Recursion for church numerals in scheme

Question

I have defined Church numeral zero and some other standard functions on church numerals according to Wikipedia definitions as following:

(define n0 (λ (f x) x))

(define newtrue
  (λ(m n) m))

(define newfalse
  (λ(m n) n))

(define iszero
  (λ(m) (m (λ(x) newfalse) newtrue)))

(define ifthenelse
  (λ(a b c) (a b c)))

Using these, I write a recursion loop as:

(((λ(r) (λ(n) (ifthenelse (iszero n) n ((r r) n))))
   (λ(r) (λ(n) (ifthenelse (iszero n) n ((r r) n))))) n0)

Now for argument n0 as above, this should return n0, without going into recursion. But it doesn't. Why?

Note 1: This recursion loop works perfectly fine with ordinary numerals and ordinary functions:

(((λ(r) (λ(n) (if (= 0 n) n ((r r) n))))
   (λ(r) (λ(n) (if (= 0 n) n ((r r) n))))) 0)

This returns 0 as it should.

Note 2: Functions ifthenelse, iszero, newtrue, newfalse also work fine on their own.

score 0 · Accepted Answer · answered Oct 02 '15 at 05:02

0

The reason for the loop is that the semantics of functions in Scheme is to always evaluate all its arguments.

In your second example:

(((λ(r) (λ(n) (if (= 0 n) n ((r r) n))))
   (λ(r) (λ(n) (if (= 0 n) n ((r r) n))))) 0)

you are using if, which is a special form that does not evaluate the third argument if the condition is true. So, it evaluates (= 0 n) and returns immediately 0, since the condition is #true, without evaluating the third argument ((r r) n).

Instead, in the first example:

(((λ(r) (λ(n) (ifthenelse (iszero n) n ((r r) n))))
   (λ(r) (λ(n) (ifthenelse (iszero n) n ((r r) n))))) n0)

ifthenelse is a function, so that, according to the evaluation rules of Scheme, all its arguments are evaluated, including ((r r) n), and this causes an endless loop.

answered Oct 02 '15 at 05:02

Renzo

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Can you suggest how to make my `ifthenelse` work like usual `if`? – user5284834 Oct 02 '15 at 05:17
The evaluation rules of Lisp languages are not totally equivalent to the Beta reduction rules of lambda calculus, so that what you are trying to do is not possible, for what I know. – Renzo Oct 02 '15 at 05:22
Hooray! Was able to do it using the lazy evaluation feature of scheme. – user5284834 Oct 02 '15 at 10:22
Excellent! Post as an answer to your own question. – Renzo Oct 02 '15 at 10:24

Recursion for church numerals in scheme

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