannot get away with a simple solution that does nothing more than textual manipulation of your programs. For example, the standard C preprocessor is a macro language, but it is fundamentally limited to very simple situations. This is still a hot topic these days, with modern languages trying out different solutions (or giving up and claiming that macros are evil).

Here is an example that was written by Mark Jason Dominus (“Higher Order Perl”), in a Perl mailing list post among further discussion on macros in Lisp vs other languages, including Perl’s source transformers that are supposed to fill a similar role.

The example starts with writing the following simple macro:

This doesn’t quite work because

expands to

which is 2, but you wanted 0.02. So you need this instead:

but this breaks because

expands to

which is 3, but you wanted 4. So you need this instead:

But what about

which expands to

or an expensive expression? So you need this instead:

but now it only works for ints; you can’t do square(3.5) any more. To really fix this you have to use nonstandard extensions, something like:

or more like:

And that’s just to get trivial macros, like “square()”, to work.

---

You should be able to appreciate now the tremendous power of macros. This is why there are so many “primitive features” of programming languages that can be considered as merely library functionality given a good macro system. For example, people are used to think about OOP as some inherent property of a language — but in Racket there are at least two very different object systems that it comes with, and several others in user-distributed code. All of these are implemented as a library which provides the functionality as well as the necessary syntax in the form of macros. So the basic principle is to have a small core language with powerful constructs, and make it easy to express complex ideas using these constructs.

This is an important point to consider before starting a new DSL (reminder: domain specific language) — if you need something that looks like a simple DSL but might grow to a full language, you can extend an existing language with macros to have the features you want, and you will always be able to grow to use the full language if necessary. This is particularly easy with Racket, but possible in other languages too.

---

Side note: the principle of a powerful but simple code language and easy extensions is not limited to using macros — other factors are involved, like first-class functions. In fact, “first class”-ness can help in many situations, for example: single inheritance + classes as first-class values can be used instead of multiple inheritance.

Tuesday, November 26th Types

PLAI §24

In our Toy language implementation, there are certain situations that are not covered. For example,

is not a problem, but

will eventually use Racket’s addition function on a boolean value, which will crash our evaluator. Assuming that we go back to the simple language we once had, where there were no booleans, we can still run into errors — except now these are the errors that our code raises:

or

or

In any case, it would be good to avoid such errors right from the start — it seems like we should be able to identify such bad code and not even try to run it. One thing that we can do is do a little more work at parse time, and declare the {1 2 3} program fragment as invalid. We can even try to forbid

in the same way, but what should we do with this? —

The validity of this depends on how it is used. The same goes for some invalid expressions — the above bogus expression can be fine if it’s in a context that shadows <:

Finally, consider this:

where mystery contains something like random or read. In general, knowing whether a piece of code will run with no errors is a problem that is equivalent to the halting problem — and because of this, there is no way to create an “exact” type system: they are all either too restrictive (rejecting programs that would run with no errors) or too permissive (accepting programs that might crash). This is a very practical issue — type safety means a lot less bugs in the system. A good type system is still an actively researched problem.

Tuesday, November 26th What is a Type?

PLAI §25

A type is any property of a program (or an expression) that can be determined without running the program. (This is different than what is considered a type in Racket which is a property that is known only at run-time, which means that before run-time we know nothing so in essence we have a single type (in the static sense).) Specifically, we want to use types in a way that predicts some aspects of the program’s behavior, for example, whether a program will crash.

Usually, types are being used as the kind of value that an expression can evaluate to, not the precise value itself. For example, we might have two kinds of values — functions and numbers, and we know that addition always operates on numbers, therefore

is a type error. Note that to determine this we don’t care about the actual function, just the fact that it is a function.

Important: types can discriminate certain programs as invalid, but they cannot discriminate correct programs from incorrect ones. For example, there is no way for any type system to know that this:

is an incorrect decrease-by-one function.

In general, type systems try to get to the optimal point where as much information as possible is known, yet the language is not too restricted, no significant computing resources are wasted, and programmers don’t spend much time annotating their code.

Why would you want to use a type system?

• Catch errors even in code that you don’t execute, for example, when your tests are too weak (but they do not substitute proper test suites).

• They help reduce the time spent on debugging (when they detect legitimate errors, rather than force you to change your code).

• As we have seen, they help in documenting code (but they do not substitute proper documentation).

• Compilers can use type information to make programs run faster.

• They encourage a more organized code development process. For example, our use of define-type and cases (inspired by ML) help guide your code. (But note that the actual code can be as disorganized as usual, typechecking or not…)

Tuesday, November 26th Our Types — The Picky Language

The first thing we need to do is to agree on what types are. Earlier, we talked about two types: numbers and functions (ignore booleans or anything else for now), we will use these two types for now.

In general, this means that we are using the Types are Sets meaning for types, and specifically, we will be implmenting a type system known as a Hindley-Milner system. This is not what Typed Racket is using. In fact, one of the main differences is that in our type system each binding has exactly one type, whereas in Typed Racket an identifier can have different types in different places in the code. An example of this is something that we’ve talked about earlier:

(: foo : (U String Number) -> Number)
(define (foo x)          ; \ these `x`s have a
  (if (number? x)        ; / (U Number String) type
    (+ x 1)              ; > this one is a Number
    (string-length x)))  ; > and this one is a String

A type system is presented as a collection of rules called “type judgments”, which describe how to determine the type of an expression. Beside the types and the judgments, a type system specification needs a (decidable) algorithm that can assign types to expressions.

Such a specification should have one rule for every kind of syntactic construct, so when we get a program we can determine the precise type of any expression. Also, these judgments are usually recursive since a type judgment will almost always rely on the types of sub-expressions (if any).

For our restricted system, we have two rules that we can easily specify:

(These rules are actually “axioms”, since the state facts that are true by themselves, with no need for any further work.)

And what about an identifier? Well, it is clear that we need to keep some form of an environment that will keep an account of types assigned to identifiers (note: all of this is not at run-time). This environment is used in all type judgments, and usually written as a capital Greek Gamma character (in some places G is used to stick to ASCII texts).

The conventional way to write the above two axioms is:

The first one is read as “Gamma proves that n has the type Number”. Note that this is a syntactic environment, much like DE-ENVs that you have seen in homework.

So, we can write a rule for identifiers that simply has the type assigned by the environment:

We now need a rule for addition and a rule for application (note: we’re using a very limited subset of our old language, where arithmetic operators are not function applications). Addition is easy: if we can prove that both a and b are numbers in some environment Γ, then we know that {+ a b} is a number in the same environment. We write this as follows:

Now, what about application? We need to refer to some arbitrary type now, and the common letter for that is a Greek lowercase tau:

that is — if we can prove that f is a function, and that v is a value of some type τₐ, then … ??? Well, we need to know more about f: we need to know what type it consumes and what type it returns. So a simple function is not enough — we need some sort of a function type that specifies both input and output types. We will use the notation that was seen throughout the semester and dump function. Now we can write:

which makes sense — if you take a function of type τ₁->τ₂ and you feed it what it expects, you get the obvious output type. But going back to the language — where do we get these new arrow types from? We will modify the language and require that every function specifies its input and output type (and assume we have only one argument functions). For example, we will write something like this for a function that is the curried version of addition:

So: the revised syntax for the limited language that contains only additions, applications and single-argument functions, and for fun — go back to using the call keyword is. The syntax we get is:

and the typing rules are:

But we’re still missing a big part — the current axiomatic rule for a fun expression is too weak. If we use it, we conclude that these expressions:

are valid, as well concluding that this program:

is valid, and should return a number. What’s missing? We need to check that the body part of the function is correct, so the rule for typing a fun is no longer a simple axiom but rather a type judgment. Here is how we check the body instead of blindly believing program annotations:

That is — we want to make sure that if x has type τ₁, then the body expression E has type τ₂, and if we can prove this, then we can trust these annotations.

There is an important relationship between this rule and the call rule for application:

• In this rule we assume that the input will have the right type and guarantee (via a proof) that the output will have the right type.

• In the application rule, we guarantee (by a proof) an input of the right type and assume a result of the right type.

(Side note: Racket comes with a contract system that can identify type errors dynamically, and assign blame to either the caller or the callee — and these correspond to these two sides.)

Note that, as we said, number is really just a property of a certain kind of values, we don’t know exactly what numbers are actually used. In the same way, the arrow function types don’t tell us exactly what function it is, for example, (Number -> Number) can indicate a function that adds three to its argument, subtracts seven, or multiplies it by 7619. But it certainly contains much more than the previous naive function type. (Consider also Typed Racket here: it goes much further in expressing facts about code.)

For reference, here is the complete BNF and typing rules:

---

Examples of using types (abbreviate Number as Num) — first, a simple example:

and a little more involved one:

Finally, try a buggy program like

and see where it is impossible to continue.

The main thing here is that to know that this is a type error, we have to prove that there is no judgment for a certain type (in this case, no way to prove that a fun expression has a Num type), which we (humans) can only do by inspecting all of the rules. Because of this, we need to also add an algorithm to our type system, one that we can follow and determine when it gives up.

Tuesday, November 26th Typing control

PLAI §26

We will now extend our typed Picky language to have a conditional expression, and predicates. First, we extend the BNF with a predicate expression, and we also need a type for the results:

Initially, we use the same rules, and add the obvious type for the predicate:

And what should the rule for if look like? Well, to make sure that the condition is a boolean, it should be something of this form:

What would be the types of t and e? A natural choice would be to let the programmer use any two types:

But what would the return type be? This is still a problem. (BTW, some kind of a union would be nice, but it has some strong implications that we will not discuss.) In addition, we will have a problem detecting possible errors like:

Since we know nothing about the condition, we can just as well be conservative and force both arms to have the same type. The rule is therefore:

— using the same letter indicates that we expect the types to be identical, unlike the previous attempt. Consequentially, this type system is fundamentally weaker than Typed Racket which we use in this class.

Here is the complete language specification with this extension:

Tuesday, November 26th Extending Picky

In general, we can extend this language in one of two ways. For example, lets say that we want to add the with form. One way to add it is what we did above — simply add it to the language, and write the rule for it. In this case, we get:

Note how this rule encapsulates information about the scope of with. Also note that we need to specify the types for the bound values.

Another way to achieve this extension is if we add with as a derived rule. We know that when we see a

expression, we can just translate it into

So we could achieve this extension by using a rewrite rule to translate all with expressions into calls of anonymous functions (eg, using the with-stx facility that we have seen recently). This could be done formally: begin with the with form, translate to the call form, and finally show the necessary goals to prove its type. The only thing to be aware of is the need to translate the types too, and there is one type that is missing from the typed-with version above — the output type of the function. This is an indication that we don’t really need to specify function output types — we can just deduce them from the code, provided that we know the input type to the function.

Indeed, if we do this on a general template for a with expression, then we end up with the same goals that need to be proved as in the above rule:

---

Conclusion — we’ve seen type judgment rules, and using them in proof trees. Note that in these trees there is a clear difference between rules that have no preconditions — there are axioms that are always true (eg, a numeral is always of type num).

The general way of proving a type seems similar to evaluation of an expression, but there is a huge difference — nothing is really getting evaluated. As an example, we always go into the body of a function expression, which is done to get the function’s type, and this is later used anywhere this function is used — when you evaluate this:

you first create a closure which means that you don’t touch the body of the function, and later you use it twice. In contrast, when you prove the type of this expression, you immediately go into the body of the function which you have to do to prove that it has the expected Number->Number type, and then you just use this type twice.

Finally, we have seen the importance of using the same type letters to enforce types, and in the case of typing an if statement this had a major role: specifying that the two arms can be any two types, or the same type.

Tuesday, November 26th Implementing Picky

The following is a simple implementation of the Picky language. It is based on the environments-based Flang implementation. Note the two main functions here — typecheck and typecheck*.

One thing that is very obvious when you look at the examples is that this language is way too verbose to be practical — types are repeated over and over again. If you look carefully at the typechecking fragments for the two relevant expressions — fun and with — you can see that we can actually get rid of almost all of the type annotations.

