-[[!toc]]
+[[!toc levels=3]]
-#Recursion: fixed points in the lambda calculus##
+#Recursion: fixed points in the Lambda Calculus#
Sometimes when you type in a web search, Google will suggest
alternatives. For instance, if you type in "Lingusitics", it will ask
you "Did you mean Linguistics?". But the engineers at Google have
added some playfulness to the system. For instance, if you search for
"anagram", Google asks you "Did you mean: nag a ram?" And if you
-search for "recursion", Google asks: "Did you mean: recursion?"
+[search for "recursion"](http://www.google.com/search?q=recursion), Google asks: "Did you mean: recursion?"
##What is the "rec" part of "letrec" doing?##
-How could we compute the length of a list? Without worrying yet about what lambda-calculus implementation we're using for the list, the basic idea would be to define this recursively:
+How could we compute the length of a list? Without worrying yet about what Lambda Calculus encoding we're using for the list, the basic idea is to define this recursively:
-> the empty list has length 0
+> the empty list has length 0
-> any non-empty list has length 1 + (the length of its tail)
+> any non-empty list has length 1 + (the length of its tail)
In OCaml, you'd define that like this:
- let rec length = fun lst ->
- if lst == [] then 0 else 1 + length (tail lst)
- in ... (* here you go on to use the function "length" *)
+ let rec length = fun xs ->
+ if xs = [] then 0
+ else 1 + length (List.tl xs)
+ in ... (* here you go on to use the function "length" *)
In Scheme you'd define it like this:
- (letrec [(length
- (lambda (lst) (if (null? lst) 0 [+ 1 (length (cdr lst))] )) )]
- ... ; here you go on to use the function "length"
- )
+ (letrec [(length (lambda (xs)
+ (if (null? xs) 0
+ (+ 1 (length (cdr xs))) )))]
+ ... ; here you go on to use the function "length"
+ )
Some comments on this:
-1. `null?` is Scheme's way of saying `isempty`. That is, `(null? lst)` returns true (which Scheme writes as `#t`) iff `lst` is the empty list (which Scheme writes as `'()` or `(list)`).
+1. `null?` is Scheme's way of saying `empty?`. That is, `(null? xs)` returns true (which Scheme writes as `#t`) iff `xs` is the empty list (which Scheme writes as `'()` or `(list)`).
-2. `cdr` is function that gets the tail of a Scheme list. (By definition, it's the function for getting the second member of an ordered pair. It just turns out to return the tail of a list because of the particular way Scheme implements lists.)
+2. `cdr` is function that gets the tail of a Scheme list. (By definition, it's the function for getting the second member of a [[dotted pair|week3_unit#imp]]. As we discussed in notes for last week, it just turns out to return the tail of a list because of the particular way Scheme implements lists.) `List.tl` is the function that gets the tail of an OCaml list.
-3. I use `length` instead of the convention we've been following so far of hyphenated names, as in `make-list`, because we're discussing OCaml code here, too, and OCaml doesn't permit the hyphenated variable names. OCaml requires variables to always start with a lower-case letter (or `_`), and then continue with only letters, numbers, `_` or `'`. Most other programming languages are similar. Scheme is very relaxed, and permits you to use `-`, `?`, `/`, and all sorts of other crazy characters in your variable names.
-
-4. I alternate between `[ ]`s and `( )`s in the Scheme code just to make it more readable. These have no syntactic difference.
+3. We alternate between `[ ]`s and `( )`s in the Scheme code just to make it more readable. These have no syntactic difference.
The main question for us to dwell on here is: What are the `let rec` in the OCaml code and the `letrec` in the Scheme code?
-Answer: These work like the `let` expressions we've already seen, except that they let you use the variable `length` *inside* the body of the function being bound to it---with the understanding that it will there refer to the same function that you're then in the process of binding to `length`. So our recursively-defined function works the way we'd expect it to. In OCaml:
+Answer: These work a lot like `let` expressions, except that they let you use the variable `length` *inside* the body of the function being bound to it --- with the understanding that it will there be bound to *the same function* that you're *then* in the process of binding `length` to. So our recursively-defined function works the way we'd expect it to. Here is OCaml:
- let rec length = fun lst ->
- if lst == [] then 0 else 1 + length (tail lst)
- in length [20; 30]
- (* this evaluates to 2 *)
+ let rec length = fun xs ->
+ if xs = [] then 0
+ else 1 + length (List.tl xs)
+ in length [20; 30]
+ (* this evaluates to 2 *)
-In Scheme:
+Here is Scheme:
- (letrec [(length
- (lambda (lst) (if (null? lst) 0 [+ 1 (length (cdr lst))] )) )]
- (length (list 20 30)))
- ; this evaluates to 2
+ (letrec [(length (lambda (xs)
+ (if (null? xs) 0
+ (+ 1 (length (cdr xs))) )))]
+ (length (list 20 30)))
+ ; this evaluates to 2
If you instead use an ordinary `let` (or `let*`), here's what would happen, in OCaml:
- let length = fun lst ->
- if lst == [] then 0 else 1 + length (tail lst)
- in length [20; 30]
- (* fails with error "Unbound value length" *)
+ let length = fun xs ->
+ if xs = [] then 0
+ else 1 + length (List.tl xs)
+ in length [20; 30]
+ (* fails with error "Unbound value length" *)
Here's Scheme:
- (let* [(length
- (lambda (lst) (if (null? lst) 0 [+ 1 (length (cdr lst))] )) )]
- (length (list 20 30)))
- ; fails with error "reference to undefined identifier: length"
+ (let* [(length (lambda (xs)
+ (if (null? xs) 0
+ (+ 1 (length (cdr xs))) )))]
+ (length (list 20 30)))
+ ; fails with error "reference to undefined identifier: length"
Why? Because we said that constructions of this form:
- let length = A
- in B
+ let
+ length match/= A
+ in B
really were just another way of saying:
- (\length. B) A
+ (\length. B) A
and so the occurrences of `length` in A *aren't bound by the `\length` that wraps B*. Those occurrences are free.
