Church Numerals

From Rosetta Code
Task
Church Numerals
You are encouraged to solve this task according to the task description, using any language you may know.
Task

In the Church encoding of natural numbers, the number N is encoded by a function that applies its first argument N times to its second argument.

  • Church zero always returns the identity function, regardless of its first argument. In other words, the first argument is not applied to the second argument at all.
  • Church one applies its first argument f just once to its second argument x, yielding f(x)
  • Church two applies its first argument f twice to its second argument x, yielding f(f(x))
  • and each successive Church numeral applies its first argument one additional time to its second argument, f(f(f(x))), f(f(f(f(x)))) ... The Church numeral 4, for example, returns a quadruple composition of the function supplied as its first argument.


Arithmetic operations on natural numbers can be similarly represented as functions on Church numerals.

In your language define:

  • Church Zero,
  • a Church successor function (a function on a Church numeral which returns the next Church numeral in the series),
  • functions for Addition, Multiplication and Exponentiation over Church numerals,
  • a function to convert integers to corresponding Church numerals,
  • and a function to convert Church numerals to corresponding integers.


You should:

  • Derive Church numerals three and four in terms of Church zero and a Church successor function.
  • use Church numeral arithmetic to obtain the the sum and the product of Church 3 and Church 4,
  • similarly obtain 4^3 and 3^4 in terms of Church numerals, using a Church numeral exponentiation function,
  • convert each result back to an integer, and return it or print it to the console.


AppleScript[edit]

Implementing churchFromInt as a fold seems to protect Applescript from overflowing its (famously shallow) stack with even quite low Church numerals.

on run
set cThree to churchFromInt(3)
set cFour to churchFromInt(4)
 
map(intFromChurch, ¬
{churchAdd(cThree, cFour), churchMult(cThree, cFour), ¬
churchExp(cFour, cThree), churchExp(cThree, cFour)})
end run
 
-- churchZero :: (a -> a) -> a -> a
on churchZero(f, x)
x
end churchZero
 
-- churchSucc :: ((a -> a) -> a -> a) -> (a -> a) -> a -> a
on churchSucc(n)
script
on |λ|(f)
script
property mf : mReturn(f)
on |λ|(x)
mf's |λ|(mReturn(n)'s |λ|(mf)'s |λ|(x))
end |λ|
end script
end |λ|
end script
end churchSucc
 
-- churchFromInt(n) :: Int -> (b -> b) -> b -> b
on churchFromInt(n)
script
on |λ|(f)
foldr(my compose, my |id|, replicate(n, f))
end |λ|
end script
end churchFromInt
 
-- intFromChurch :: ((Int -> Int) -> Int -> Int) -> Int
on intFromChurch(cn)
mReturn(cn)'s |λ|(my succ)'s |λ|(0)
end intFromChurch
 
on churchAdd(m, n)
script
on |λ|(f)
script
property mf : mReturn(m)
property nf : mReturn(n)
on |λ|(x)
nf's |λ|(f)'s |λ|(mf's |λ|(f)'s |λ|(x))
end |λ|
end script
end |λ|
end script
end churchAdd
 
on churchMult(m, n)
script
on |λ|(f)
script
property mf : mReturn(m)
property nf : mReturn(n)
on |λ|(x)
mf's |λ|(nf's |λ|(f))'s |λ|(x)
end |λ|
end script
end |λ|
end script
end churchMult
 
on churchExp(m, n)
n's |λ|(m)
end churchExp
 
 
-- GENERIC -----------------------------------------------------------
 
-- compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
on compose(f, g)
script
property mf : mReturn(f)
property mg : mReturn(g)
on |λ|(x)
mf's |λ|(mg's |λ|(x))
end |λ|
end script
end compose
 
-- id :: a -> a
on |id|(x)
x
end |id|
 
-- foldr :: (a -> b -> b) -> b -> [a] -> b
on foldr(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from lng to 1 by -1
set v to |λ|(item i of xs, v, i, xs)
end repeat
return v
end tell
end foldr
 