Tuesday, November 26th Improving Picky

The following version does that, there are no types mentioned except for the input type for a function. Note that we can do that at this point because our language is so simple that many pieces of code have a specific type. (For example, if we add polymorphism things get more complicated.)

Finally, an obvious question is whether we can get rid of all of the type declarations. The main point here is that we need to somehow be able to typecheck expressions and assign “temporary types” to them that will later on change — for example, when we typecheck this:

we need to somehow decide that the named expression has a general function type, with no commitment on the actual input and output types — and then change them after we typecheck the body. (We could try to resolve that somehow by typechecking the body first, but that will not work, since the body must be checked with some type assigned to the identifier, or it will fail.)

Even better…

This can be done using type variables — things that contain boxes that can be used to change types as typecheck progresses. The following version does that. (Also, it gets rid of the typecheck* thing, since it can be achieved by using a type-variable and a call to typecheck.) Note the interesting tests at the end.

Here are two other interesting things to try out — in particular, the type that is shown in the error message is interesting:

More specifically, it is interesting to try the following to see explicitly what our typechecker infers for {fun {x} {call x x}}:

To see it clearly, we can replace each (?T #&...) with the ... that it contains:

and to clarify further, convert the FunT to an infix -> and the #f to a <?> and use α for the unknown “type variable” that is represented by the #1 (which is used twice):

This shows us that the type is recursive.

Sidenote#1: You can now go back to the code and look at type->string, which is an attempt to implement a nice string representation for types. Can you see now why it cannot work (at least not without more complex code)?

Sidenote#2: Compare the above with OCaml, which can infer such types when started with a -rectypes flag:

# let foo = fun x -> x x ;;
val foo : ('a -> 'b as 'a) -> 'b = <fun>

The type here is identical to our type: 'a and 'b should be read as α and β resp., and as is used in the same way that Racket shows a cyclic structure using #0#. As for the question of why OCaml doesn’t always behave as if the -rectypes flag is given, the answer is that its type checker might fall into the same trap that ours does — it gets stuck with:

# let foo = (fun x -> x x) (fun x -> x x) ;;

The α that we see here is “kind of” in a direction of something that resembles a polymorphic type, but we really don’t have polymorphism in our language: each box can be filled just one time with one type, and from then on that type is used in all further uses of the same box type. For example, note the type error we get with:

Tuesday, November 26th Typing Recursion

We already know that without recursion life can be very boring… So we obviously want to be able to have recursive functions — but the question is how will they interact with our type system. One thing that we have seen is that by just having functions we get recursion. This was achieved by the Y combinator function. It seems like the same should apply to our simple typed language. The core of the Y combinator was using an expression similar to Omega that generates the infinite loop that is needed. In our language:

This expression was impossible to evaluate completely since it never terminates, but it served as a basis for the Y combinator so we need to be able to perform this kind of infinite loop. Now, consider the type of the first x — it’s used in a call expression as a function, so its type must be a function type, say τ₁->τ₂. In addition, its argument is x itself so its type is also τ₁ — this means that we have:

and from this we get:

And this is a type that does not exist in our type system, since we can only have finite types. Therefore, we have a proof by contradiction that this expression cannot be typed in our system.

This is closely related to the fact that the typed language we have described so far is “strongly normalizing”: no matter what program you write, it will always terminate! To see this, very informally, consider this language without functions — this is clearly a language where all programs terminate, since the only way to create a loop is through function applications. Now add functions and function application — in the typing rules for the resulting language, each fun creates a function type (creates an arrow), and each function application consumes a function type (deletes one arrow) — since types are finite, the number of arrows is finite, which means that the number of possible applications is finite, so all programs must run in finite time.

Note that when we discussed how to type the Y combinator we needed to use a Rec constructor — something that the current type system has. Using that, we could have easily solve the τ₁ = τ₁ -> τ₂ equation with (Rec τ₁ (τ₁ -> τ₂)).

In the our language, therefore, the halting problem doesn’t even exist, since all programs (that are properly typed) are guaranteed to halt. This property is useful in many real-life situations (consider firewall rules, configuration files, devices with embedded code). But the language that we get is very limited as a result — we really want the power to shoot our feet…

Tuesday, November 26th Extending Picky with recursion

As we have seen, our language is strongly normalizing, which means that to get general recursion, we must introduce a new construct (unlike previously, when we didn’t really need one). We can do this as we previously did — by adding a new construct to the language, or we can somehow extend the (sub) language of type descriptions to allow a new kind of type that can be used to solve the τ₁ = τ₁ -> τ₂ equation. An example of this solution would be similar to the Rec type constructor in Typed Racket: a new type constructor that allows a type to refer to itself — and using (Rec τ₁ (τ₁ -> τ₂)) as the solution. However, this complicates things: type descriptions are no longer unique, since we have Num, (Rec this Num), and (Rec this (Rec that Num)) that are all equal.

For simplicity we will now take the first route and add rec — an explicit recursive binder form to the language (as with with, we’re going back to rec rather than bindrec to keep things simple).

First, the new BNF:

We now need to add a typing judgment for rec expressions. What should it look like?

rec is similar to all the other local binding forms, like with, it can be seen as a combination of a function and an application. So we need to check the two things that those rules checked — first, check that the body expression has the right type assuming that the type annotation given to x is valid:

Now, we also want to add the other side — making sure that the τ₁ type annotation is valid:

But that will not be possible in general — V is an expression that usually includes x itself — that’s the whole point. The conclusion is that we should use a similar trick to the one that we used to specify evaluation of recursive binders — the same environment is used for both the named expression and for the body expression:

You can also see now that if this rule adds an arrow type to the Γ type environment (i.e., τ₁ is an arrow), then it is doing so in a way that makes it possible to use it over and over, making it possible to run infinite loops in this language.

Our complete language specification is below.

Tuesday, November 26th Typing Data

PLAI §27

An important concept that we have avoided so far is user-defined types. This issue exists in practically all languages, including the ones we did so far, since a language without the ability to create new user-defined types is a language with a major problem. (As a side note, we did talk about mimicking an object system using plain closures, but it turns out that this is insufficient as a replacement for true user-defined types — you can kind of see that in the Schlac language, where the lack of all types mean that there is no type error.)

In the context of a statically typed language, this issue is even more important. Specifically, we talked about typing recursive code, but we should also consider typing recursive data. For example, we will start with a length function in an extension of the language that has empty?, rest, and NumCons and NumEmpty constructors:

But adding all of these new functions as built-ins is getting messy: we want our language to have a form for defining new kinds of data. In this example — we want to be able to define the NumList type for lists of numbers. We therefore extend the language with a new with-type form for creating new user-defined types, using variants in a similar way to our own course language:

We assume here that the NumList definition provides us with a number of new built-ins — NumEmpty and NumCons constructors, and assume also a cases form that can be used to both test a value and access its components (with the constructors serving as patterns). This makes the code a little different than what we started with:

The question is what should the ??? be filled with? Clearly, recursive data types are very common and we need to support them. The scope of with-type should therefore be similar to rec, except that it works at the type level: the new type is available for its own definition. This is the complete code now:

(Note that in the course language we can do just that, and in addition, the Rec type constructor can be used to make up recursive types.)

An important property that we would like this type to have is for it to be well founded: that we’d never get stuck in some kind of type-level infinite loop. To see that this holds in this example, note that some of the variants are self-referential (only NumCons here), but there is at least one that is not (NumEmpty) — if there wasn’t any simple variant, then we would have no way to construct instances of this type to begin with!

[As a side note, if the language has lazy semantics, we could use such types — for example:

Reasoning about such programs requires more than just induction though.]

Tuesday, November 26th Judgments for recursive types

If we want to have a language that is basically similar to the course language, then — as seen above — we’d use a similar cases expression. How should we type-check such expressions? In this case, we want to verify this:

Similarly to the judgment for if expressions, we require that the two result expressions are numbers. Indeed, you can think about cases as a more primitive tool that has the functionality of if — in other words, given such user-defined types we could implement booleans as a new type and and implement if using cases. For example, wrap programs with:

and translate {if E1 E2 E3} to {cases E1 [{True} E2] [{False} E3]}.

Continuing with typing cases, we now have:

But this will not work — we have no type for r here, so we can’t prove the second subgoal. We need to consider the NumList type definition as something that, in addition to the new built-ins, provides us with type judgments for these built-ins. In the case of the NumCons variant, we know that using {NumCons x r} is a pattern that matches NumList values that are a result of this variant constructor but it also binds x and r to the values of the two fields, and since all uses of the constructor are verified, the fields have the declared types. This means that we need to extend Γ in this rule so we’re able to prove the two subgoals. Note that we do the same for the NumEmpty case, except that there are no new bindings there.

Finally, we need to verify that the value itself — l — has the right type: that it is a NumList.

But why NumList and not some other defined type? This judgment needs to do a little more work: it should inspect all of the variants that are used in the branches, find the type that defines them, then use that type as the subgoal. Furthermore, to make the type checker more useful, it can check that we have complete coverage of the variants, and that no variant is used twice:

Note that how this is different from the version in the textbook — it has a type-case expression with the type name mentioned explicitly — for example: {type-case l NumList {{NumEmpty} 0} ...}. This is essentially the same as having each defined type come with its own cases expression. Our rule needs to do a little more work, but overall it is a little easier to use. (And the same goes for the actual implementation of the two languages.)

In addition to cases, we should also have typing judgments for the constructors. These are much simpler, for example:

Alternatively, we could add the constructors as new functions instead of new special forms — so in the Picky language they’d be used in call expressions. The with-type will then create the bindings for its scope at runtime, and for the typechecker it will add the relevant types to Γ:

(This requires functions of any arity, of course.) Using accessor functions could be similarly simpler than cases, but less convenient for users.

Note about representation: a by-product of our type checker is that whenever we have a NumList value, we know that it must be an instance of either NumEmpty or NumCons. Therefore, we could represent such values as a wrapped value container, with a single bit that distinguishes the two. This is in contrast to dynamically typed languages like Racket, where every new type needs to have its own globally unique tag.

extra] [Tuesday, November 26th “Runaway” instances

Consider this code:

We now know how to type check its validity, but what about the type of this whole expression? The obvious choice would be NumList:

There is a subtle but important problem here: the expression evaluates to a NumList, but we can no longer use this value, since we’re out of the scope of the NumList type definition! In other words, we would typecheck a program that is pretty much useless.

Even if we were to allow such a value to flow to a different context with a NumList type definition, we wouldn’t want the two to be confused — following the principle of lexical scope, we’d want each type definition to be unique to its own scope even if it has the same concrete name. For example, using NumList as the type of the inner with-type here:

would make it wrong.

(In fact, we might want to have a new type even if the value goes outside of this scope and back in. The default struct definitions in Racket have exactly this property — they’re generative — which means that each “call” to define-struct creates a new type, so:

returns two instances of two different foo types!)

One way to resolve this is to just forbid the type from escaping the scope of its definition — so we would forbid the type of the expression from being NumList, which makes

invalid. But that’s not enough — what about returning a compound value that contains an instance of NumList? For example — what if we return a list or a function with a NumList instance?

Obviously, we would need to extend this restriction: the resulting type should not mention the defined type at all — not even in lists or functions or anything else. This is actually easy to do: if the overall expression is type-checked in the surrounding lexical scope, then it is type-checked in the surrounding type environment (Γ), and that environment has nothing in it about NumList (well, nothing about this NumList).

Note that this is, very roughly speaking, what our course language does: define-type can only define new types when it is used at the top-level.

This works fine with the above assumption that such a value would be completely useless — but there are aspects of such values that are useful. Such types are close to things that are known as “existential types”, and they are for defining opaque values that you can do nothing with except pass them around, and only code in a specific lexical context can actually use them. For example, you could lump together the value with a function that can work on this value. If it wasn’t for the define-type top-level restriction, we could write the following:

There is nothing that we can do with resulting Foo instance (we don’t even have a way to name it) — but in the result of the above function we get also a function that could work on such values, even ones from different calls:

Since such kind of values are related to hiding information, they’re useful (among other things) when talking about module systems (and object systems), where you want to have a local scope for a piece of code with bindings that are not available outside it.