-We can verify this by wrapping the whole expression in a more outer binding of `length` to some other function, say the constant function from any list to the integer 99:
+We can verify this by wrapping the whole expression in a more outer binding of `length` to some other function, say the constant function from any list to the integer `99`:
- let length = fun lst -> 99
- in let length = fun lst ->
- if lst == [] then 0 else 1 + length (tail lst)
- in length [20; 30]
- (* evaluates to 1 + 99 *)
+ let length = fun xs -> 99
+ in let length = fun xs ->
+ if xs = [] then 0
+ else 1 + length (List.tl xs)
+ in length [20; 30]
+ (* evaluates to 1 + 99 *)
-Here the use of `length` in `1 + length (tail lst)` can clearly be seen to be bound by the outermost `let`.
+Here the use of `length` in `1 + length (List.tl xs)` can clearly be seen to be bound by the outermost `let`.
-And indeed, if you tried to define `length` in the lambda calculus, how would you do it?
+And indeed, if you tried to define `length` in the Lambda Calculus, how would you do it?
- \lst. (isempty lst) zero (add one (length (extract-tail lst)))
+ \xs. (empty? xs) 0 (succ (length (tail xs)))
-We've defined all of `isempty`, `zero`, `add`, `one`, and `extract-tail` in earlier discussion. But what about `length`? That's not yet defined! In fact, that's the very formula we're trying here to specify.
+We've defined all of `empty?`, `0`, `succ`, and `tail` in earlier discussion. But what about `length`? That's not yet defined! In fact, that's the very formula we're trying here to specify.
What we really want to do is something like this:
- \lst. (isempty lst) zero (add one (... (extract-tail lst)))
+ \xs. (empty? xs) 0 (succ ((...) (tail xs)))
where this very same formula occupies the `...` position:
- \lst. (isempty lst) zero (add one (
- \lst. (isempty lst) zero (add one (... (extract-tail lst)))
- (extract-tail lst)))
+ \xs. (empty? xs) 0 (succ (
+ \xs. (empty? xs) 0 (succ ((...) (tail xs)))
+ ) (tail xs)))
but as you can see, we'd still have to plug the formula back into itself again, and again, and again... No dice.
So how could we do it? And how do OCaml and Scheme manage to do it, with their `let rec` and `letrec`?
-1. OCaml and Scheme do it using a trick. Well, not a trick. Actually an impressive, conceptually deep technique, which we haven't yet developed. Since we want to build up all the techniques we're using by hand, then, we shouldn't permit ourselves to rely on `let rec` or `letrec` until we thoroughly understand what's going on under the hood.
-
-2. If you tried this in Scheme:
+1. OCaml and Scheme do it using a trick. Well, not a trick. Actually an impressive, conceptually deep technique, which we haven't yet developed. Since we want to build up all the techniques we're using by hand, then, we shouldn't permit ourselves to rely on `let rec` or `letrec` until we thoroughly understand what's going on under the hood.
- (define length
- (lambda (lst) (if (null? lst) 0 [+ 1 (length (cdr lst))] )) )
+2. If you tried this in Scheme:
- (length (list 20 30))
+ (define length (lambda (xs)
+ (if (null? xs) 0
+ (+ 1 (length (cdr xs))) )))
+
+ (length (list 20 30))
- You'd find that it works! This is because `define` in Scheme is really shorthand for `letrec`, not for plain `let` or `let*`. So we should regard this as cheating, too.
+ You'd find that it works! This is because `define` in Scheme is really shorthand for `letrec`, not for plain `let` or `let*`. So we should regard this as cheating, too.
-3. In fact, it *is* possible to define the `length` function in the lambda calculus despite these obstacles. This depends on using the "version 3" implementation of lists, and exploiting its internal structure: that it takes a function and a base value and returns the result of folding that function over the list, with that base value. So we could use this as a definition of `length`:
+3. In fact, it *is* possible to define the `length` function in the Lambda Calculus despite these obstacles, without yet knowing how to implement `letrec` in general. We've already seen how to do it, using our right-fold (or left-fold) encoding for lists, and exploiting their internal structure. Those encodings take a function and a seed value and returns the result of folding that function over the list, with that seed value. So we could use this as a definition of `length`:
- \lst. lst (\x sofar. successor sofar) zero
+ \xs. xs (\x sofar. succ sofar) 0
- What's happening here? We start with the value zero, then we apply the function `\x sofar. successor sofar` to the two arguments <code>x<sub>n</sub></code> and `zero`, where <code>x<sub>n</sub></code> is the last element of the list. This gives us `successor zero`, or `one`. That's the value we've accumuluted "so far." Then we go apply the function `\x sofar. successor sofar` to the two arguments <code>x<sub>n-1</sub></code> and the value `one` that we've accumulated "so far." This gives us `two`. We continue until we get to the start of the list. The value we've then built up "so far" will be the length of the list.
+ What's happening here? We start with the value `0`, then we apply the function `\x sofar. succ sofar` to the two arguments <code>x<sub>n</sub></code> and `0`, where <code>x<sub>n</sub></code> is the last element of the list. This gives us `succ 0`, or `1`. That's the value we've accumulated "so far." Then we go apply the function `\x sofar. succ sofar` to the two arguments <code>x<sub>n-1</sub></code> and the value `1` that we've accumulated "so far." This gives us `2`. We continue until we get to the start of the list. The value we've then built up "so far" will be the length of the list.
-We can use similar techniques to define many recursive operations on
-lists and numbers. The reason we can do this is that our "version 3,"
-fold-based implementation of lists, and Church's implementations of
+ We can use similar techniques to define many recursive operations on
+lists and numbers. The reason we can do this is that our
+fold-based encoding of lists, and Church's encodings of
numbers, have a internal structure that *mirrors* the common recursive
operations we'd use lists and numbers for. In a sense, the recursive
structure of the `length` operation is built into the data
structure we are using to represent the list. The non-recursive
-version of length exploits this embedding of the recursion into
+definition of length, above, exploits this embedding of the recursion into
the data type.
-This is one of the themes of the course: using data structures to
-encode the state of some recursive operation. See discussions of the
+ This illustrates what will be one of the recurring themes of the course: using data structures to
+encode the state of some recursive operation. See our discussions later this semester of the
[[zipper]] technique, and [[defunctionalization]].
