`T` has a fixed point, then there exists some `X` such that `X <~~>
TX` (that's what it means to *have* a fixed point).
-<pre>
-let W = \x.T(xx) in
-let X = WW in
-X = WW = (\x.T(xx))W = T(WW) = TX
-</pre>
+<pre><code>let L = \x. T (x x) in
+let X = L L in
+X ≡ L L ≡ (\x. T (x x)) L ~~> T (L L) ≡ T X
+</code></pre>
Please slow down and make sure that you understand what justified each
of the equalities in the last line.
-#Q: How do you know that for any term `T`, `YT` is a fixed point of `T`?#
+#Q: How do you know that for any term `T`, `Y T` is a fixed point of `T`?#
A: Note that in the proof given in the previous answer, we chose `T`
-and then set `X = WW = (\x.T(xx))(\x.T(xx))`. If we abstract over
-`T`, we get the Y combinator, `\T.(\x.T(xx))(\x.T(xx))`. No matter
-what argument `T` we feed Y, it returns some `X` that is a fixed point
+and then set `X = L L = (\x. T (x x)) (\x. T (x x))`. If we abstract over
+`T`, we get the Y combinator, `\T. (\x. T (x x)) (\x. T (x x))`. No matter
+what argument `T` we feed `Y`, it returns some `X` that is a fixed point
of `T`, by the reasoning in the previous answer.
#Q: So if every term has a fixed point, even `Y` has fixed point.#
A: Right:
- let Y = \T.(\x.T(xx))(\x.T(xx)) in
- Y Y = \T.(\x.T(xx))(\x.T(xx)) Y
- = (\x.Y(xx))(\x.Y(xx))
- = Y((\x.Y(xx))(\x.Y(xx)))
- = Y(Y((\x.Y(xx))(\x.Y(xx))))
- = Y(Y(Y(...(Y(YY))...)))
+<pre><code>let Y = \T. (\x. T (x x)) (\x. T (x x)) in
+Y Y ≡ \T. (\x. T (x x)) (\x. T (x x)) Y
+~~> (\x. Y (x x)) (\x. Y (x x))
+~~> Y ((\x. Y (x x)) (\x. Y (x x)))
+~~> Y (Y ((\x. Y (x x)) (\x. Y (x x))))
+~~> Y (Y (Y (...(Y (Y Y))...)))</code></pre>
#Q: Ouch! Stop hurting my brain.#
-A: Let's come at it from the direction of arithmetic. Recall that we
+A: Is that a question?
+
+Let's come at it from the direction of arithmetic. Recall that we
claimed that even `succ`---the function that added one to any
number---had a fixed point. How could there be an X such that X = X+1?
That would imply that
- X = succ X = succ (succ X) = succ (succ (succ (X))) = succ (... (succ X)...)
+ X <~~> succ X <~~> succ (succ X) <~~> succ (succ (succ (X))) <~~> succ (... (succ X)...)
In other words, the fixed point of `succ` is a term that is its own
successor. Let's just check that `X = succ X`:
- let succ = \n s z. s (n s z) in
- let X = (\x.succ(xx))(\x.succ(xx)) in
- succ X
- = succ ((\x.succ(xx))(\x.succ(xx)))
- = succ (succ ((\x.succ(xx))(\x.succ(xx))))
- = succ (succ X)
+<pre><code>let succ = \n s z. s (n s z) in
+let X = (\x. succ (x x)) (\x. succ (x x)) in
+succ X
+≡ succ ( (\x. succ (x x)) (\x. succ (x x)) )
+~~> succ (succ ( (\x. succ (x x)) (\x. succ (x x))))
+≡ succ (succ X)</code></pre>
-You should see the close similarity with YY here.
+You should see the close similarity with `Y Y` here.