Tuesday, November 26th Type soundness

PLAI §28

Having a type checker is obviously very useful — but to be able to rely on it, we need to provide some kind of a formal account of the kind of guarantees that we get by using one. Specifically, we want to guarantee that a program that type-checks is guaranteed to never fail with a type error. Such type errors in Racket result in an exception — but in C they can result in anything. In our simple Picky implementation, we still need to check the resulting value in run:

A soundness proof for this would show that checking the result (in cases) is not needed. However, the check must be there since Typed Racket (or any other typechecker) is far from making up and verifying such a proof by itsef.

In this context we have a specific meaning for “fail with a type error”, but these failures can be very different based on the kind of properties that your type checker verifies. This property of a type system is called soundness: a sound type system is one that will never allow such errors for type-checked code:

For any program p, if we can type-check p : τ, then p will evaluate to a value that is in the type τ.

The importance of this can be seen in that it is the only connection between the type system and code execution. Without it, a type system is a bunch of syntactic rules that are completely disconnected from how the program runs. (Note also that — “in the type” — works for the (common) case where types are sets of values.)

But this statement isn’t exactly what we need — it states a property that is too strong: what if execution gets stuck in an infinite loop? (That wasn’t needed before we introduced rec, where we could extend the conclusion part to: “… then p will terminate and evaluate to a value that is in the type τ”.) We therefore need to revise it:

For any program p, if we can type-check p : τ, and if p terminates and returns v, then v is in the type τ.

But there are still problems with this. Some programs evaluate to a value, some get stuck in an infinite loop, and some … throw an error. Even with type checking, there are still cases when we get runtime errors. For example, in practically all statically typed languages the length of a list is not encoded in its type, so {first null} would throw an error. (It’s possible to encode more information like that in types, but there is a downside to this too: putting more information in the type system means that things get less flexible, and it becomes more difficult to write programs since you’re moving towards proving more facts about them.)

Even if we were to encode list lengths in the type, we would still have runtime errors: opening a missing file, writing to a read-only file fetching a non-existent URL, etc, so we must find some way to account for these errors. Some “solutions” are:

• For all cases where an error should be raised, just return some value (of the appropriate type). For example, (first l) could return 0 if the list is empty; (substring "foo" 10 20) would return “huh?”, etc. It seems like a dangerous way to resolve the issue, but in fact that’s what most C library calls do: return some bogus value (for example, malloc() returns NULL when there is no available memory), and possibly set some global flag that specifies the exact error. (The main problem with this is that C programmers often don’t check all of these conditions, leading to propagating undetected errors further down — and all of this is a very rich source of security issues.)

• For all cases where an error should be raised, just get stuck into an infinite loop. This approach is obviously impractical — but it is actually popular in some theoretical circles. The reason for that is that theory people will often talk about “domains”, and to express facts about computation on these domains, they’re extended with a “bottom” value that represents a diverging computation. Since this introduction is costly in terms of work that it requires, adding one more such value can lead to more effort than re-using the same “bottom” value.

• Raise an exception. This works out better than the above two extremes, and it is the approach taken by practically all modern languages.

So, assuming exceptions, we need to further refine what it means for a type system to be sound:

For any program p, if we can type-check p : τ, and if p terminates without exceptions and returns v, then v is in the type τ.

An important thing to note here is that languages can have very different ideas about where to raise an exception. For example, Scheme implementations often have a trivial type-checker and throw runtime exceptions when there is a type error. On the other hand, there are systems that express much more in their type system, leaving much less room for runtime exceptions.

A soundness proof ties together a particular type system with the statement that it is sound. As such, it is where you tie the knot between type checking (which happens at the syntactic level) and execution (dealing with runtime values). These are two things that are usually separate — we’ve seen throughout the course many examples for things that could be done only at runtime, and things that should happen completely on the syntax. eval is the important semantic function that connects the two worlds (compile also did this, when we converted our evaluator to a compiler) — and in here, it is the soundness proof that makes the connection.

To demonstrate the kind of differences between the two sides, consider an if expression — when it is executed, only one branch is evaluated, and the other is irrelevant, but when we check its type, both sides need to be verified. The same goes for a function whose execution get stuck in an infinite loop: the type checker will not get into a loop since it is not executing the code, only scans the (finite) syntax.

The bottom line here is that type soundness is really a claim that the type system provides some guarantees about the runtime behavior of programs, and its proof demonstrates that these guarantees do hold. A fundamental problem with the type system of C and C++ is that it is not sound: these languages have a type system, but it does not provide such runtime guarantees. (In fact, C is even worse in that it really has two type systems: there is the system that C programmers usually interact with, which has a conventional set of type — including even higher-order function types; and there is the machine-level type system, which only talks about various bit lengths of data. For example, using “%s” in a printf() format string will blindly copy characters from the address pointed to by the argument until it reaches a 0 character — even if the actual argument is really a floating point number or a function.)

Note that people often talk about “strongly typed languages”. This term is often meaningless in that different people take it to mean different things: it is sometimes used for a language that “has a static type checker”, or a language that “has a non-trivial type checker”, and sometimes it means that a language has a sound type system. For most people, however, it means some vague idea like “a language like C or Pascal or Java” rather than some concrete definition.

extra] [Tuesday, November 26th Explicit polymorphism

PLAI §29

Consider the length definition that we had — it is specific for NumLists, so rename it to lengthNum:

To simplify things, assume that types are previously defined, and that we have an even more Racket-like language where we simply write a define form:

What would happen if, for example, we want to take the length of a list of booleans? We won’t be able to use the above code since we’d get a type error. Instead, we’d need a separate definition for the other kind of length:

We’ve designed a statically typed language that is effective in catching a large number of errors, but it turns out that it’s too restrictive — we cannot implement a single generic length function. Given that our type system allows an infinite number of types, this is a major problem, since every new type that we’ll want to use in a list requires writing a new definition for a length function that is specific to this type.

One way to address the problem would be to somehow add a new length primitive function, with specific type rules to make it apply to all possible types. (Note that the same holds for the list type too — we need a new type definition for each of these, so this solution implies a new primitive type that will do the same generic trick.) This is obviously a bad idea: there are other functions that will need the same treatment (append, reverse, map, fold, etc), and there are other types with similar problems (any new container type). A good language should allow writing such a length function inside the language, rather than changing the language for every new addition.

Going back to the code, a good question to ask is what is it exactly that is different between the two length functions? The answer is that there’s very little that is different. To see this, we can take the code and replace all occurrences of Num or Bool by some ???. Even better — this is actually abstracting over the type, so we can use a familiar type variable, τ:

This is a kind of a very low-level “abstraction” — we replace parts of the text — parts of identifiers — with a kind of a syntactic meta variable. But the nature of this abstraction is something that should look familiar — it’s abstracting over the code, so it’s similar to a macro. It’s not really a macro in the usual sense — making it a real macro involves answering questions like what does length evaluate to (in the macro system that we’ve seen, a macro is not something that is a value in itself), and how can we use these macros in the cases patterns. But still, the similarity should provide a good intuition about what goes on — and in particular the basic fact is the same: this is an abstraction that happens at the syntax level, since typechecking is something that happens at that level.

To make things more manageable, we’ll want to avoid the abstraction over parts of identifiers, so we’ll move all of the meta type variables, and make them into arguments, using 〈...〉 brackets to stand for “meta level applications”:

Now, the first “〈τ〉” is actually a kind of an input to length, it’s a binding that has the other τs in its scope. So we need to have the syntax reflect this somehow — and since fun is the way that we write such abstractions, it seems like a good choice:

But this is very confused and completely broken. The new abstraction is not something that is implemented as a function — otherwise we’ll need to somehow represent type values within our type system. (Trying that has some deep problems — for example, if we have such type values, then it needs to have a type too; and if we add some Type for this, then Type itself should be a value — one that has itself as its type!)

So instead of fun, we need a new kind of a syntactic, type-level abstraction. This is something that is acts as a function that gets used by the type checker. The common way to write such functions is with a capital lambda — Λ. Since we already use Greek letters for things that are related to types, we’ll use that as is (again, with "〈〉"s), instead of a confusing capitalized Lambda (or a similarly confusing Fun):

and to use this length we’ll need to instantiate it with a specific type:

Note that we have several kinds of meta-applications, with slightly different intentions:

• length〈τ〉 is the recursive call, which needs to keep using the same type that initiated the length call. It makes sense to have it there, since length is itself a type abstraction.

• List〈τ〉 is using List as if it’s also this kind of an abstraction, except that instead of abstracting over some generic code, it abstracts over a generic type. This makes sense too: it naturally leads to a generic definition of List that works for all types since it is also an abstraction.

• Finally there are Empty〈τ〉 and Cons〈τ〉 that are used for patterns. This might not be necessary, since they are expected to be variants of the List〈τ〉 type. But if we were doing this without pattern matching (for example, see the book) then we’d need null? and rest functions. In that case, the meta application would make sense — null?〈τ〉 and rest〈τ〉 are the τ-specific versions of these functions which we get with this meta-application, in the same way that using length needs an explicit type.

Actually, the last item points at one way in which the above sample calls:

are broken — we should also have a type argument for list:

or, given that we’re in the limited picky language:

Such a language is called “parametrically polymorphic with explicit type parameters” — it’s polymorphic since it applies to any type, and it’s explicit since we have to specify the types in all places.

extra] [Tuesday, November 26th Polymorphism in the type description language

Given our definition for length, the type of length〈Num〉 is obvious:

but what would be the type of length by itself? If it was a function (which was a broken idea we’ve seen), then we would write:

But this is broken in the same way: the first arrow is fundamentally different than the second — one is used for a Λ, and the other for a fun. In fact, the arrows are even more different, because the two τs are very different: the first one binds the second. So the first arrow is bogus — instead of an arrow we need some way to say that this is a type that “for all τ” is “List〈τ〉 -> Num”. The common way to write this should be very familiar:

Finally, τ is usually used as a meta type variable; for these types the convention is to use the first few Greek letters, so we get:

And some more examples:

where × stands for multiple arguments (which isn’t mentioned explicitly in Typed Racket).

extra] [Tuesday, November 26th Type judgments for explicit polymorphism and execution

Given our notation for polymorphic functions, it looks like we’re introducing a runtime overhead. For example, our length definition:

looks like it now requires another curried call for each iteration through the list. This would be bad for two reasons: first, one of the main goals of static type checking is to avoid runtime work, so adding work is highly undesirable. An even bigger problem is that types are fundamentally a syntactic thing — they should not exist at runtime, so we don’t want to perform these type applications at runtime simply because we don’t want types to exist at runtime. If you think about it, then every traditional compiler that typechecks code does so while compiling, not when the resulting compiled program runs. (A recent exception in various languages are “dynamic” types that are used in a way that is similar to plain (untyped) Racket.)

This means that we want to eliminate these applications in the typechecker. Even better: instead of complicating the typechecker, we can begin by applying all of the type meta-applications, and get a result that does not have any such applications or any type variables left — then use the simple typechecker on the result. This process is called “type elaboration”.

As usual, there are two new formal rules for dealing with these abstractions — one for type abstractions and another for type applications. Starting from the latter:

which means that when we encounter a type application E〈τ₂〉 where E has a polymorphic type ∀α.τ, then we substitute the type variable α with the input type τ₂. Note that this means that conceptually, the typechecker is creating all of the different (monomorphic) length versions, but we don’t need all of them for execution — having checked the types, we can have a single length function which would be similar to the function that Racket uses (i.e., the same “low level” code with types erased).