-As we said before, it does take some ingenuity to define functions like `extract-tail` or `predecessor` for these implementations. However it can be done. (And it's not *that* difficult.) Given those functions, we can go on to define other functions like numeric equality, subtraction, and so on, just by exploiting the structure already present in our implementations of lists and numbers.
+As we've seen, it does take some ingenuity to define functions like `tail` or `pred` for our right-fold encoding of lists. However it can be done. (And it's not *that* difficult.) Given those functions, we can go on to define other functions like numeric equality, subtraction, and so on, just by exploiting the structure already present in our implementation of lists and numbers.
-With sufficient ingenuity, a great many functions can be defined in the same way. For example, the factorial function is straightforward. The function which returns the nth term in the Fibonacci series is a bit more difficult, but also achievable.
+With sufficient ingenuity, a great many functions can be defined in the same way. For example, the factorial function is straightforward. The function which returns the *n*th term in the Fibonacci series is a bit more difficult, but also achievable.
##Some functions require full-fledged recursive definitions##
However, some computable functions are just not definable in this
way. We can't, for example, define a function that tells us, for
-whatever function `f` we supply it, what is the smallest integer `x`
-where `f x` is `true`. (You may be thinking: but that
-smallest-integer function is not a proper algorithm, since it is not
-guaranteed to halt in any finite amount of time for every argument.
-This is the famous [[!wikipedia Halting problem]]. But the fact that
-an implementation may not terminate doesn't mean that such a function
-isn't well-defined. The point of interest here is that its definition
-requires recursion in the function definition.)
+whatever function `f` we supply it, what is the smallest natural number `x`
+where `f x` is `true` (even if `f` itself is a function we do already know how to define).
Neither do the resources we've so far developed suffice to define the
-[[!wikipedia Ackermann function]]:
+[[!wikipedia Ackermann function]]. In OCaml:
- A(m,n) =
- | when m == 0 -> n + 1
- | else when n == 0 -> A(m-1,1)
- | else -> A(m-1, A(m,n-1))
+ let rec A = fun (m,n) ->
+ if m = 0 then n + 1
+ else if n = 0 then A(m-1,1)
+ else A(m-1, A(m,n-1));;
- A(0,y) = y+1
- A(1,y) = 2+(y+3) - 3
- A(2,y) = 2(y+3) - 3
- A(3,y) = 2^(y+3) - 3
- A(4,y) = 2^(2^(2^...2)) [where there are y+3 2s] - 3
- ...
+ A(0,y) = y+1
+ A(1,y) = 2+(y+3) - 3
+ A(2,y) = 2(y+3) - 3
+ A(3,y) = 2^(y+3) - 3
+ A(4,y) = 2^(2^(2^...2)) (* where there are y+3 2s *) - 3
+ ...
Many simpler functions always *could* be defined using the resources we've so far developed, although those definitions won't always be very efficient or easily intelligible.
-But functions like the Ackermann function require us to develop a more general technique for doing recursion---and having developed it, it will often be easier to use it even in the cases where, in principle, we didn't have to.
+But functions like the Ackermann function require us to develop a more general technique for doing recursion --- and having developed it, it will often be easier to use it even in the cases where, in principle, we didn't have to.
##Using fixed-point combinators to define recursive functions##
###Fixed points###
-In general, we call a **fixed point** of a function f any value *x*
-such that f <em>x</em> is equivalent to *x*. For example,
-consider the squaring function `sqare` that maps natural numbers to their squares.
+In mathematics, a **fixed point** of a function `f` is any value `ξ`
+such that `f ξ` is equivalent to `ξ`. For example,
+consider the squaring function `square` that maps natural numbers to their squares.
`square 2 = 4`, so `2` is not a fixed point. But `square 1 = 1`, so `1` is a
-fixed point of the squaring function.
+fixed point of the squaring function. (Can you think of another?)
There are many beautiful theorems guaranteeing the existence of a
fixed point for various classes of interesting functions. For
-instance, imainge that you are looking at a map of Manhattan, and you
-are standing somewhere in Manhattan. The the [[!wikipedia Brouwer
-fixed point]] guarantees that there is a spot on the map that is
-directly above the corresponding spot in Manhattan. It's the spot
-where the blue you-are-here dot should be.
+instance, imagine that you are looking at a map of Manhattan, and you
+are standing somewhere in Manhattan. Then the [[!wikipedia Brouwer
+fixed-point theorem]] guarantees that there is a spot on the map that is
+directly above the corresponding spot in Manhattan. It's the spot on the map
+where the blue you-are-here dot should go.
-Whether a function has a fixed point depends on the set of arguments
+Whether a function has a fixed point depends on the domain of arguments
it is defined for. For instance, consider the successor function `succ`
that maps each natural number to its successor. If we limit our
attention to the natural numbers, then this function has no fixed
point. (See the discussion below concerning a way of understanding
-the successor function on which it does have a fixed point.)
+the successor function on which it *does* have a fixed point.)
+
+In the Lambda Calculus, we say a fixed point of a term `f` is any *term* `ξ` such that:
+
+ ξ <~~> f ξ
-In the lambda calculus, we say a fixed point of an expression `f` is any formula `X` such that:
+This is a bit different than the general mathematical definition, in that here we're saying it is *terms* that are fixed points, not *values*. We like to think that some lambda terms represent values, such as our term `\f z. z` representing the numerical value zero (and also the truth-value false, and also the empty list... on the other hand, we never did explicitly agree that those three values are all the same thing, did we?). But some terms in the Lambda Calculus don't even have a normal form. We don't want to count them as *values*. Yet the way we're proposing to use the notion of a fixed point here, they too are allowed to be fixed points, and to have fixed points of their own.
- X <~~> f X
+Note that `M <~~> N` doesn't entail that `M` and `N` have a normal form (though if they do, they will have the same normal form). It just requires that there be some term that they both reduce to. It may be that *that* term itself never stops being reducible.
You should be able to immediately provide a fixed point of the
-identity combinator I. In fact, you should be able to provide a whole
-bunch of distinct fixed points.
+identity combinator `I`. In fact, you should be able to provide a
+whole bunch of distinct fixed points.