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
 
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
 
-- Egyptian multiplication - progressively doubling a list, appending
-- stages of doubling to an accumulator where needed for binary
-- assembly of a target length
-- replicate :: Int -> a -> [a]
on replicate(n, a)
set out to {}
if n < 1 then return out
set dbl to {a}
 
repeat while (n > 1)
if (n mod 2) > 0 then set out to out & dbl
set n to (n div 2)
set dbl to (dbl & dbl)
end repeat
return out & dbl
end replicate
 
-- succ :: Int -> Int
on succ(x)
1 + x
end succ
Output:
{7, 12, 64, 81}

C#[edit]

using System;
 
public delegate Numeral Numeral(Numeral f);
 
public static class ChurchNumeral
{
public static readonly Numeral Zero = f => x => x;
 
public static Numeral Successor(this Numeral n) => f => x => f(n(f)(x));
public static Numeral Add(this Numeral m, Numeral n) => f => x => m(f)(n(f)(x));
public static Numeral Multiply(this Numeral m, Numeral n) => f => m(n(f));
public static Numeral Pow(this Numeral m, Numeral n) => n(m);
 
public static Numeral FromInt(int i) => i < 0 ? throw new ArgumentException("Negative church numeral.")
: i == 0 ? Zero : Successor(FromInt(i - 1));
 
public static int ToInt(this Numeral f) {
int count = 0;
f(x => { count++; return x; })(null);
return count;
}
 
public static void Main2() {
Numeral c3 = FromInt(3);
Numeral c4 = c3.Successor();
int sum = c3.Add(c4).ToInt();
int product = c3.Multiply(c4).ToInt();
int exp43 = c4.Pow(c3).ToInt();
int exp34 = c3.Pow(c4).ToInt();
Console.WriteLine($"{sum} {product} {exp43} {exp34}");
}
 
}
Output:
7 12 64 81

Clojure[edit]

Translation of: Perl 6
(defn zero  [f]   identity)
(defn succ [n] (fn [f] (fn [x] (f ((n f) x)))))
(defn add [n,m] (fn [f] (fn [x] ((m f)((n f) x)))))
(defn mult [n,m] (fn [f] (fn [x] ((m (n f)) x))))
(defn power [b,e] (e b))
 
(defn to-int [c] (let [countup (fn [i] (+ i 1))] ((c countup) 0)))
 
(defn from-int [n]
(letfn [(countdown [i] (if (zero? i) zero (succ (countdown (- i 1)))))]
(countdown n)))
 
(def three (succ (succ (succ zero))))
(def four (from-int 4))
 
(doseq [n [(add three four) (mult three four)
(power three four) (power four three)]]
(println (to-int n)))
Output:
7
12
81
64

Fōrmulæ[edit]

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Erlang[edit]

Translation of: Perl 6
-module(church).
-export([main/1, zero/1]).
zero(_) -> fun(F) -> F end.
succ(N) -> fun(F) -> fun(X) -> F((N(F))(X)) end end.
add(N,M) -> fun(F) -> fun(X) -> (M(F))((N(F))(X)) end end.
mult(N,M) -> fun(F) -> fun(X) -> (M(N(F)))(X) end end.
power(B,E) -> E(B).
 
to_int(C) -> CountUp = fun(I) -> I + 1 end, (C(CountUp))(0).
 
from_int(0) -> fun church:zero/1;
from_int(I) -> succ(from_int(I-1)).
 
main(_) ->
Zero = fun church:zero/1,
Three = succ(succ(succ(Zero))),
Four = from_int(4),
lists:map(fun(C) -> io:fwrite("~w~n",[to_int(C)]) end,
[add(Three,Four), mult(Three,Four),
power(Three,Four), power(Four,Three)]).
 