To see how this works, consider our length use, which has a type of ∀α. List〈α〉 -> Num. We get the following proof that ends in the exact type of length (remember that when you prove you climb up):

The second rule for type abstractions is:

This rule means that to typecheck a type abstraction, we need to check the body while binding the type variable α — but it’s not bound to some specific type. Instead, it’s left unspecified (or non-deterministic) — and typechecking is expected to succeed without requiring an actual type. If some specific type is actually required, then typechecking should fail. The intuition behind this is that a polymorphic function can be one only if it doesn’t need some specific type — for example, {fun {x} {- {+ x 1} 1}} is an identity function, but it’s an identity that requires the input to be a number, and therefore it cannot have a polymorphic ∀α.α type like {fun {x} x}.

Another example is our length function — the actual type that the list holds better not matter, or our length function is not really polymorphic. This makes sense: to typecheck the function, this rule means that we need to typecheck the body, with α being some unknown type that cannot be used.

One thing that we need to be careful when applying any kind of abstraction (and the first rule does just that for a very simple lambda-calculus-like language) is infinite loops. But in the case of our type language, it turns out that this lambda-calculus that gets used at the meta-level is one of the strongly normalizing kinds, therefore no infinite loops happen. Intuitively, this means that we should be able to do this elaboration in just one pass over the code. Furthermore, there are no side-effects, therefore we can safely cache the results of applying type abstraction to speed things up. In the case of length, using it on a list of Num will lead to one such application, but when we later get to the recursive call we can reuse the (cached) first result.

extra] [Tuesday, November 26th Explicit polymorphism conclusions

Quoted directly from the book:

Explicit polymorphism seems extremely unwieldy: why would anyone want to program with it? There are two possible reasons. The first is that it’s the only mechanism that the language designer gives for introducing parameterized types, which aid in code reuse. The second is that the language includes some additional machinery so you don’t have to write all the types every time. In fact, C++ introduces a little of both (though much more of the former), so programmers are, in effect, manually programming with explicit polymorphism virtually every time they use the STL (Standard Template Library). Similarly, the Java 1.5 and C# languages support explicit polymorphism. But we can possibly also do better than foist this notational overhead on the programmer.

Tuesday, December 3rd Web Programming

PLAI §15

Consider web programming as a demonstration of a frequent problem. The HTTP protocol is stateless: each HTTP query can be thought of as running a program (or a function), getting a result, then killing it. This makes interactive applications hard to write.

For example, consider this behavior (which is based on a real story of a probably not-so-real bug known as “the ITA bug”):

• You go on a flight reservation website, and look at flights to Paris or London for a vacation.

• You get a list of options, and choose one for Paris and one for London, ctrl-click the first and then the second to open them in new tabs.

• You look at the descriptions and decide that you like the first one best, so you click the button to buy the ticket.

• A month later you go on your plane, and when you land you realize that you’re in the wrong country — the ticket you paid for was the second one after all…

Obviously there is some fundamental problem here — especially given that this problem plagued many websites early on (and these days these kind of problems can still be found in some places (like the registrar’s system), except that people are much more aware of it, and are much more prepared to deal with it). In an attempt to clarify what it is exactly that went wrong, we might require that each interaction will result in something that is deterministically based on what the browser window shows when the interaction is made — but even that is not always true. Consider the same scenario except with a bookstore and an “add to my cart” button. In this case you want to be able to add one item to the cart in the first window, then switch to the second window and click “add” there too: you want to end up with a cart that has both items.

The basic problem here is HTTP’s statelessness, something that both web servers and web browsers use extensively. Browsers give you navigation buttons and sometimes will not even communicate with the web server when you use them (instead, they’ll show you cached pages), they give you the ability to open multiple windows or tabs from the current one, and they allow you to “clone” the current tab. If you view each set of HTTP queries as a session — this means that web browsers allow you to go back and forth in time, explore multiple futures in parallel, and clone your current world.

These are features that the HTTP protocol intentionally allows by being stateless, and that people have learned to use effectively. A stateful protocol (like ssh, or ftp) will run in a single process (or a program, or a function) that is interacting with you directly, and this process dies only when you disconnect. A big advantage of stateful protocols is their ability to be very interactive and rely on state (eg, an editor updates a character on the screen, relying on the rest of it showing the same text); but stateless protocols can scale up better, and deal with a more hectic kind of interaction (eg, open a page on an online store, keep it open and buy the item a month later; or any of the above “time manipulation” devices).

Side-note: Some people think that Ajax is the answer to all of these problems. In reality, Ajax is layered on top of (asynchronous) web queries, so in fact it is the exact same situation. You do have an option of creating an application that works completely on the client side, but that wouldn’t be as attractive — and even if you do so, you’re still working inside a browser that can play the same time tricks.

Tuesday, December 3rd Basic web programming

PLAI §16

Obviously, writing programs to run on a web server is a profitable activity, and therefore highly desirable. But when we do so, we need to somehow cope with the web’s statelessness. To see the implications from a PL point of view we’ll use a small “toy” example that demonstrates the basic issues — an “addition” service:

• Server sends a page asking for a number,
• User types a number and hits enter,
• Server sends a second page asking for another number,
• User types a second number and hits enter,
• Server sends a page showing the sum of the two numbers.

[Such a small application is not realistic, of course: you can obviously ask for both numbers on the same page. We still use it, though, to minimize the general interaction problem to a more comprehensible core problem.]

Starting from just that, consider how you’d want to write the code for such a service. (If you have experience writing web apps, then try to forget all of that now, and focus on just this basic problem.) The plain version of what we want to implement is:

which is trivially “translated” to:

But this is never going to work. The interaction is limited to presenting the user with some data and that’s all — you cannot do any kind of interactive querying. For the purpose of making this more concrete, imagine that web-read and web-display both communicate information to the user via something like error: the information is sent and at the same time the whole computation is aborted. With this, the above code will just manage to ask for the first number and nothing else happens.

We therefore must turn this server function into three separate functions: one that shows the prompt for the first number, one that gets the value entered and shows the second prompt, and a third that shows the results page. The first two of these functions would send the information (and the respective computation dies) to the browser, including a form submission URL that will invoke the next function.

Assuming a generic “query argument” that represents the browser request, and a return value that represents a page for the browser to render, we have:

Note that f2 receives the first number directly, but f3 doesn’t. Yet, it is obviously needed to show the sum. A typical hack to get around this is to use a “hidden field” in the HTML form that f2 generates, where that field holds the second result. To make things more concrete, we’ll use some imaginary web API functions:

Which would (supposedly) result in something like the following html forms when the user enters 1 and 2:

This is often a bad solution: it gets very difficult to manage with real services where the “state” of the server consists of much more than just a single number — and it might even include values that are not expressible as part of the form (for example an open database connection or a running process). Worse, the state is all saved in the client browser — if it dies, then the interaction is gone. (Imagine doing your taxes, and praying that the browser won’t crash a third time.)

Another common approach is to store the state information on the server, and use a small handle (eg, in a cookie) to identify the state, then each function can use the cookie to retrieve the current state of the service — but this is exactly how we get to the above bugs. It will fail with any of the mentioned time-manipulation features.

Tuesday, December 3rd Continuations: Web Programming

To try and get a better solution, we’ll re-start with the original expression:

and assuming that web-read works as a regular function, we need to begin with executing the first read:

We then need to take that result and plug it into an expression that will read the second number and sum the results — that’s the same as the first expression, except that instead of the first web-read we use a “hole”:

where <*> marks the point where we need to plug the result of the first question into. A better way to explain this hole is to make the expression into a function:

We can split the second and third interactions in the same way. First we can assemble the above two bits of code into an expression that has the same meaning as the original one:

And now we can continue doing this and split the body of the consumer:

into a “reader” and the rest of the computation (using a new hole):

Doing all of this gives us:

And now we can proceed to the main trick. Conceptually, we’d like to think about web-read as something that is implemented in a simple way:

except that the “real” thing would throw an error and die once the prompt is printed. The trick is one that we’ve already seen: we can turn the code inside-out by making the above “hole functions” be an argument to the reading function — a consumer callback for what needs to be done once the number is read. This callback is called a continuation, and we’ll use a /k suffix for names of functions that expect a continuation (k is a common name for a continuation argument):

This is not too different from the previous version — the only difference is that we make the function take a consumer function as an input, and hand it what we read instead of just returning it. It makes things a little easier, since we pass the hole function to web-read/k, and it will invoke it when needed:

You might notice that this looks too complicated; we could get exactly the same result with:

but then there’s not much point to having web-read/k at all… So why have it? Remember that the main problem is that in the context of a web server we think of web-read as something that throws an error and kills the computation. So if we use such a web-read/k with a continuation, we can make it save this continuation in some global state, so it can be used later when there is a value.

As a side note, all of this might start looking very familiar to you if you have any experience working with callback-heavy code. In fact, consider the fact that the continuation (or k) is basically just a callback, so the above is roughly:

We’ll talk more about JavaScript later.

Tuesday, December 3rd Simulating web reading

We can now actually try all of this in plain Racket by simulating web interactions. This is useful to look at the core problem while avoiding the whole web mess that is uninteresting for the purpose of our discussion. The main feature that we need to emulate is statelessness — and as we’ve discussed, we can simulate that using error to guarantee that the process is properly killed for each interaction. We will do this in web-display which simulates sending the results to the client and therefore terminates the server process:

More importantly, we need to do it in web-read/k — but in this case, we need more than just an error — we need a way to store the k so the computation can be resumed later. To continue with the web analogy we do this in two steps: error is used to display the information (the input prompt), and the user action of entering a number and submitting it will be simulated by calling a function. Since the computation is killed after we show the prompt, the way to implement this is by making the user call a toplevel submit function — and before throwing the interaction error, we’ll save the k continuation in a global box:

submit uses the saved continuation:

For safety, we’ll initialize resumer with a function that throws an error (a real one, not intended for interactions), make web-display reset it to the same function, and also make submit do so after grabbing its value — meaning that submit can only be used after a web-read/k. And for convenience, we’ll use raise-user-error instead of error, which is a Racket function that throws an error without a stack trace (since our errors are intended). It’s also helpful to disable debugging in DrRacket, so it won’t take us back to the code over and over again.

We can now try out our code for the addition server, using plain argument names instead of <*>s:

and see how everything works. You can also try now the bogus expression that we mentioned:

and see how it breaks: the first web-read/k saves the identity function as the global resumer, losing the rest of the computation.

---

Again, this should be familiar: we’ve taken a simple compound expression and “linearized” it as a sequence of an input operation and a continuation receiver for its result. This is essentially the same thing that we used for dealing with inputs in the lazy language — and the similarity is not a coincidence. The problem that we faced there was very different (representing IO as values that describe it), but it originates from a similar situation — some computation goes on (in whatever way the lazy language decides to evaluate it), and when we have a need to read something we must return a description of this read that contains “the rest of the computation” to the eager part of the interpreter that executes the IO. Once we get the user input, we send it to this computation remainder, which can return another read request, and so on.

Based on this intuition, we can guess that this can work for any piece of code, and that we can even come up with a nicer “flat” syntax for it. For example, here is a simple macro that flattens a sequence of reads and a final display:

and using it:

However, we’ll avoid such cuteness to make the transformation more explicit for the sake of the discussion. Eventually, we’ll see how things can become even better than that (as done in Racket): we can get to write plain-looking Racket expressions and avoid even the need for an imperative form for the code. In fact, it’s easy to write this addition server using Racket’s web server framework, and the core of the code looks very simple:

There is not much more than that — it has two utilities, page creates a well-formed web page, and web-read performs the reading. The main piece of magic there is in send/suspend which makes the web server capture the computation’s continuation and store it in a hash table, to be retrieved when the user visits the given URL. Here’s the full code:

Tuesday, December 3rd More Web Transformations

PLAI §17

Tuesday, December 3rd Transforming a recursive function

Going back to transforming code, we did the transformation on a simple expression — and as you’d guess, it’s possible to make it work for more complex code, even recursive functions. Let’s start with some simple function that sums up a bunch of numbers, given a list of prompts for these numbers. Since it’s a function, it’s a reusable piece of code that can be used in multiple places, and to demonstrate that, we add a web-display with a specific list of prompts.