With a little thought, you should be able to provide a fixed point of
-the false combinator, KI. Here's how to find it: recall that KI
-throws away its first argument, and always returns I. Therefore, if
-we give it I as an argument, it will throw away the argument, and
-return I. So KII ~~> I, which is all it takes for I to qualify as a
-fixed point of KI.
+the false combinator, `KI`. Here's how to find it: recall that `KI`
+throws away its first argument, and always returns `I`. Therefore, if
+we give it `I` as an argument, it will throw away the argument, and
+return `I`. So `KII` ~~> `I`, which is all it takes for `I` to qualify as a
+fixed point of `KI`.
-What about K? Does it have a fixed point? You might not think so,
+What about `K`? Does it have a fixed point? You might not think so,
after trying on paper for a while.
-However, it's a theorem of the lambda calculus that every formula has
-a fixed point. In fact, it will have infinitely many, non-equivalent
+However, it's a theorem of the Lambda Calculus that *every* lambda term has
+a fixed point. Even bare variables like `x`! In fact, they will have infinitely many, non-equivalent
fixed points. And we don't just know that they exist: for any given
formula, we can explicit define many of them.
-Yes, even the formula that you're using the define the successor
-function will have a fixed point. Isn't that weird? Think about how it
-might be true. We'll return to this point below.
+(As we mentioned, even the formula that you're using the define
+the successor function will have a fixed point. Isn't that weird? There's some `ξ` such that it is equivalent to `succ ξ`?
+Think about how it might be true. We'll return to this point below.)
-###How fixed points help definie recursive functions###
-Recall our initial, abortive attempt above to define the `length` function in the lambda calculus. We said "What we really want to do is something like this:
+###How fixed points help define recursive functions###
- \list. if empty list then zero else add one (... (tail lst))
+Recall our initial, abortive attempt above to define the `length` function in the Lambda Calculus. We said:
-where this very same formula occupies the `...` position."
+> What we really want to do is something like this:
-Imagine replacing the `...` with some function that computes the
-length function. Call that function `length`. Then we have
+> \xs. (empty? xs) 0 (succ ((...) (tail xs)))
- \list. if empty list then zero else add one (length (tail lst))
+> where this very same formula occupies the `...` position...
+
+Imagine replacing the `...` with some expression `LENGTH` that computes the
+length function. Then we have
+
+ \xs. (empty? xs) 0 (succ (LENGTH (tail xs)))
At this point, we have a definition of the length function, though
it's not complete, since we don't know what value to use for the
-symbol `length`. Technically, it has the status of an unbound
+symbol `LENGTH`. Technically, it has the status of an unbound
variable.
-Imagine now binding the mysterious variable:
+Imagine now binding the mysterious variable, and calling the resulting
+term `h`:
- h := \length \list . if empty list then zero else add one (length (tail list))
+ h ≡ \length \xs. (empty? xs) 0 (succ (length (tail xs)))
Now we have no unbound variables, and we have complete non-recursive
-definitions of each of the other symbols.
+definitions of each of the other symbols (`empty?`, `0`, `succ`, and `tail`).
-Let's call this function `h`. Then `h` takes an argument, and returns
-a function that accurately computes the length of a list---as long as
-the argument we supply is already the length function we are trying to
-define. (Dehydrated water: to reconstitute, just add water!)
+So `h` takes a `length` argument, and returns a function that accurately
+computes the length of a list --- as long as the argument we supply is
+already the length function we are trying to define. (Dehydrated
+water: to reconstitute, just add water!)
-But this is just another way of saying that we are looking for a fixed point.
-Assume that `h` has a fixed point, call it `LEN`. To say that `LEN`
-is a fixed point means that
+Here is where the discussion of fixed points becomes relevant. Saying
+that `h` is looking for an argument (call it `LENGTH`) that has the same
+behavior as the result of applying `h` to `LENGTH` is just another way of
+saying that we are looking for a fixed point for `h`:
- h LEN <~~> LEN
+ h LENGTH <~~> LENGTH
-But this means that
+Replacing `h` with its definition, we have
- (\list . if empty list then zero else add one (LEN (tail list))) <~~> LEN
+ (\xs. (empty? xs) 0 (succ (LENGTH (tail xs)))) <~~> LENGTH
+
+If we can find a value for `LENGTH` that satisfies this constraint, we'll
+have a function we can use to compute the length of an arbitrary list.
+All we have to do is find a fixed point for `h`.
-So at this point, we are going to search for fixed point.
The strategy we will present will turn out to be a general way of
finding a fixed point for any lambda term.
+
##Deriving Y, a fixed point combinator##
How shall we begin? Well, we need to find an argument to supply to
length of a list. Let's try applying `h` to itself. It won't quite
work, but examining the way in which it fails will lead to a solution.
- h h <~~> \list . if empty list then zero else 1 + h (tail list)
+ h h <~~> \xs. (empty? xs) 0 (succ (h (tail xs)))
-There's a problem. The diagnosis is that in the subexpression `h
-(tail list)`, we've applied `h` to a list, but `h` expects as its
-first argument the length function.
+The problem is that in the subexpression `h (tail list)`, we've
+applied `h` to a list, but `h` expects as its first argument the
+length function.
-So let's adjust h, calling the adjusted function H:
+So let's adjust `h`, calling the adjusted function `H`. (We'll use `u` as the variable
+that expects to be bound to the as-yet-*unknown* argument, rather than `length`. This will make it easier
+to discuss generalizations of this strategy.)
- H = \h \list . if empty list then zero else one plus ((h h) (tail list))
+ h ≡ \length \xs. (empty? xs) 0 (succ (length (tail xs)))
+ H ≡ \u \xs. (empty? xs) 0 (succ ((u u) (tail xs)))
-This is the key creative step. Since `h` is expecting a
-length-computing function as its first argument, the adjustment
-tries supplying the closest candidate avaiable, namely, `h` itself.
+This is the key creative step. Instead of applying `u` to a list, as happened
+when we self-applied `h`, `H` applies its argument `u` first to *itself*: `u u`.
+After `u` gets an argument, the *result* is ready to apply to a list, so we've solved the problem noted above with `h (tail list)`.
