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.
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!)
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 mk
s as x
s:
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.
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.
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
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 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
withQuote
, 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 (thex
s) 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: