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.