+We're not done yet, of course; we don't yet know what argument `u` to give
+to `H` that will behave in the desired way.
-We now reason about `H`. What exactly is H expecting as its first
-argument? Based on the excerpt `(h h) (tail l)`, it appears that `H`'s
-argument, `h`, should be a function that is ready to take itself as an
-argument, and that returns a function that takes a list as an
+So let's reason about `H`. What exactly is `H` expecting as its first
+argument? Based on the excerpt `(u u) (tail xs)`, it appears that
+`H`'s argument, `u`, should be a function that is ready to take itself
+as an argument, and that returns a function that takes a list as an
argument. `H` itself fits the bill:
- H H <~~> (\h \list . if empty list then zero else 1 + ((h h) (tail list))) H
- <~~> \list . if empty list then zero else 1 + ((H H) (tail list))
- == \list . if empty list then zero else 1 + ((\list . if empty list then zero else 1 + ((H H) (tail list))) (tail list))
- <~~> \list . if empty list then zero
- else 1 + (if empty (tail list) then zero else 1 + ((H H) (tail (tail list))))
+ H H <~~> (\u \xs. (empty? xs) 0 (succ ((u u) (tail xs)))) H
+ <~~> \xs. (empty? xs) 0 (succ ((H H) (tail xs)))
+ <~~> \xs. (empty? xs) 0 (succ ((
+ \xs. (empty? xs) 0 (succ ((H H) (tail xs)))
+ ) (tail xs)))
+ <~~> \xs. (empty? xs) 0 (succ (
+ (empty? (tail xs)) 0 (succ ((H H) (tail (tail xs)))) ))
We're in business!
How does the recursion work?
We've defined `H` in such a way that `H H` turns out to be the length function.
+That is, `H H` is the `LENGTH` we were looking for.
In order to evaluate `H H`, we substitute `H` into the body of the
-lambda term. Inside the lambda term, once the substitution has
+lambda term `H`. Inside that lambda term, once the substitution has
occurred, we are once again faced with evaluating `H H`. And so on.
-We've got the infinite regress we desired, defined in terms of a
+We've got the (potentially) infinite regress we desired, defined in terms of a
finite lambda term with no undefined symbols.
Since `H H` turns out to be the length function, we can think of `H`
-by itself as half of the length function (which is why we called it
-`H`, of course). Given the implementation of addition as function
-application for Church numerals, this (H H) is quite literally H + H.
-Can you think up a recursion strategy that involves "dividing" the
-recursive function into equal thirds `T`, such that the length
-function <~~> T T T?
+by itself as *half* of the length function (which is why we called it
+`H`, of course). (Thought exercise: Can you think up a recursion strategy that involves
+"dividing" the recursive function into equal thirds `T`, such that the
+length function <~~> `T T T`?)
We've starting with a particular recursive definition, and arrived at
a fixed point for that definition.
What's the general recipe?
-1. Start with any recursive definition `h` that takes itself as an arg: `h := \fn ... fn ...`
-2. Next, define `H := \f . h (f f)`
-3. Then compute `H H = ((\f . h (f f)) (\f . h (f f)))`
-4. That's the fixed point, the recursive function we're trying to define
+1. Start with a formula `h` that takes the recursive function you're seeking as an argument: `h ≡ \length. ... length ...`
+2. Next, define `H ≡ \u. h (u u)`
+3. Then compute `H H ≡ ((\u. h (u u)) (\u. h (u u)))`
+4. That's the fixed point of `h`, the recursive function you're seeking.
-So here is a general method for taking an arbitrary h-style recursive function
-and returning a fixed point for that function:
+Expressed in terms of a single formula, here is this method for taking an arbitrary `h`-style term and returning
+the recursive function that term expects as an argument, which as we've seen will be the `h`-term's fixed point:
- Y := \h. ((\f.h(ff))(\f.h(ff)))
+ Y ≡ \h. (\u. h (u u)) (\u. h (u u))
-Test:
+Let's test that `Y h` will indeed be `h`'s fixed point:
- Yh == ((\f.h(ff))(\f.h(ff)))
- <~~> h((\f.h(ff))(\f.h(ff)))
- == h(Yh)
+ Y h ≡ (\h. (\u. h (u u)) (\u. h (u u))) h
+ ~~> (\u. h (u u)) (\u. h (u u))
+ ~~> h ((\u. h (u u)) (\u. h (u u)))
-That is, Yh is a fixed point for h.
+But the argument of `h` in the last line is just the same as the second line, which <\~~> `Y h`. So the last line <\~~> `h (Y h)`. In other words, `Y h <~~> h (Y h)`. So by definition, `Y h` is a fixed point for `h`.
Works!
+###Coming at it another way###
+
+TODO
+
+
+##A fixed point for K?##
+
+Let's do one more example to illustrate. We'll do `K`, since we
+wondered above whether it had a fixed point.
+
+Before we begin, we can reason a bit about what the fixed point must
+be like. We're looking for a fixed point for `K`, i.e., `\x y. x`. The term `K`
+ignores its second argument. That means that no matter what we give
+`K` as its first argument, the result will ignore the next argument
+(that is, `KX` ignores its first argument, no matter what `X` is). So
+if `KX <~~> X`, `X` had also better ignore its first argument. But we
+also have `KX ≡ (\x y. x) X ~~> \y. X`. This means that if `X` ignores
+its first argument, then `\y.X` will ignore its first two arguments.
+So once again, if `KX <~~> X`, `X` also had better ignore (at least) its
+first two arguments. Repeating this reasoning, we realize that `X`
+must be a function that ignores as many arguments as you give it.
+
+Our expectation, then, is that our recipe for finding fixed points
+will build us a term that somehow manages to ignore arbitrarily many arguments.
+
+ h ≡ \xy.x
+ H ≡ \u.h(uu) ≡ \u.(\xy.x)(uu) ~~> \uy.uu
+ H H ≡ (\uy.uu)(\uy.uu) ~~> \y.(\uy.uu)(\uy.uu)
+
+Let's check that it is in fact a fixed point for `K`:
+
+ K(H H) ≡ (\xy.x)((\uy.uu)(\uy.uu))
+ ~~> \y.(\uy.uu)(\uy.uu)
+
+Yep, `H H` and `K(H H)` both reduce to the same term.