Output:
7
12
81
64

F#[edit]

type INumeral =
abstract Apply : ('a -> 'a) -> 'a -> 'a
 
let zero = {new INumeral with override __.Apply _ x = x}
let successor (n: INumeral) = {new INumeral with override __.Apply f x = f (n.Apply f x)}
let addition (m: INumeral) (n: INumeral) = {new INumeral with override __.Apply f x = m.Apply f (n.Apply f x)}
let multiplication (m: INumeral) (n: INumeral) = {new INumeral with override __.Apply f x = m.Apply (n.Apply f) x}
let exponential (m: INumeral) (n: INumeral) = {new INumeral with override __.Apply f x = n.Apply m.Apply f x}
 
let ntoi (n: INumeral) = n.Apply ((+) 1) 0
let iton i = List.fold (>>) id (List.replicate i successor) zero
 
let c3 = iton 3
let c4 = successor c3
 
[addition c3 c4;
multiplication c3 c4;
exponential c4 c3;
exponential c3 c4]
|> List.map ntoi
|> printfn "%A"
 
Output:
[7; 12; 64; 81]

Go[edit]

package main
 
import "fmt"
 
type any = interface{}
 
type fn func(any) any
 
type church func(fn) fn
 
func zero(f fn) fn {
return func(x any) any {
return x
}
}
 
func (c church) succ() church {
return func(f fn) fn {
return func(x any) any {
return f(c(f)(x))
}
}
}
 
func (c church) add(d church) church {
return func(f fn) fn {
return func(x any) any {
return c(f)(d(f)(x))
}
}
}
 
func (c church) mul(d church) church {
return func(f fn) fn {
return func(x any) any {
return c(d(f))(x)
}
}
}
 
func (c church) pow(d church) church {
di := d.toInt()
prod := c
for i := 1; i < di; i++ {
prod = prod.mul(c)
}
return prod
}
 
func (c church) toInt() int {
return c(incr)(0).(int)
}
 
func intToChurch(i int) church {
if i == 0 {
return zero
} else {
return intToChurch(i - 1).succ()
}
}
 
func incr(i any) any {
return i.(int) + 1
}
 
func main() {
z := church(zero)
three := z.succ().succ().succ()
four := three.succ()
 
fmt.Println("three ->", three.toInt())
fmt.Println("four ->", four.toInt())
fmt.Println("three + four ->", three.add(four).toInt())
fmt.Println("three * four ->", three.mul(four).toInt())
fmt.Println("three ^ four ->", three.pow(four).toInt())
fmt.Println("four ^ three ->", four.pow(three).toInt())
fmt.Println("5 -> five ->", intToChurch(5).toInt())
}
Output:
three        -> 3
four         -> 4
three + four -> 7
three * four -> 12
three ^ four -> 81
four ^ three -> 64
5 -> five    -> 5

Haskell[edit]

churchZero = const id
 
churchSucc = (<*>) (.)
 
churchAdd = (<*>) . fmap (.)
 
churchMult = (.)
 
churchExp = flip id
 
churchFromInt :: Int -> ((a -> a) -> a -> a)
churchFromInt 0 = churchZero
churchFromInt n = churchSucc $ churchFromInt (n - 1)
 
-- Or as a fold:
-- churchFromInt n = foldr (.) id . replicate n
 
-- Or as an iterate:
-- churchFromInt n = iterate churchSucc churchZero !! n
 
intFromChurch :: ((Int -> Int) -> Int -> Int) -> Int
intFromChurch cn = cn succ 0
 
-- TEST --------------------------------------------
[cThree, cFour] = churchFromInt <$> [3, 4]
 
main :: IO ()
main =
print $
intFromChurch <$>
[ churchAdd cThree cFour
, churchMult cThree cFour
, churchExp cFour cThree
, churchExp cThree cFour
]
Output:
[7,12,64,81]

Java[edit]

package lvijay;
 
import java.util.concurrent.atomic.AtomicInteger;
import java.util.function.Function;
 
/**
* Java 8 and above
*/

public class Church {
public static interface ChurchNum extends Function<ChurchNum, ChurchNum> {
}
 
public static ChurchNum zero() {
return f -> x -> x;
}
 
public static ChurchNum next(ChurchNum n) {
return f -> x -> f.apply(n.apply(f).apply(x));
}
 
public static ChurchNum plus(ChurchNum a) {
return b -> f -> x -> b.apply(f).apply(a.apply(f).apply(x));
}
 
public static ChurchNum pow(ChurchNum m) {
return n -> m.apply(n);
}
 
public static ChurchNum mult(ChurchNum a) {
return b -> f -> x -> b.apply(a.apply(f)).apply(x);
}
 
public static ChurchNum toChurchNum(int n) {
if (n <= 0) {
return zero();
}
return next(toChurchNum(n - 1));
}
 
public static int toInt(ChurchNum c) {
AtomicInteger counter = new AtomicInteger(0);
ChurchNum funCounter = f -> {
counter.incrementAndGet();
return f;
};
 
plus(zero()).apply(c).apply(funCounter).apply(x -> x);
 
return counter.get();
}
 
public static void main(String[] args) {
ChurchNum zero = zero();
ChurchNum three = next(next(next(zero)));
ChurchNum four = next(next(next(next(zero))));
 