#Q. So `Y` applied to `succ` returns a number that is not finite!#
It's important to bear in mind the simplest term in question is not
infinite:
- Y succ = (\f.(\x.f(xx))(\x.f(xx)))(\n s z. s (n s z))
+ Y succ = (\f. (\x. f (x x)) (\x. f (x x))) (\n s z. s (n s z))
The way that infinity enters into the picture is that this term has
no normal form: no matter how many times we perform beta reduction,
next: one choice leads to a normal form, the other choice leads to
endless reduction:
- let prefac = \f n. isZero n 1 (mult n (f (pred n))) in
+ let prefac = \f n. iszero n 1 (mult n (f (pred n))) in
let fac = Y prefac in
fac 2
= [(\f.(\x.f(xx))(\x.f(xx))) prefac] 2
= [(\x.prefac(xx))(\x.prefac(xx))] 2
= [prefac((\x.prefac(xx))(\x.prefac(xx)))] 2
= [prefac(prefac((\x.prefac(xx))(\x.prefac(xx))))] 2
- = [(\f n. isZero n 1 (mult n (f (pred n))))
+ = [(\f n. iszero n 1 (mult n (f (pred n))))
(prefac((\x.prefac(xx))(\x.prefac(xx))))] 2
- = [\n. isZero n 1 (mult n ([prefac((\x.prefac(xx))(\x.prefac(xx)))] (pred n)))] 2
- = isZero 2 1 (mult 2 ([prefac((\x.prefac(xx))(\x.prefac(xx)))] (pred 2)))
+ = [\n. iszero n 1 (mult n ([prefac((\x.prefac(xx))(\x.prefac(xx)))] (pred n)))] 2
+ = iszero 2 1 (mult 2 ([prefac((\x.prefac(xx))(\x.prefac(xx)))] (pred 2)))
= mult 2 ([prefac((\x.prefac(xx))(\x.prefac(xx)))] 1)
...
= mult 2 (mult 1 ([prefac((\x.prefac(xx))(\x.prefac(xx)))] 0))
- = mult 2 (mult 1 (isZero 0 1 ([prefac((\x.prefac(xx))(\x.prefac(xx)))] (pred 0))))
+ = mult 2 (mult 1 (iszero 0 1 ([prefac((\x.prefac(xx))(\x.prefac(xx)))] (pred 0))))
= mult 2 (mult 1 1)
= mult 2 1
= 2
The crucial step is the third from the last. We have our choice of
-either evaluating the test `isZero 0 1 ...`, which evaluates to `1`,
+either evaluating the test `iszero 0 1 ...`, which evaluates to `1`,
no matter what the ... contains;
or we can evaluate the `Y` pump, `(\x.prefac(xx))(\x.prefac(xx))`, to
produce another copy of `prefac`. If we postpone evaluting the
-`isZero` test, we'll pump out copy after copy of `prefac`, and never
+`iszero` test, we'll pump out copy after copy of `prefac`, and never
realize that we've bottomed out in the recursion. But if we adopt a
leftmost/call-by-name/normal-order evaluation strategy, we'll always
-start with the `isZero` predicate, and only produce a fresh copy of
+start with the `iszero` predicate, and only produce a fresh copy of
`prefac` if we are forced to.
#Q. You claimed that the Ackerman function couldn't be implemented using our primitive recursion techniques (such as the techniques that allow us to define addition and multiplication). But you haven't shown that it is possible to define the Ackerman function using full recursion.#
| else when n == 0 -> A(m-1,1)
| else -> A(m-1, A(m,n-1))
-let A = Y (\A m n. isZero m (succ n) (isZero n (A (pred m) 1) (A (pred m) (A m (pred n))))) in
+let A = Y (\A m n. iszero m (succ n) (iszero n (A (pred m) 1) (A (pred m) (A m (pred n))))) in
</pre>
For instance,
along to the continue-leftwards handler.
A **version 5** list encodes the kind of fold operation we're envisaging here, in
-the same way that v3 (and [v4](/advanced/#v4)) lists encoded the simpler fold operation.
+the same way that v3 (and [v4](/advanced/#index1h1)) lists encoded the simpler fold operation.
Roughly, the list `[5;4;3;2;1]` would look like this:
larger_computation
3. To extract tails efficiently, too, it'd be nice to fuse the apparatus developed
- in these v5 lists with the ideas from [v4](/advanced/#v4) lists.
+ in these v5 lists with the ideas from [v4](/advanced/#index1h1) lists.
But that also is left as an exercise.