+
+To see what this fixed point does, let's reduce it a bit more:
+
+ H H ≡ (\uy.uu)(\uy.uu)
+ ~~> \y.(\uy.uu)(\uy.uu)
+ ~~> \yy.(\uy.uu)(\uy.uu)
+ ~~> \yyy.(\uy.uu)(\uy.uu)
+
+Sure enough, this fixed point ignores an endless, arbitrarily-long series of
+arguments. It's a write-only memory, a black hole.
+
+Now that we have one fixed point, we can find others, for instance,
+
+ (\uy.[uu]u) (\uy.uuu)
+ ~~> \y. [(\uy.uuu) (\uy.uuu)] (\uy.uuu)
+ ~~> \y. [\y. (\uy.uuu) (\uy.uuu) (\uy.uuu)] (\uy.uuu)
+ ~~> \yyy. (\uy.uuu) (\uy.uuu) (\uy.uuu) (\uy.uuu) (\uy.uuu)
+
+Continuing in this way, you can now find an infinite number of fixed
+points, all of which have the crucial property of ignoring an infinite
+series of arguments.
##What is a fixed point for the successor function?##
-Well, you might think, only some of the formulas that we might give to the `successor` as arguments would really represent numbers. If we said something like:
+As we've seen, the recipe just given for finding a fixed point worked
+great for our `h`, which we wrote as a definition for the length
+function. But the recipe doesn't make any assumptions about the
+internal structure of the term it works with. That means it can
+find a fixed point for literally any lambda term whatsoever.
+
+In particular, what could the fixed point for (our encoding of) the
+successor function possibly be like?
+
+Well, you might think, only some of the formulas that we might give to `succ` as arguments would really represent numbers. If we said something like:
+
+ succ pair
- successor make-pair
+who knows what we'd get back? Perhaps there's some non-number-representing formula such that when we feed it to `succ` as an argument, we get the same formula back.
-who knows what we'd get back? Perhaps there's some non-number-representing formula such that when we feed it to `successor` as an argument, we get the same formula back.
+Yes! That's exactly right. And which formula this is will depend on the particular way you've encoded the successor function.
-Yes! That's exactly right. And which formula this is will depend on the particular way you've implemented the successor function.
+One (by now obvious) upshot is that the recipes that enable us to name
+fixed points for any given formula `h` aren't *guaranteed* to give us
+*terminating, normalizing* fixed points. They might give us formulas `ξ` such that
+neither `ξ` nor `h ξ` have normal forms. (Indeed, what they give us
+for the `square` function isn't any of the Church numerals, but is
+rather an expression with no normal form.) However, if we take care we
+can ensure that we *do* get terminating fixed points. And this gives
+us a principled, fully general strategy for doing recursion. It lets
+us define even functions like the Ackermann function, which were until
+now out of our reach. It would also let us define list
+functions on [[the encodings we discussed last week|week3_lists#other-lists]], where it
+wasn't always clear how to force the computation to "keep going."
-Moreover, the recipes that enable us to name fixed points for any
-given formula aren't *guaranteed* to give us *terminating* fixed
-points. They might give us formulas X such that neither `X` nor `f X`
-have normal forms. (Indeed, what they give us for the square function
-isn't any of the Church numerals, but is rather an expression with no
-normal form.) However, if we take care we can ensure that we *do* get
-terminating fixed points. And this gives us a principled, fully
-general strategy for doing recursion. It lets us define even functions
-like the Ackermann function, which were until now out of our reach. It
-would also let us define arithmetic and list functions on the "version
-1" and "version 2" implementations, where it wasn't always clear how
-to force the computation to "keep going."
+###Varieties of fixed-point combinators###
OK, so how do we make use of this?
-Many fixed-point combinators have been discovered. (And some
+Many fixed-point combinators have been discovered. (And as we've seen, some
fixed-point combinators give us models for building infinitely many
more, non-equivalent fixed-point combinators.)
Two of the simplest:
-<pre><code>Θ′ ≡ (\u f. f (\n. u u f n)) (\u f. f (\n. u u f n))
-Y′ ≡ \f. (\u. f (\n. u u n)) (\u. f (\n. u u n))</code></pre>
+ Θ′ ≡ (\u h. h (\n. u u h n)) (\u h. h (\n. u u h n))
+ Y′ ≡ \h. (\u. h (\n. u u n)) (\u. h (\n. u u n))
-<code>Θ′</code> has the advantage that <code>f (Θ′ f)</code> really *reduces to* <code>Θ′ f</code>. Whereas <code>f (Y′ f)</code> is only *convertible with* <code>Y′ f</code>; that is, there's a common formula they both reduce to. For most purposes, though, either will do.
+Applying either of these to a term `h` gives a fixed point `ξ` for `h`, meaning that `h ξ` <~~> `ξ`. `Θ′` has the advantage that `h (Θ′ h)` really *reduces to* `Θ′ h`. Whereas `h (Y′ h)` is only *convertible with* `Y′ h`; that is, there's a common formula they both reduce to. For most purposes, though, either will do.
-You may notice that both of these formulas have eta-redexes inside them: why can't we simplify the two `\n. u u f n` inside <code>Θ′</code> to just `u u f`? And similarly for <code>Y′</code>?
+You may notice that both of these formulas have eta-redexes inside them: why can't we simplify the two `\n. u u h n` inside `Θ′` to just `u u h`? And similarly for `Y′`?
Indeed you can, getting the simpler:
-<pre><code>Θ ≡ (\u f. f (u u f)) (\u f. f (u u f))
-Y ≡ \f. (\u. f (u u)) (\u. f (u u))</code></pre>
+ Θ ≡ (\u h. h (u u h)) (\u h. h (u u h))
+ Y ≡ \h. (\u. h (u u)) (\u. h (u u))
-I stated the more complex formulas for the following reason: in a language whose evaluation order is *call-by-value*, the evaluation of <code>Θ (\self. BODY)</code> and `Y (\self. BODY)` will in general not terminate. But evaluation of the eta-unreduced primed versions will.