We begin by converting the web-read to its continuation version:

But using web-read/k immediately terminates the running computation, which means that when sum is called on the last line, the surrounding web-display will be lost, and therefore this will not work. The way to solve this is to make sum itself take a continuation, which we’ll get in a similar way — by rewriting it as a sum/k function, and then we can make our sample use pull in the web-display into the callback as we’ve done before:

We also need to deal with the recursive sum call and change it to a sum/k. Clearly, the continuation is the same continuation that the original sum was called with, so we need to pass it on in the recursive call too:

But there is another problem now: the addition is done outside of the continuation, therefore it will be lost as soon as there’s a second web-read/k call. In other words, computation bits that are outside of any continuations are going to disappear, and therefore they must be encoded as an explicit part of the continuation. The solution is therefore to move the addition into the continuation:

Note that with this code every new continuation is bigger — it contains the previous continuation (note that “contains” here is done by making it part of the closure), and it also contains one new addition.

But if the continuation is only getting bigger, then how do we ever get a result out of this? Put differently, when we reach the end of the prompt list, what do we do? — Clearly, we just return 0, but that silently drops the continuation that we worked so hard to accumulate. This means that just returning 0 is wrong — instead, we should send the 0 to the pending continuation:

This makes sense now, and the code works as expected. This sum/k is a utility to be used in a web server application, and such applications need to be transformed in a similar way to what we’re doing. Therefore, our own sum/k is a function that expects to be invoked from such transformed code — so it needs to have an argument for the waiting receiver, and it needs to pass that receiver around (accumulating more functionality into it) until it’s done.

As a side note, web-display is the only thing that is used in the toplevel continuation, so we could have used it directly without a lambda wrapper:

Tuesday, December 3rd Using sum/k

To get some more experience with this transformation, we’ll try to convert some code that uses the above sum/k. For example, lets add a multiplier argument that will get multiplied by the sum of the given numbers. Begin with the simple code. This is an actual application, so we’re writing just an expression to do the computation and show the result, not a function.

We now need to turn the two function calls into their */k form. Since we covered sum/k just now, begin with that. The first step is to inspect its continuation: this is the same code after we replace the sum call with a hole:

Now take this expression, make it into a function by abstracting over the hole and call it n, and pass that to sum/k:

(Note that this is getting rather mechanical now.) Now for the web-read part, we need to identify its continuation — that’s the expression that surrounds it inside the first continuation function, and we’ll use m for the new hole:

As above, abstract over m to get a continuation, and pass it into web-read/k:

and we’re done. An interesting question here is what would happen if instead of the above, we start with the web-read and then get to the sum? We’d end up with a different version:

Note how these options differ — one reads the multiplier first, and the other reads it last.

Side-note: if in the last step of turning web-read to web-read/k we consider the whole expression when we formulate the continuation, then we get to the same code. But this isn’t really right, since it is converting code that is already-converted.

In other words, our conversion results in code that fixes a specific evaluation order for the original expression. The way that the inputs happen in the original expression

is unspecified in the code — it only happens to be left-to-right implicitly, because Racket evaluates function arguments in that order. However, the converted code does not depend on how Racket evaluates function arguments. (Can you see a similar conclusion here about strictness?)

Note also another property of the converted code: every intermediate result has a name now. This makes sense, since another way to fix the evaluation order is to do just that. For example, convert the above to either

or

This is also a good way to see why this kind of conversion can be a useful tool in compiling code: the resulting code is in a kind of a low-level form that makes it easy to translate to assembly form, where function calls are eliminated, and instead there are only jumps (since all calls are tail-calls). In other words, the above can be seen as a piece of code that is close to something like:

and it’s almost visible in the original converted code if we format it in a very weird way:

Tuesday, December 3rd Converting stateful code

Another case to consider is applying this transformation to code that uses mutation with some state. For example, here’s some simple account code that keeps track of a balance state:

(Note that there is no web-display here, since it’s an infinite loop.) As usual, the fact that this function is expected to be used by a web application means that it should receive a continuation:

Again, we need to convert the web-read into web-read/k by abstracting out its continuation. We’ll take the set-box! expression and create a continuation out of it:

and using change as the name for the continuation argument, we get:

And finally, we translate the loop call to pass along the same continuation it received (it seems suspicious, but there’s nothing else that could be used there):

But if we try to run this — (account/k web-display) — we don’t get any result at all: it reads one number and then just stops without the usual request to continue, and without showing any result. The lack of printed result is a hint for the problem — it must be the void return value of the set-box!. Again, we need to remember that invoking a web-read/k kills any pending computation and the following (resume) will restart its continuation — but the recursive call is not part of the loop.

The problem is the continuation that we formulated:

which should actually contain the recursive call too:

In other words, the recursive call was left outside of the continuation, and therefore it was lost when the fake server terminated the computation on a web-read/k — so it must move into the continuation as well:

and the code now works. The only suspicious thing that we’re still left with is the loop that passes k unchanged — but this actually is the right thing to do here. The original loop had a tail-recursive call that didn’t pass along any new argument values, since the infinite loop is doing its job via mutations to the box and nothing else was done in the looping call. The continuation of the original call is therefore also the continuation of the second call, etc. All of these continuations are closing over a single box and this binding does not change (it cannot change if we don’t use a set!); instead, the boxed value is what changes through the loop.

Tuesday, December 3rd Converting higher order functions

Next we try an even more challenging transformation: a higher order function. To get a chance to see more interesting examples, we’ll have some more code in this case.

For example, say that we want to compute the sum of squares of a list. First, the simple code (as above, there’s no need to wrap a web-display around the whole thing, just make it return the result):

Again, we can begin with web-read — we want to convert it to the continuation version, which means that we need to convert read-number to get one too. This transformation is refreshingly trivial:

This is an interesting point — it’s a simple definition that just passes k on, as is. The reason for this is similar to the simple continuation passing of the imperative loop: the pre-translation read-number is doing a simple tail call to web-read, so the evaluation context of the two is identical. The only difference is the prompt argument, and that’s the same format call.

Of course things would be different if format itself required a web interaction, since then we’d need some format/k, but without that things are really simple. The same goes for the two utility functions — sum and square: they’re not performing any web interaction so it seems likely that they’ll stay the same.

We now get to the main expression, which should obviously change since it needs to call read-number/k, so it needs to send it some continuation. By now, it should be clear that passing an identity function as a continuation is going to break the surrounding context once the running computation is killed for the web interaction. We need to somehow generate a top-level identity continuation and propagate it inside, and the sum call should be in that continuation together with the web-display call. Actually, if we do the usual thing and write the expression with a <*> hole, we get:

and continuing with the mechanical transformation that we know, we need to abstract over this expression+hole into a function, then pass it as an argument to read-number/k:

But that can’t work in this case — we need to send read-number/k a prompt, but we can’t get a specific one since there is a list of them. In fact, this is related to a more serious problem — pulling out read-number/k like this is obviously broken since it means that it gets called only once, instead, we need to call it once for each prompt value.

The solution in this case is to convert map too:

and of course we should move web-display and sum into that continuation:

We can now use read-number/k, but the question is what should it get for it’s continuation?

Clearly, map/k will need to somehow communicate some continuation to the mapped function, which in turn will send it to read-number/k. This means that the mapped function should get converted too, and gain a k argument. To do this, we’ll first make things convenient and have a name for it (this is only for convenience, we could just as well convert the lambda directly):

Then convert it in the now-obvious way:

Everything is in place now — except for map/k, of course. We’ll start with the definition of plain map:

The first thing in turning it into a map/k is adding a k input,

and now we need to face the fact that the f input is itself one with a continuation — an f/k:

Consider now the single f call — that should turn into a call to f/k with some continuation:

but since f/k will involve a web interaction, it will lead to killing the cons around it. The solution is to move that cons into the continuation that is handed to f/k — and as usual, this involves the second cons argument — the continuation is derived from replacing the f/k call by a hole:

and abstracting that hole, we get:

We now do exactly the same for the recursive map call — it should use map/k with f/k and some continuation:

and we need to move the surrounding cons yet again into this continuation. The holed expression is:

and abstracting that and moving it into the map/k continuation we get:

There are just one more problem with this — the k argument is never used. This implies two changes, since it needs to be used once in each of the conditional branches. Can you see where it should be added? (Try to do this before reading the code below.)

The complete code follows:

Tuesday, December 3rd Highlevel Overview on Continuations

Very roughly speaking, the transformation we made turns a function call like

into

This is the essence of the solution to the statelessness problem: to remember where we left off, we conveniently flip the expression inside-out so that its context is all stored in its continuation. One thing to note is that we did this only for functions that had some kind of web interaction, either directly or indirectly (since in the indirect case they still need to carry around the continuation).

If we wanted to make this process a completely mechanical one, then we wouldn’t have been able to make this distinction. After all, a function like map is perfectly fine as it is, unless it happens to be used with a continuation-carrying function — and that’s something that we know only at runtime. We would therefore need to transform all function calls as above, which in turn means that all functions would need to get an extra continuation argument.

Here are a few things to note about such fully-transformed code:

• All function calls in such code are tail calls. There is no single call with some context around it that is left for the time when the call is done. This is the exact property that makes it useful for a stateless interaction: such contexts are bad since a web interaction will mean that the context is discarded and lost. (In our pretend system, this is done by throwing an error.) Having no non-tail context means that capturing the continuation argument is sufficient, and no context gets lost.

• An implication of this, when you consider how the language is implemented, is that there is no need to have anything “on the stack” to execute fully transformed code. (If you’d use the stepper on such code, there would be no accumulation of context.) So is this some radical new way of computing without a stack? Not really: if you think about it, continuation arguments hold the exact same information that is traditionally put on the stack. (There is therefore an interesting relationship between continuations and runtime stacks, and in fact, one way of making it possible to grab continuations without doing such a transformation is to capture the current stack.)

• The evaluation order is fixed. Obviously, if Racket guarantees a left-to-right evaluation, then the order is always fixed — but in the fully transformed code there are no function calls where this makes any difference. If Racket were to change, the transformed code would still retain the order it uses. More specifically, when we do the transformation, we control the order of evaluation by choosing how to proceed at every point. For example, if we have:

  (...stuff... (f1 ...args...) (f2 ...args...) ...more-stuff...)

  then it’s up to use to choose whether to pull f1 first, or maybe we’d want to start with f2.

  But there’s more: the resulting code is independent of the evaluation strategy of the language. Even if the language is lazy, the transformed code is still executing things in the same order. (Alternatively, we could convert things so that the resulting computation corresponds to a lazy evaluation strategy even in a strict language.)

• In other words, the converted code is completely sequential. The conversion process requires choosing left-to-right or delaying some evaluations (or all), but the resulting code is free from any of these and has exactly one specific (sequential) order. You can therefore see how this kind of transformation is something that a compiler would want to do, since the resulting sequential code is easier for execution on a sequential base (like machine code, or C code). Another way to see this is that we have explicit names for each and every intermediate result — so the converted code would have a direct mapping between identifiers and machine registers (unlike “plain” code where some of these are implicit and compilation needs to make up names).

• The transformation is a global one. Not only do we have to transform the first top-level expression that makes up the web application, we also need to convert every function that is mentioned in the code, and in functions that those functions mentioned, etc. Even worse, the converted code is very different from the original version, since everything is shuffled around — in a way that matches the sequential execution, but it’s very hard to even see the original intention through all of these explicit continuations and the new intermediate result names.

  The upshot of this is that it’s not really something that we need to do manually, but instead we’d like it to be done automatically for us, by the compiler of the language.

What we did here is the tedious way of getting continuations: we basically implemented them by massaging our code, turning it inside-out into code with the right shape. The problem with this is that the resulting code is no longer similar to what we had originally written, which makes it more difficult to debug and to maintain. We therefore would like to have this done in some automatic way, ideally in a way that means that we can leave our plain original code as is.

Tuesday, December 3rd An Automatic Continuation Converter

PLAI §18

The converted code that we produced manually above is said to be written in “Continuation Passing Style”, or CPS. What we’re looking for is for a way to generate such code automatically — a way to “CPS” a given source code. When you think about it, this process is essentially a source to source function which should be bolted onto the compiler or evaluator. In fact, if we want to do this in Racket, then this description makes it sound a lot like a macro — and indeed it could be implemented as such.

