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The integers (and other constants) can be defined in
the Lambda Calculus; it is not necessary to provide
them as "built in". However this "implementation" of
integers is very inefficient and so they invariably are
built into programming languages based on the Lambda Calculus.
Here is a Lambda Calculus definition of non-negative integers
and some operators on them:-
let rec
ZERO = lambda s. lambda z. z,
ONE = lambda s. lambda z. s(z),
TWO = lambda s. lambda z. s(s(z)),
THREE= lambda s. lambda z. s(s(s(z))),{etc.}
PLUS = lambda x. lambda y.
lambda s. lambda z. x s (y s z),
{traditional defn of + }
SUCC = lambda x. lambda s. lambda z. s(x s z),
{successor function}
PLUSb = lambda x. x SUCC,
{a nicer alternative defn of +, PLUS}
TIMES = lambda x. lambda y. x (PLUS y) ZERO,
PRED = lambda n. lambda s. lambda z.
{NB. PRED ZERO = ZERO}
let s2 = lambda n. lambda f. f(n s),
z2 = lambda f. z
in n s2 z2 (lambda x.x),
ISZERO = lambda n. n (lambda x. false) true,
LE = lambda x. lambda y. ISZERO (MONUS x y),
{ ? x <= y ? }
MONUS = lambda a. lambda b. b PRED a,
{NB. assumes a >= b >= 0}
DIVMOD = lambda x. lambda y.
let rec dm = lambda q. lambda x.
if LE y x then {y <= x}
dm (SUCC q) (MONUS x y)
else pair q x
in dm ZERO x,
DIV = lambda x. lambda y. DIVMOD x y fst,
MOD = lambda x. lambda y. DIVMOD x y snd,
pair = lambda fst. lambda snd. lambda sel. sel fst snd,
fst = lambda x. lambda y. x, {see}
snd = lambda x. lambda y. y, {Bool}
PRINT = lambda n. n (lambda m. 'I'::m) '.'
in let rec {e.g.}
four = MONUS (TIMES THREE TWO) (PLUS ONE ONE),
eight = PLUSb four four
in
PRINT (DIV eight THREE)
{ Define (non -ve) Int From Scratch. }
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For example:
PLUS ONE TWO
= (λ x. λ y. λ s. λ z. x s (y s z))
(λ s. λ z. s(z))
(λ s. λ z. s(s(z)))
= (λ y. λ s. λ z.
((λ s'. λ z'. s'(z')) s (y s z)))
(λ s. λ z. s(s(z)))
= (λ y. λ s. λ z. (s(y s z)))
(λ s. λ z. s(s(z)))
= λ s. λ z.
s( (λ s". λ z". s"(s"(z"))) s z)
= λ s. λ z. s(s(s(z)))
= THREE
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Integers are PRINTed in unary notation.
(Well, you try defining a binary or decimal print routine this way!-).
- Also see
[Boolean] and
[Lists].
- Thanks to Joel R. for DIVMOD, TIMES and LE,
the latter nicely making the point that
"you cannot do something less than 0 times."
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window on the wide world:
λ ...
| :: | list cons |
| nil | the [ ] list |
| null | predicate |
| hd | head (1st) |
| tl | tail (rest) |
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