System.out.println("3+4=" + toInt(plus(three).apply(four))); // prints 7
System.out.println("4+3=" + toInt(plus(four).apply(three))); // prints 7
 
System.out.println("3*4=" + toInt(mult(three).apply(four))); // prints 12
System.out.println("4*3=" + toInt(mult(four).apply(three))); // prints 12
 
// exponentiation. note the reversed order!
System.out.println("3^4=" + toInt(pow(four).apply(three))); // prints 81
System.out.println("4^3=" + toInt(pow(three).apply(four))); // prints 64
 
System.out.println(" 8=" + toInt(toChurchNum(8))); // prints 8
}
}
 
Output:
3+4=7
4+3=7
3*4=12
4*3=12
3^4=81
4^3=64
  8=8

JavaScript[edit]

(() => {
'use strict';
 
const churchZero = f => identity;
 
const churchSucc = n => f => compose(f)(n(f));
 
const churchAdd = m => n => f => compose(n(f))(m(f));
 
const churchMult = m => n => f => n(m(f));
 
const churchExp = m => n => n(m);
 
const intFromChurch = n => n(succ)(0);
 
const churchFromInt = n =>
f => foldl(compose)(
identity
)(replicate(n)(f));
 
// Or, by explicit recursion:
const churchFromInt_ = x => {
const go = i =>
0 === i ? (
churchZero
) : churchSucc(go(i - 1));
return go(x);
};
 
 
// TEST -----------------------------------------------
// main :: IO ()
const main = () => {
const [cThree, cFour] = map(churchFromInt)([3, 4]);
 
return map(intFromChurch)([
churchAdd(cThree)(cFour),
churchMult(cThree)(cFour),
churchExp(cFour)(cThree),
churchExp(cThree)(cFour),
]);
};
 
 
// GENERIC FUNCTIONS ----------------------------------
 
// compose (>>>) :: (a -> b) -> (b -> c) -> a -> c
const compose = f => g => x => f(g(x));
 
// foldl :: (a -> b -> a) -> a -> [b] -> a
const foldl = f => a => xs =>
xs.reduce(uncurry(f), a);
 
// identity :: a -> a
const identity = x => x;
 
// map :: (a -> b) -> [a] -> [b]
const map = f => xs =>
(Array.isArray(xs) ? (
xs
) : xs.split('')).map(f);
 
// replicate :: Int -> a -> [a]
const replicate = n => x =>
Array.from({
length: n
}, () => x);
 
// succ :: Enum a => a -> a
const succ = x => 1 + x;
 
// uncurry :: (a -> b -> c) -> ((a, b) -> c)
const uncurry = f =>
function() {
const
args = Array.from(arguments),
a = 1 < args.length ? (
args
) : args[0]; // Tuple object.
return f(a[0])(a[1]);
};
 
// MAIN ---
console.log(JSON.stringify(main()));
})();
Output:
[7,12,64,81]

Julia[edit]

We could overload the Base operators, but that is not needed here.

 
id(x) = x -> x
zero() = x -> id(x)
add(m) = n -> (f -> (x -> n(f)(m(f)(x))))
mult(m) = n -> (f -> (x -> n(m(f))(x)))
exp(m) = n -> n(m)
succ(i::Int) = i + 1
succ(cn) = f -> (x -> f(cn(f)(x)))
church2int(cn) = cn(succ)(0)
int2church(n) = n < 0 ? throw("negative Church numeral") : (n == 0 ? zero() : succ(int2church(n - 1)))
 
function runtests()
church3 = int2church(3)
church4 = int2church(4)
println("Church 3 + Church 4 = ", church2int(add(church3)(church4)))
println("Church 3 * Church 4 = ", church2int(mult(church3)(church4)))
println("Church 4 ^ Church 3 = ", church2int(exp(church4)(church3)))
println("Church 3 ^ Church 4 = ", church2int(exp(church3)(church4)))
end
 
runtests()
 