Note that “CPS” has two related but distinct meanings here: you could have code that is written “in CPS style”, which means that it handles explicit continuations. Uses of this term usually refer to using continuation functions in some places in the code, not for fully transformed code. The other meaning is used for fully-CPS-ed code, which is almost never written directly. In addition, “CPS” is often used as a verb — either the manual process of refactoring code into passing some continuations explicitly (in the first case), or the automatic process of fully converting code (in the second one).

Before we get to the actual implementation, consider how we did the translation — there was a difference in how we handled plain top-level expressions and library functions. In addition, we had some more discounts in the manual process — one such discount was that we didn’t treat all value expressions as possible computations that require conversion. For example, in a function application, we took the function sub-expression as a simple value and left it as is, but for an automatic translation we need to convert that expression too since it might itself be a more complicated expression.

Instead of these special cases and shortcuts, we’ll do something more uniform: we will translate every expression into a function. This function will accept a receiver (= a continuation) and will pass it the value of the expression. This will be done for all expressions, even simple ones like plain numbers, for example, we will translate the 5 expression into (lambda (k) (k 5)), and the same goes for other constants and plain identifiers. Since we’re specifying a transformation here, we will treat it as a kind of a meta function and use a CPS[x] to make it easy to talk about:

When we convert a primitive function application, we still do the usual thing, which is now easier to see as a general rule — using CPS[?] as the meta function that does the transformation:

In the above, you can see that (CPS[E] (lambda (v) …)) can be read as “evaluate E and bind the result to v”. (But note that the CPS conversion is not doing any evaluation, it just reorders code to determine how it gets evaluated when it later runs — so “compute” might be a better term to use here.) With this in mind, we can deal with other function applications: evaluate the function form, evaluate the argument form, then apply the first value on the second value, and finally wrap everything with a (lambda (k) …) and return the result to this continuation:

But this is the rule that we should use for primitive non-continuation functions only — it’s similar to what we did with + (except that we skipped evaluating + since it’s known). Instead, we’re dealing here with functions that are defined in the “web language” (in the code that is being converted), and as we’ve seen, these functions get a k argument which they use to return the result to. That was the whole point: pass k on to functions, and have them return the value directly to the k context. So the last part of the above should be fixed:

There’s a flip side to this transformation — whenever a function is created with a lambda form, we need to add a k argument to it, and make it return its value to it. Then, we need to “lift” the whole function as usual, using the same transformation we used for other values in the above. We’ll use k for the latter continuation argument, and cont for the former:

It is interesting to note the two continuations in the translated result: the first one (using k) is the continuation for the function value, and the second one (using cont) is the continuation used when the function is applied. Comparing this to our evaluators — we can say that the first roughly corresponds to evaluating a function form to get a closure, and the second corresponds to evaluating the body of a function when it’s applied, which means that cont is the dynamic continuation that matches the dynamic context in which the function is executed. Inspecting the CPS-ed form of the identity function is unsurprising: it simply passes its first argument (the “real” one) into the continuation since that’s how we return values in this system:

Note the reduction of a trivial application — doing this systematic conversion leads to many of them.

We now get to the transformation of the form that is the main reason we started with all of this — web-read. This transformation is simple, it just passes along the continuation to web-read/k:

We also need to deal with web-display — we changed the function calling protocol by adding a continuation argument, but web-display is defined outside of the CPS-ed language so it doesn’t have that argument. Another way of fixing it could be to move its definition into the language, but then we’ll still need to have a special treatment for the error that it uses.

As you can see, all of these transformations are simple rewrites. We can use a simple syntax-rules macro to implement this transformation, essentially creating a DSL by translating code into plain Racket. Note that in the specification above we’ve implicitly used some parts of the input as keywords — lambda, +, web-read, and define — this is reflected in the macro code. The order of the rules is important, for example, we need to match first on (web-read E) and then on the more generic (E1 E2), and we ensure that the last default lifting of values has a simple expression by matching on (x …) before that.

The transformation that this code implements is one of the oldest CPS transformations — it is called the Fischer Call by Value CPS transformation, and is due Michael Fischer. There has been much more research into such transformations — the Fischer translation, while easy to understand due to its uniformity, introduces significant overhead in the form of many new functions in its result. Some of these are easy to optimize — for example, things like ((lambda (k) (k v)) E) could be optimized to just (E v) assuming a left-to-right evaluation order or proving that E has no side-effects (and Racket performs this optimization and several others), but some of the overhead is not easily optimized. There have been several other CPS transformations, in an attempt to avoid such overhead.

Finally, trying to run code using this macro can be a little awkward. We need to explicitly wrap all values in definitions by a CPS, and we need to invoke top-level expressions with a particular continuation — web-display in our context. We can do all of that with a convenience macro that will transform a number of definitions followed by an optional expression.

Note the use of begin — usually, it is intended for sequential execution, but it is also used in macro result expressions when we need a macro to produce multiple expressions (since the result of a macro must be a single S-expression) — this is why it’s used here, not for sequencing side-effects.

The interesting thing that this macro does is set up a proper continuation for definitions and top-level expressions. In the latter case, it passes web-display as the continuation, and in the former case, it passes the identity function as the continuation — which is used to “lower” the lifted value from its continuation form into a plain value. Using the identity function as a continuation is not really correct: it means that if evaluating the expression to be bound performs some web interaction, then the definition will be aborted, leaving the identifier unbound. The way to solve this is by arranging for the definition operation to be done in the continuation, for example, we can get closer to this using an explicit mutation step:

But there are two main problems with this: first, the rest of the code — (CPS-code more ...) — should also be done in the continuation, which will defeat the global definitions. We could try to use the continuation to get the scope:

but that breaks recursive definitions. In any case, the second problem is that this is not accurate even if we solved the above: we really need to have parts of the Racket definition mechanism exposed to make it work. So we’ll settle with the simple version as an approximation. It works fine if we use definitions only for functions, and invoke them in toplevel expressions.

For reference, the complete code at this point follows.

Here is a quick example of using this:

Note that this code uses web-display, which is not really needed since CPS-code would use it as the top-level continuation. (Can you see why it works the same either way?) So this is even closer to a plain program:

A slightly more complicated example:

Using this for the other examples is not possible with the current state of the translation macro. These example will require extending the CPS transformation with functions of any arity, multiple expressions in a body, and it recognize additional primitive functions. None of these is difficult, it will just make it more verbose.

Tuesday, December 3rd Continuations as a Language Feature

This is conceptually between PLAI §18 and PLAI §19

In the list of CPS transformation rules there were two rules that deserve additional attention in how they deal with their continuation.

First, note the rule for web-display:

— it simply ignores its continuation. This means that whenever web-display is used, the rest of the computation is simply discarded, which seems wrong — it’s the kind of problem that we’ve encountered several times when we discussed the transforming web application code. Of course, this doesn’t matter much for our implementation of web-display since it aborts the computation anyway using error — but what if we did that intentionally? We would get a kind of an “abort now” construct: we can implement this as a new abort form that does just that:

You could try that — (CPS-code (+ 1 2)) produces 3 as “web output”, but (CPS-code (+ 1 (abort 2))) simply returns 2. In fact, it doesn’t matter how complicated the code is — as soon as it encounters an abort the whole computation is discarded and we immediately get its result, for example, try this:

it reads the first number and then it immediately returns 999. This seems like a potentially useful feature, except that it’s a little too “untamed” — it aborts the program completely, getting all the way back to the top-level with a result. (It’s actually quite similar to throwing an exception, only without a way to catch it.) It would be more useful to somehow control the part of the computation that gets aborted instead.

That leads to the second exceptional form in our translator: web-read. If you look closely at all of our transformation rules, you’ll see that the continuation argument is never made accessible to user code — the k argument is always generated by the macro (and inaccessible to user code due to the hygienic macro system). The continuation is only passed as the extra argument to user functions, but in the rule that adds this argument:

the new cont argument is introduced by the macro so it is inaccessible as well. The only place where the k argument is actually used is in the web-read rule, where it is sent to the resulting web-read/k call. (This makes sense, since web reading is how we mimic web interaction, and therefore it is the only reason for CPS-ing our code.) However, in our fake web framework this function is a given built-in, so the continuation is still not accessible for user code.

What if we pass the continuation argument to a user function in a way that intentionally exposes it? We can achieve this by writing a function that is similar to web-read/k, except that it will somehow pass the continuation to user code. A simple way to do that is to have the new function take a function value as its primary input, and call this function with the continuation (which is still received as the implicit second argument):

This is close, but it fails because it doesn’t follow our translated function calling protocol, where every function receives two inputs — the original argument and the continuation. Because of this, we need to call f with a second continuation value, which is k as well:

But we also fail to follow the calling protocol by passing k as is: it is a continuation value, which in our CPS system is a one-argument function. (In fact, this is another indication that continuations are not accessible to user code — they don’t follow the same function calling protocol.) It is best to think about continuations as meta values that are not part of the user language just yet. To make it usable, we need to wrap it so we get the usual two-argument function which user code can call:

This explicit wrapping is related to the fact that continuations are a kind of meta-level value — and the wrapping is needed to “lower” it to the user’s world. (This is sometimes called “reification”: a meta value is reified as a user value.)

Using this new definition, we can write code that can access its own continuation as a plain value. Here is a simple example that grabs the top-level continuation and labels it abort, then uses it in the same way we’ve used the above abort:

But we can grab any continuation we want, not just the top-level one:

Side note: how come we didn’t need a new CPS translation rule for this function? There is no need for one, since call-k is already written in a way that follows our calling convention, and no translation rule is needed. In fact, no such rule is needed for web-read too — except for changing the call to web-read/k, it does exactly the same thing that a function call does, so we can simply rename web-read/k as web-read and drop the rule. (Note that the rewritten function call will have a (CPS web-read) — but CPS-ing an identifier results in the identifier itself.) The same holds for web-display — we just need to make it adhere to the calling convention and add a k input which is ignored. One minor complication is that web-display is also used as a continuation value for a top-level expression in CPS-code — so we need to wrap it there.

The resulting code follows:

Obviously, given call-k we could implement web-read/k in user code: call-k makes the current continuation available and going on from there is simple (it will require a little richer language, so we will do that in a minute). In fact, there is no real reason to stick to the fake web framework to play with continuations. (Note: since we don’t throw an error to display the results, we can also allow multiple non-definition expressions in CPS-code.)

Tuesday, December 3rd Continuations in Racket

As we have seen, CPS-ing code makes it possible to implement web applications with a convenient interface. This is fine in theory, but in practice it suffers from some problems. Some of these problems are technicalities: it relies on proper implementation of tail calls (since all calls are tail calls), and it represents the computation stack as a chain of closures and therefore prevents the usual optimizations. But there is one problem that is much more serious: it is a global transformation, and as such, it requires access to the complete program code. As an example, consider how CPS-code deals with definitions: it uses an identity function as the continuation, but that wasn’t the proper way to do them, since it would break if computing the value performs some web interaction. A good solution would instead put the side-effect that define performs in the continuation — but this side effect is not even available for us when we work inside Racket.

Because of this, the proper way to make continuations available is for the language implementation itself to provide it. There are a few languages that do just that — and Scheme has pioneered this as part of the core requirements that the standard dictates: a Scheme implementation needs to provide call-with-current-continuation, which is the same tool as our call-k. Usually it is also provided with a shorter name, call/cc. Here are our two examples, re-done with Racket’s built-in call/cc:

[Side note: continuations as we see here are still provided only by a few “fringe” functional languages. However, they are slowly making their way into more mainstream languages — Ruby has these continuations too, and several other languages provide more limited variations, like generators in Python. On the other hand, Racket provides a much richer functionality: it has delimited continuations (which represents only a part of a computation context), and its continuations are also composable — a property that goes beyond what we see here.]