+We stated the more complex formulas for the following reason: in a language whose evaluation order is *call-by-value*, the evaluation of `Θ (\body. BODY)` and `Y (\body. BODY)` will in general not terminate. But evaluation of the eta-unreduced primed versions will.
-Of course, if you define your `\self. BODY` stupidly, your formula will never terminate. For example, it doesn't matter what fixed point combinator you use for <code>Ψ</code> in:
+Of course, if you define your `\body. BODY` stupidly, your formula will never terminate. For example, it doesn't matter what fixed point combinator you use for `Ψ` in:
-<pre><code>Ψ (\self. \n. self n)</code></pre>
+ Ψ (\body. \n. body n)
When you try to evaluate the application of that to some argument `M`, it's going to try to give you back:
- (\n. self n) M
+ (\n. body n) M
-where `self` is equivalent to the very formula `\n. self n` that contains it. So the evaluation will proceed:
+where `body` is equivalent to the very formula `\n. body n` that contains it. So the evaluation will proceed:
- (\n. self n) M ~~>
- self M ~~>
- (\n. self n) M ~~>
- self M ~~>
- ...
+ (\n. body n) M ~~>
+ body M <~~>
+ (\n. body n) M ~~>
+ body M <~~>
+ ...
You've written an infinite loop!
However, when we evaluate the application of our:
-<pre><code>Ψ (\self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) ))</code></pre>
+ Ψ (\body. (\xs. (empty? xs) 0 (succ (body (tail xs))) ))
-to some list `L`, we're not going to go into an infinite evaluation loop of that sort. At each cycle, we're going to be evaluating the application of:
+to some list, we're not going to go into an infinite evaluation loop of that sort. At each cycle, we're going to be evaluating the application of:
- \lst. (isempty lst) zero (add one (self (extract-tail lst)))
+ \xs. (empty? xs) 0 (succ (body (tail xs)))
-to *the tail* of the list we were evaluating its application to at the previous stage. Assuming our lists are finite (and the implementations we're using don't permit otherwise), at some point one will get a list whose tail is empty, and then the evaluation of that formula to that tail will return `zero`. So the recursion eventually bottoms out in a base value.
+to *the tail* of the list we were evaluating its application to at the previous stage. Assuming our lists are finite (and the encodings we've been using so far don't permit otherwise), at some point one will get a list whose tail is empty, and then the evaluation of that formula to that tail will return `0`. So the recursion eventually bottoms out in a base value.
##Fixed-point Combinators Are a Bit Intoxicating##
-
+[[tatto|/images/y-combinator-fixed.jpg]]
There's a tendency for people to say "Y-combinator" to refer to fixed-point combinators generally. We'll probably fall into that usage ourselves. Speaking correctly, though, the Y-combinator is only one of many fixed-point combinators.
-I used <code>Ψ</code> above to stand in for an arbitrary fixed-point combinator. I don't know of any broad conventions for this. But this seems a useful one.
+We used `Ψ` above to stand in for an arbitrary fixed-point combinator. We don't know of any broad conventions for this. But this seems a useful one.
As we said, there are many other fixed-point combinators as well. For example, Jan Willem Klop pointed out that if we define `L` to be:
- \a b c d e f g h i j k l m n o p q s t u v w x y z r. (r (t h i s i s a f i x e d p o i n t c o m b i n a t o r))
+ \a b c d e f g h i j k l m n o p q s t u v w x y z r. (r (t h i s i s a f i x e d p o i n t c o m b i n a t o r))
then this is a fixed-point combinator:
- L L L L L L L L L L L L L L L L L L L L L L L L L L
+ L L L L L L L L L L L L L L L L L L L L L L L L L L
##Watching Y in action##
sink true true false ~~> I
sink true true true false ~~> I
-So we make `sink = Y (\f b. b f I)`:
+So we make `sink = Y (\s b. b s I)`:
1. sink false
- 2. Y (\fb.bfI) false
- 3. (\f. (\h. f (h h)) (\h. f (h h))) (\fb.bfI) false
- 4. (\h. [\fb.bfI] (h h)) (\h. [\fb.bfI] (h h)) false
- 5. [\fb.bfI] ((\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))) false
- 6. (\b.b[(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))]I) false
- 7. false [(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))] I
+ 2. Y (\sb.bsI) false
+ 3. (\h. (\u. h [u u]) (\u. h (u u))) (\sb.bsI) false
+ 4. (\u. (\sb.bsI) [u u]) (\u. (\sb.bsI) (u u)) false
+ 5. (\sb.bsI) [(\u. (\sb.bsI) (u u)) (\u. (\sb.bsI) (u u))] false
+ 6. (\b. b [(\u. (\sb.bsI) (u u))(\u. (\sb.bsI) (u u))] I) false
+ 7. false [(\u. (\sb.bsI) (u u))(\u. (\sb.bsI) (u u))] I
--------------------------------------------
8. I
Now we try the next most complex example:
1. sink true false
- 2. Y (\fb.bfI) true false
- 3. (\f. (\h. f (h h)) (\h. f (h h))) (\fb.bfI) true false
- 4. (\h. [\fb.bfI] (h h)) (\h. [\fb.bfI] (h h)) true false
- 5. [\fb.bfI] ((\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))) true false
- 6. (\b.b[(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))]I) true false
- 7. true [(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))] I false
- 8. [(\h. [\fb.bsI] (h h))(\h. [\fb.bsI] (h h))] false
+ 2. Y (\sb.bsI) true false
+ 3. (\h. (\u. h [u u]) (\u. h (u u))) (\sb.bsI) true false
+ 4. (\u. (\sb.bsI) [u u]) (\u. (\sb.bsI) (u u)) true false
+ 5. (\sb.bsI) [(\u. (\sb.bsI) (u u)) (\u. (\sb.bsI) (u u))] true false
+ 6. (\b. b [(\u. (\sb.bsI) (u u)) (\u. (\sb.bsI) (u u))] I) true false
+ 7. true [(\u. (\sb.bsI) (u u)) (\u. (\sb.bsI) (u u))] I false
+ 8. [(\u. (\sb.bsI) (u u)) (\u. (\sb.bsI) (u u))] false
We've now arrived at line (4) of the first computation, so the result
-is again I.
