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.

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!)

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.

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.

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

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:

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:

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?

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.