Racket also comes with a more convenient let/cc form, which exposes the “grab the current continuation” pattern more succinctly — it’s a simple macro definition:

and the two examples become:

When it gets to choosing an implementation strategy, there are two common approaches: one is to do the CPS transformation at the compiler level, and another is to capture the actual runtime stack and wrap it in an applicable continuation objects. The former can lead to very efficient compilation of continuation-heavy programs, but the latter makes it easier to deal with foreign functions (consider higher order functions that are given as a library where you don’t have its source) and allows using the normal runtime stack that CPUs are using very efficiently. Racket implements continuations with the latter approach mainly for these reasons.

To see how these continuations expose some of the implementation details that we normally don’t have access to, consider grabbing the continuation of a definition expression:

Note that using a top-level (let/cc abort …code…) is not really aborting for a reason that is related to this: a true abort must capture the continuation before any computation begins. A natural place to do this is in the REPL implementation.

Finally, we can use these to re-implement our fake web framework, using Racket’s continuations instead of performing our own transformation. The only thing that requires continuations is our web-read — and using the Racket facilities we can implement it as follows:

Note that this kind of an implementation is no longer a “language” — it is implemented as a plain library now, demonstrating the flexibility that having continuations in our language enables. While this is still just our fake toy framework, it is the core way in which the Racket web server is implemented (see the “addition server” implementation above), using a hash table that maps freshly made URLs to stored continuations. The complete code follows:

Using this, you can try out some of the earlier examples, which now become much simpler since there is no need to do any CPS-ing. For example, the code that required transforming map into a map/k can now use the plain map directly. In fact, that’s the exact code we started that example with — no changes needed:

Note how web-read is executed directly — it is a plain library function.

extra] [Tuesday, December 3rd Playing with Continuations

PLAI §19

So far we’ve seen a number of “tricks” that can be done with continuations. The simplest was aborting a computation — here’s an implementation of functions with a return that can be used to exit the function early:

[Side note: This is a cheap demonstration. If we rewrite the loop tail-recursively, then aborting it is simple — just return 0 instead of continuing the loop. And that’s not a coincidence, aborting from a tail-calling loop is easy, and CPS makes such aborts possible by making only tail calls.]

But such uses of continuations are simple because they’re used only to “jump out” of some (dynamic) context. More exotic uses of continuations rely on the ability to jump into a previously captured continuation. In fact, our web-read implementation does just that (and more). The main difference is that in the former case the continuation is used exactly once — either explicitly by using it, or implicitly by returning a value (without aborting). If a continuation can be used after the corresponding computation is over, then why not use it over and over again… For example, we can try an infinite loop by capturing a continuation and later use it as a jump target:

This almost works — we get two printouts so clearly the jump was successful. The problem is that the captured loop continuation is the one that expects a value to bind to loop itself, so the second attempted call has 'something as the value of loop, obviously, leading to an error. This can be used as a hint for a solution — simply pass the continuation to itself:

Another way around this problem is to capture the continuation that is just after the binding — but we can’t do that (try it…). Instead, we can use side-effects:

Note: the 'something value goes to the continuation which makes it the result of the (let/cc ...) expression — which means that it’s never actually used now.

This might seem like a solution that is not as “clean”, since it uses mutation — but note that the problem that we’re solving stems from a continuation that exposes the mutation that Racket performs when it creates a new binding.

Here’s an example of a loop that does something a little more interesting in a goto-like kind of way:

Note: in this example the continuation is called without any inputs. How is this possible? As we’ve seen, the 'something value in the last example is the never-used result of the let/cc expression. In this case, the continuation is called with no input, which means that the let/cc expression evaluates to … nothing! This is not just some void value, but no value at all. The complete story is that in Racket expressions can evaluate to multiple values, and in this case, it’s no values at all.

Given such examples it’s no wonder that continuations tend to have a reputation for being “similar to goto in their power”. This reputation has some vague highlevel justification in that both features can produce obscure “spaghetti code” — but in practice they are very different. On one hand continuations are more limited: unlike goto, you can only jump to a continuation that you have already “visited”. On the other hand, jumping to a continuation is doing much more than jumping to a goto label, the latter changes the next instruction to execute (the “program counter” register), but the former changes current computation context (in low level terms, both the PC register and the stack). (See also the setjmp() and longjmp() functions in C, or the context related functions (getcontext(), setcontext(), swapcontext()).)

To demonstrate how different continuations are from plain gotos, we’ll start with a variation of the above loop — instead of performing the loop we just store it in a global box, and we return the counter value instead of printing it:

Now, the first time we call (foo), we get 1 as expected, and then we can call (unbox loop) to re-invoke the continuation and get the following numbers:

[Interesting experiment: try doing the same, but use (list (foo)) as the first interaction, and the same ((unbox loop)) later.]

The difference between this use and a goto is now evident: we’re not just just jumping to a label — we’re jumping back into a computation that returns the next number. In fact, the continuation can include a context that is outside of foo, for example, we can invoke the continuation from a different function, and loop can be used to search for a specific number:

and now (bar) returns 11. The loop is now essentially going over the obvious part of foo but also over parts of bar. Here’s an example that makes it even more obvious:

Since the y binding becomes part of the loop. Our foo can be considered as a kind of a producer for natural numbers that can be used to find a specific number, invoking the loop continuation to try the next number when the last one isn’t the one we want.

extra] [Tuesday, December 3rd The Ambiguous Operator: amb

Our foo is actually a very limited version of something that is known as “McCarthy’s Ambiguous Operator”, usually named amb. This operator is used to perform a kind of a backtrack-able choice among several values.

To develop our foo into such an amb, we begin by renaming foo as amb and loop as fail, and instead of returning natural numbers in sequence we’ll have it take a list of values and return values from this list. Also, we will use mutable variables instead of boxes to reduce clutter (a feature that we’ve mostly ignored so far). The resulting code is:

Of course, we also need to check that we actually have values to return:

The resulting amb can be used in a similar way to the earlier foo:

This is somewhat useful, but searching through a simple list of values is not too exciting. Specifically, we can have only one search at a time. Making it possible to have multiple searches is not too hard: instead of a single failure continuation, store a stack of them, where each new amb pushes a new one on it.

We define failures as this stack and push a new failure continuation in each amb. fail becomes a function that simply invokes the most recent failure continuation, if one exists.

This is close, but there’s still something missing. When we run out of options from the choices list, we shouldn’t just throw an error — instead, we should invoke the previous failure continuation, if there is one. In other words, we want to use fail, but before we do, we need to pop up the top-most failure continuation since it is the one that we are currently dealing with:

Note the addition of a tiny assert utility, something that is commonly done with amb. We can now play with this code as before:

But of course the new feature is more impressive, for example, find two numbers that sum up to 6 and the first is the square of the second:

Find a Pythagorean triplet:

Specifying the list of integers is tedious, but easily abstracted into a function:

A more impressive demonstration is finding a solution to tests known as “Self-referential Aptitude Test”, for example, here’s one such test (by Christian Schulte and Gert Smolka) — it’s a 10-question multiple choice test:

1. The first question whose answer is b is question (a) 2; (b) 3; (c) 4; (d) 5; (e) 6.
2. The only two consecutive questions with identical answers are questions (a) 2 and 3; (b) 3 and 4; (c) 4 and 5; (d) 5 and 6; (e) 6 and 7.
3. The answer to this question is the same as the answer to question (a) 1; (b) 2; (c) 4; (d) 7; (e) 6.
4. The number of questions with the answer a is (a) 0; (b) 1; (c) 2; (d) 3; (e) 4.
5. The answer to this question is the same as the answer to question (a) 10; (b) 9; (c) 8; (d) 7; (e) 6.
6. The number of questions with answer a equals the number of questions with answer (a) b; (b) c; (c) d; (d) e; (e) none of the above.
7. Alphabetically, the answer to this question and the answer to the following question are (a) 4 apart; (b) 3 apart; (c) 2 apart; (d) 1 apart; (e) the same.
8. The number of questions whose answers are vowels is (a) 2; (b) 3; (c) 4; (d) 5; (e) 6.
9. The number of questions whose answer is a consonant is (a) a prime; (b) a factorial; (c) a square; (d) a cube; (e) divisible by 5.
10. The answer to this question is (a) a; (b) b; (c) c; (d) d; (e) e.

and the solution is pretty much a straightforward translation:

Note that the solution is simple because of the freedom we get with continuations: the search is not a sophisticated one, but we’re free to introduce ambiguity points anywhere that fits, and mix assertions with other code without worrying about control flow (as you do in an implementation that uses explicit loops). On the other hand, it is not too efficient since it uses a naive search strategy. (This could be improved somewhat by deferring ambiguous points, for example, don’t assign q7, q8, q9, and q10 before the first question; but much of the cost comes from the strategy for implementing continuation in Racket, which makes capturing continuations a relatively expensive operation.)

When we started out with the modified loop, we had a representation of an arbitrary natural number — but with the move to lists of choices we lost the ability to deal with such infinite choices. Getting it back is simple: delay the evaluation of the amb expressions. We can do that by switching to a list of thunks instead. The change in the code is in the result: just return the result of calling choice instead of returning it directly. We can then rename amb to amb/thunks and reimplement amb as a macro that wraps all of its sub-forms in thunks.

With this, we can implement code that computes choices rather than having them listed:

or even ones that are infinite:

As with any infinite sequence, there are traps to avoid. In this case, trying to write code that can find any Pythagorean triplet as:

will not work. The problem is that the search loop will keep incrementing c, and therefore will not find any solution. The search can work if only the top-most choice is infinite:

The complete code follows:

As a bonus, the code includes a collect tool that can be used to collect a number of results — it uses fail to iterate until a sufficient number of values is collected. A simple version is:

(Question: why does this code use mutation to collect the results?)

But since this might run into a premature failure, the actual version in the code installs its own failure continuation that simply aborts the collection loop. To try it out:

extra] [Tuesday, December 3rd Generators (and Producers)

Another popular facility that is related to continuations is generators. The idea is to split code into separate “producers” and “consumers”, where the computation is interleaved between the two. This simplifies some notoriously difficult problems. It is also a twist on the idea of co-routines, where two functions transfer control back and forth as needed. (Co-routines can be developed further into a “cooperative threading” system, but we will not cover that here.)

A classical example that we have mentioned previously is the “same fringe” problem. One of the easy solutions that we talked about was to run two processes that spit out the tree leaves, and a third process that grabs both outputs as they come and compares them. Using a lazy language allowed a very similar solution, where the two processes are essentially represented as two lazy lists. But with continuations we can find a solution that works in a strict language too, and in fact, one that is very close to the two processes metaphor.

The fact that continuations can support such a solution shouldn’t be surprising: as with the kind of server-client interactions that we’ve seen with the web language, and as with the amb tricks, the main theme is the same — the idea of suspending computation. (Intuitively, this also explains why a lazy language is related: it is essentially making all computations suspendable in a sense.)

To implement generators, we begin with a simple code that we want to eventually use:

where yield is expected to behave similarly to a return — it should make the function return 1 when called, and then somehow return 2 and 3 on subsequent calls. To make it easier to develop, we’ll make yield an argument to the producer:

To use this producer, we need to find a proper value to call it with. Sending it an identity, (lambda (x) x), is clearly not going to work: it will make all yields executed on the first call, returning the last value. Instead, we need some way to abort the computation on the first yield. This, of course, can be done with a continuation, which we should send as the value of the yield argument. And indeed,

returns 1 as we want. But if we use this expression again, we get more 1s as results:

The problem is obvious: our producer starts every time from scratch, always sending the first value to the given continuation. Instead, we need to make it somehow save where it stopped — its own continuation — and on subsequent calls it should resume from that point. We start with adding a resume continuation to save our position into:

Next, we need to make it so that each use of yield will save its continuation as the place to resume from:

But this is still broken in an obvious way: every time we invoke this function, we define a new local resume which is always #f, leaving us with the same behavior. We need resume to persist across calls — which we can get by “pulling it out” using a let:

And this actually works:

(Tracing how it works is a good exercise.)