+is again `I`.
You should be able to see that `sink` will consume as many `true`s as
-we throw at it, then turn into the identity function after it
+we throw at it, then turn into the identity function when it
encounters the first `false`.
-The key to the recursion is that, thanks to Y, the definition of
+The key to the recursion is that, thanks to `Y`, the definition of
`sink` contains within it the ability to fully regenerate itself as
many times as is necessary. The key to *ending* the recursion is that
the behavior of `sink` is sensitive to the nature of the input: if the
That's about as simple as recursion gets.
+<!-- TODO Perhaps move rest to new document? -->
+
##Application to the truth teller/liar paradoxes##
###Base cases, and their lack###
3! = 3 * 2! = 3 * 2 * 1! = 3 * 2 * 1 * 0! = 3 * 2 * 1 * 0 * -1! ...
-That's why it's crucial to declare that 0! = 1, in which case the
+That's why it's crucial to declare that `0!` = `1`, in which case the
recursive rule does not apply. In our terms,
- fac = Y (\fac n. iszero n 1 (fac (predecessor n)))
+ fact ≡ Y (\fact n. (zero? n) 1 (fact (pred n)))
-If `n` is 0, `fac` reduces to 1, without computing the recursive case.
+If `n` is `0`, `fact` reduces to `1`, without computing the recursive case.
-Curry originally called `Y` the paradoxical combinator, and discussed
+Curry originally called `Y` the "paradoxical" combinator, and discussed
it in connection with certain well-known paradoxes from the philosophy
-literature. The truth teller paradox has the flavor of a recursive
-function without a base case: the truth-teller paradox (and related
-paradoxes).
+literature. The truth-teller paradox has the flavor of a recursive
+function without a base case:
(1) This sentence is true.
like the ones we have been working with here. Then the sentence *John
is John* might denote the function `\x y. x`, our `true`.
+<!-- Jim says: I haven't yet followed the next chunk to my satisfaction -->
+
Then (1) denotes a function from whatever the referent of *this
sentence* is to a boolean. So (1) denotes `\f. f true false`, where
the argument `f` is the referent of *this sentence*. Of course, if
the identity function. Here's a fixed point for the identity
function:
-<pre><code>Y I
-(\f. (\h. f (h h)) (\h. f (h h))) I
-(\h. I (h h)) (\h. I (h h)))
-(\h. (h h)) (\h. (h h)))
-ω ω
-&Omega
-</code></pre>
+ Y I ≡
+ (\h. (\u. h (u u)) (\u. h (u u))) I ~~>
+ (\u. I (u u)) (\u. I (u u))) ~~>
+ (\u. (u u)) (\u. (u u))) ≡
+ ω ω
+ Ω
Oh. Well! That feels right. The meaning of *This sentence is true*
in a context in which *this sentence* refers to the sentence in which
-it occurs is <code>Ω</code>, our prototypical infinite loop...
+it occurs is `Ω`, our prototypical infinite loop...
What about the liar paradox?
(2) This sentence is false.
Used in a context in which *this sentence* refers to the utterance of
-(2) in which it occurs, (2) will denote a fixed point for `\f.neg f`,
-or `\f l r. f r l`, which is the `C` combinator. So in such a
+(2) in which that noun phrase occurs, (2) will denote a fixed point for `\f. neg f`,
+or `\f y n. f n y`, which is the `C` combinator. So in such a
context, (2) might denote
Y C
- (\f. (\h. f (h h)) (\h. f (h h))) I
- (\h. C (h h)) (\h. C (h h)))
- C ((\h. C (h h)) (\h. C (h h)))
- C (C ((\h. C (h h))(\h. C (h h))))
- C (C (C ((\h. C (h h))(\h. C (h h)))))
+ (\h. (\u. h (u u)) (\u. h (u u))) C
+ (\u. C (u u)) (\u. C (u u)))
+ C ((\u. C (u u)) (\u. C (u u)))
+ C (C ((\u. C (u u)) (\u. C (u u))))
+ C (C (C ((\u. C (u u)) (\u. C (u u)))))
...
And infinite sequence of `C`s, each one negating the remainder of the
You should be cautious about feeling too comfortable with
these results. Thinking again of the truth-teller paradox, yes,
-<code>Ω</code> is *a* fixed point for `I`, and perhaps it has
-some a privileged status among all the fixed points for `I`, being the
-one delivered by Y and all (though it is not obvious why Y should have
-any special status).
+`Ω` is *a* fixed point for `I`, and perhaps it has
+some privileged status among all the fixed points for `I`, being the
+one delivered by `Y` and all (though it is not obvious why `Y` should have
+any special status, versus other fixed point combinators).
But one could ask: look, literally every formula is a fixed point for
`I`, since
X <~~> I X
-for any choice of X whatsoever.
+for any choice of `X` whatsoever.
-So the Y combinator is only guaranteed to give us one fixed point out
-of infinitely many---and not always the intuitively most useful
-one. (For instance, the squaring function has zero as a fixed point,
-since 0 * 0 = 0, and 1 as a fixed point, since 1 * 1 = 1, but `Y
-(\x. mul x x)` doesn't give us 0 or 1.) So with respect to the
+So the `Y` combinator is only guaranteed to give us one fixed point out
+of infinitely many --- and not always the intuitively most useful
+one. (For instance, the squaring function `\x. mul x x` has `0` as a fixed point,
+since `square 0 <~~> 0`, and `1` as a fixed point, since `square 1 <~~> 1`, but `Y
+(\x. mul x x)` doesn't give us `0` or `1`.) So with respect to the
truth-teller paradox, why in the reasoning we've
just gone through should we be reaching for just this fixed point at
just this juncture?
[Jacobson 1999](http://www.springerlink.com/content/j706674r4w217jj5/),
pronouns denote (you guessed it!) identity functions...
+<!-- Jim says: haven't made clear how we got from the self-referential (3) to I. -->
+
Ultimately, in the context of this course, these paradoxes are more
useful as a way of gaining leverage on the concepts of fixed points
and recursion, rather than the other way around.