Before we continue, we’ll clean things up a little. First, to make it easier to get values from the producer, we can write a little helper:

Next, we can define a local helper inside the producer to improve it in a similar way by making up a yield that wraps the raw-yield input continuation (also flip the condition):

And we can further abstract out the general producer code from the specific 1-2-3 producer that we started with. The complete code is now:

When we now evaluate (get producer) three times, we get back the three values in the correct order. But there is a subtle bug here, first try this (after re-running!):

Seems that this is stuck in an infinite loop. To see where the problem is, re-run to reset the producer, and then we can see the following interaction:

This looks weird… Here’s a more clarifying example:

Can you see what’s wrong now? It seems that all three invocations of the producer use the same continuation — the first one, specifically, the (list <*>) continuation. This also explains why we run into an infinite loop with (list (get producer) (get producer)) — the first continuation is:

so when we get the first 1 result we plug it in and proceed to evaluate the second (get producer), but that re-invokes the first continuation again, getting into an infinite loop. We need to look closely at our make-producer to see the problem:

When (make-producer (lambda (yield) ...)) is first called, resume is initialized to #f, and the result is the (lambda (raw-yield) ...), which is bound to the global producer. Next, we call this function, and since resume is #f, we apply the producer on our yield — which is a closure that has a reference to the raw-yield that we received — the continuation that was used in this first call. The problem is that on subsequent calls resume will contain a continuation which it is called, but this will jump back to that first closure with the original raw-yield, so instead of returning to the current calling context, we re-return to the first context — the same first continuation. The code can be structured slightly to make this a little more obvious: push the yield definition into the only place it is used (the first call):

yield is not used elsewhere, so this code has exactly the same meaning as the previous version. You can see now that when the producer is first used, it gets a raw-yield continuation which is kept in a newly made closure — and even though the following calls have different continuations, we keep invoking the first one. These calls get new continuations as their raw-yield input, but they ignore them. It just happened that the when we evaluated (get producer) three times on the REPL, all calls had essentially the same continuation (the P part of the REPL), so it seemed like things are working fine.

To fix this, we must avoid calling the same initial raw-yield every time: we must change it with each call so it is the right one. We can do this with another mutation — introduce another state variable that will refer to the correct raw-yield, and update it on every call to the producer. Here’s one way to do this:

Using this, we get well-behaved results:

or (again, after restarting the producer by re-running the code):

Side-note: a different way to achieve this is to realize that when we invoke resume, we’re calling the continuation that was captured by the let/cc expression. Currently, we’re sending just 'blah to that continuation, but we could send raw-yield there instead. With that, we can make that continuation be the target of setting the return-to-caller state variable. (This is how PLAI solves this problem.)

Continuing with our previous code, and getting the yield back into a a more convenient definition form, we have this complete code:

There is still a small problem with this code:

Tracking this problem is another good exercise, and finding a solution is easy. (For example, throwing an error when the producer is exhausted, or returning 'done, or returning the return value of the producer function.)

extra] [Tuesday, December 3rd Delimited Continuations

While the continuations that we have seen are a useful tool, they are often “too global” — they capture the complete computation context. But in many cases we don’t want that, instead, we want to capture a specific context. In fact, this is exactly why producer code got complicated: we needed to keep capturing the return-to-caller continuation to make it possible to return to the correct context rather than re-invoking the initial (and wrong) context.

Additional work on continuations resulted in a feature that is known as “delimited continuations”. These kind of continuations are more convenient in that they don’t capture the complete context — just a potion of it up to a specific point. To see how this works, we’ll restart with a relatively simple producer definition:

This producer is essentially the same as one that we’ve seen before: it seems to work in that it returns the desired values for every call:

But fails in that it always returns to the initial context:

Fixing this will lead us down the same path we’ve just been through: the problem is that generator is essentially an indirection “trampoline” function that goes to whatever cont currently holds, and except for the initial value of cont the other values are continuations that are captured inside yield, meaning that the calls are all using the same ret continuation that was grabbed once, at the beginning. To fix it, we will need to re-capture a return continuation on every use of yield, which we can do by modifying the ret binding, giving us a working version:

This pattern of grabbing the current continuation and then jumping to another — (let/cc k (set! cont k) (ret value)) — is pretty common, enough that there is a specific construct that does something similar: control. Translating the let/cc form to it produces:

A notable difference here is that we don’t use a ret continuation. Instead, another feature of the control form is that the value returns to a specific point back in the current computation context that is marked with a prompt. (Note that the control and prompt bindings are not included in the default racket language, we need to get them from a library: (require racket/control).) The fully translated code simply uses this prompt in place of the outer capture of the ret continuation:

We also need to translate the (lambda () (let/cc r (set! ret r) (k))) expression — but there is no ret to modify. Instead, we get the same effect by another use of prompt which is essentially modifying the implicitly used return continuation:

This looks like the previous version, but there’s an obvious advantage: since there is no ret binding that we need to maintain, we can pull out the yield definition to a more convenient place:

Note that this is an important change, since the producer machinery can now be abstracted into a make-producer function, as we’ve done before:

This is, again, a common pattern in such looping constructs — where the continuation of the loop keeps modifying the prompt as we do in the thunk assigned to cont. There are two other operators that are similar to control and prompt, which re-instate the point to return to automatically. Confusingly, they have completely different name: shift and reset. In the case of our code, we simply do the straightforward translation, and drop the extra wrapping step inside the value assigned to cont since that is done automatically. The resulting definition becomes even shorter now:

(Question: which set of forms is the more powerful one?)

It even looks like this code works reasonably well when the producer is exhausted:

But the problem is still there, except a but more subtle. We can see it if we add a side-effect:

and now we get:

This can be solved in the same way as we’ve discussed earlier — for example, grab the result value of the producer (which means that we get the value only after it’s exhausted), then repeat returning that value. A particularly easy way to do this is to set cont to a thunk that returns the value — since the resulting generator function simply invokes it, we get the desired behavior of returning the last value on further calls:

and now we get the improved behavior:

Tuesday, December 3rd Continuation Conclusions

Continuations are often viewed as a feature that is too complicated to understand and/or are hard to implement. As a result, very few languages provide general first-class continuations. Yet, they are an extremely useful tool since they enable implementing new kinds of control operators as user-written libraries. The “user” part is important here: if you want to implement producers (or a convenient web-read, or an ambiguous operator, or any number of other uses) in a language that doesn’t have continuations your options are very limited. You can ask for the new feature and wait for the language implementors to provide it, or you can CPS the relevant code (and the latter option is possible only if you have complete control over the whole code source to transform). With continuations, as we have seen, it is not only possible to build such libraries, the resulting functionality is as if the language has the desired feature already built-in. For example, Racket comes with a generator library that is very similar to Python generators — but in contrast to Python, it is implemented completely in user code. (In fact, the implementation is very close to the delimited continuations version that we’ve seen last.)

Obviously, in cases where you don’t have continuations and you need them (or rather when you need some functionality that is implementable via continuations), you will likely resort to the CPS approach, in some limited version. For example, the Racket documentation search page allows input to be typed while the search is happening.

This is a feature that by itself is not available in JavaScript — it is as if there are two threads running (one for the search and one to handle input), where JS is single-threaded on principle. This was implemented by making the search code side-effect free, then CPS-ing the code, then mimic threads by running the search for a bit, then storing its (manually built) continuation, handling possible new input, then resuming the search via this continuation. An approach that solves a similar problem using a very different approach is node.js — a JavaScript-based server where all IO is achieved via functions that receive callback functions, resulting in a style of code that is essentially writing CPSed code. For example, it is similar in principle to write code like:

or a concrete node.js example — to swap two files, you could write:

and if you want to follow the convention of providing a “convenient” synchronous version, you would also add:

As we have seen in the web server application example, this style of programming tends to be “infectious”, where a function that deals with these callback-based functions will itself consume a callback —

You should be able to see now what is happening here, without even mentioning the word “continuation” in the docs… See also this Node vs Apache video and read this extended JS rant. Quote:

No one ever for a second thought that a programmer would write actual code like that. And then Node came along and all of the sudden here we are pretending to be compiler back-ends. Where did we go wrong?

(Actually, JavaScript has gotten better with promises, and then even better with async/await — but these are new, so it is actually common to find libraries that provide two such versions and a third promise-based one, and even a fourth one, using async. See for example the replace-in-file package on NPM.)

Finally, as mentioned a few times, there has been extensive research into many aspects of continuations. Different CPS approaches, different implementation strategies, a zoo-full of control operators, assigning types to continuation-related functions and connections between continuations and types, even connections between CPS and certain proof techniques. Some research is still going on, though not as hot as it was — but more importantly, many modern languages “discover” the utility of having continuations, sometimes in some more limited forms (eg, Python and JavaScript generators), and sometimes in full form (eg, Ruby’s callcc).

Tuesday, December 3rd Sidenote: JavaScript: Continuations, Promises, and Async/Await

JavaScript went through a long road that is directly related to continuations. The motivating language feature that started all of this is the early decision to make the language single-threaded. This feature does make code easier to handle WRT side effects of all kinds. For example, consider:

In JS, this code is guaranteed to show exactly the two output lines, with the expected values of 1 and 2. This is not guaranteed in any language that has threads (or any form of parallelism), since other threads might kick in at any point, possibly producing more output or mutating the value of x.

For several years JS was used only as a lightweight scripting language for simple tasks on web pages, and life was simple with the single threaded-ness nature of the language. Note that the environment around the language (= the browser) was very early quite thread-heavy, with different threads used for text layout, rendering, loading, user interactions, etc. But then came Node and JS was used for more traditional uses, with a web server being one of the first use cases. This meand that there was a a need for some way to handle “multiple threads” — it’s impractical to wait for any server-side interaction to be done before serving the next.

This was addressed by designing Node as a program that makes heavy use of callback functions and IO events. In cases where you really want a linear sequence of operations you can use *Sync functions, such as the one we’ve seen above:

But you can also use the non-Sync version which allows to switch between different active jobs — translating the above to (ignoring errors for simplicity):

This could be better in some cases — for example, if the filesystem is a remote-mounted directory, each rename will likely take noticeable time. Using callbacks means that after firing each rename request the whole process is free to do any other work, and later resume the execution with the next operation when the last request is done.

But there is a major problem with this code: we can be more efficient since the main process can do anything while waiting for the three operations, but if you want to run this function to swap two files then you likely have some code that needs to run after executing it — code that expects the name swapping to have happened:

The problem is that the function call returns immediately, and just registers the first rename operation. There might be a small chance (very unlikely) for the first rename to have happened, but even if it happens fast, the callback is guaranteed to not be called (again, since the core language is single-threaded), making this code completely broken.

This means that for such uses we must switch to using callbacks in the main function too:

And the uses too:

This is the familiar manual CPS-ing of code that was for many years something that all JS programmes needed to use.

As we’ve seen, the problem here is that writing such code is difficult, and even more so working with such code when it needs to be extended, debugged, etc. Eventually, the JS world settled on using “promises” to simplify all of this: the idea is that instead of building “towers of callbacks”, we abstract them as promises that are threaded using a .then property for the chain of operations.

Note that here too, the changes in the function body are reflated in how the whole function is used: the body uses promises, and therefore the whole function returns a promise.

This is a little better in that we don’t suffer from nested code towers, but it’s still not too convenient. The last step in this evolution was the introduction of async/await — language features that declare a promise-ified function which can wait for promises to be executed in a more-or-less plain looking code:

(The implementation of this feature is interesting: it relies on using generator functions, and using the fact that you can send values into a generator when you resume it — and all of this is, roughly speaking, used to implement a threading system.)

This looks almost like normal code, but note that while doing so we lose the simplification of a single-threaded language. Uses of await are equivalent to function calls that can switch the context to other code which might introduce unexpected side-effects. This is not really getting back to a proper multi-threaded language like Racket: it’s rather a language with “cooperative threads”. You still have the advantage that if you don’t use await, then code executes sequentially and uninterrupted.

Whether it is useful or not to complicate these cases just to be able to write non-async code is a question…