Output:

Church 3 + Church 4 = 7 Church 3 * Church 4 = 12 Church 4 ^ Church 3 = 64 Church 3 ^ Church 4 = 81

Lambdatalk[edit]

 
{def succ {lambda {:n :f :x} {:f {:n :f :x}}}}
{def add {lambda {:n :m :f :x} {{:n :f} {:m :f :x}}}}
{def mul {lambda {:n :m :f} {:m {:n :f}}}}
{def power {lambda {:n :m} {:m :n}}}
 
{def church {lambda {:n} {{:n {+ {lambda {:x} {+ :x 1}}}} 0}}}
 
{def zero {lambda {:f :x} :x}}
{def three {succ {succ {succ zero}}}}
{def four {succ {succ {succ {succ zero}}}}}
 
3+4 = {church {add {three} {four}}} -> 7
3*4 = {church {mul {three} {four}}} -> 12
3^4 = {church {power {three} {four}}} -> 81
4^3 = {church {power {four} {three}}} -> 64
 

Lua[edit]

 
function churchZero()
return function(x) return x end
end
 
function churchSucc(c)
return function(f)
return function(x)
return f(c(f)(x))
end
end
end
 
function churchAdd(c, d)
return function(f)
return function(x)
return c(f)(d(f)(x))
end
end
end
 
function churchMul(c, d)
return function(f)
return c(d(f))
end
end
 
function churchExp(c, e)
return e(c)
end
 
function numToChurch(n)
local ret = churchZero
for i = 1, n do
ret = succ(ret)
end
return ret
end
 
function churchToNum(c)
return c(function(x) return x + 1 end)(0)
end
 
three = churchSucc(churchSucc(churchSucc(churchZero)))
four = churchSucc(churchSucc(churchSucc(churchSucc(churchZero))))
 
print("'three'\t=", churchToNum(three))
print("'four' \t=", churchToNum(four))
print("'three' * 'four' =", churchToNum(churchMul(three, four)))
print("'three' + 'four' =", churchToNum(churchAdd(three, four)))
print("'three' ^ 'four' =", churchToNum(churchExp(three, four)))
print("'four' ^ 'three' =", churchToNum(churchExp(four, three)))
Output:
'three' =   3
'four'  =   4
'three' * 'four' =  12
'three' + 'four' =  7
'three' ^ 'four' =  81
'four' ^ 'three' =  64

Perl[edit]

Translation of: Perl 6
use 5.020;
use feature qw<signatures>;
no warnings qw<experimental::signatures>;
 
use constant zero => sub ($f) {
sub ($x) { $x }};
 
use constant succ => sub ($n) {
sub ($f) {
sub ($x) { $f->($n->($f)($x)) }}};
 
use constant add => sub ($n) {
sub ($m) {
sub ($f) {
sub ($x) { $m->($f)($n->($f)($x)) }}}};
 
use constant mult => sub ($n) {
sub ($m) {
sub ($f) {
sub ($x) { $m->($n->($f))($x) }}}};
 
use constant power => sub ($b) {
sub ($e) { $e->($b) }};
 
use constant countup => sub ($i) { $i + 1 };
use constant countdown => sub ($i) { $i == 0 ? zero : succ->( __SUB__->($i - 1) ) };
use constant to_int => sub ($f) { $f->(countup)->(0) };
use constant from_int => sub ($x) { countdown->($x) };
 
use constant three => succ->(succ->(succ->(zero)));
use constant four => from_int->(4);
 
say join ' ', map { to_int->($_) } (
add ->( three )->( four ),
mult ->( three )->( four ),
power->( four )->( three ),
power->( three )->( four ),
);
Output:
7 12 64 81

Perl 6[edit]

Traditional subs and sigils[edit]

Translation of: Python
constant $zero  = sub (Code $f) {
sub ( $x) { $x }}
 
constant $succ = sub (Code $n) {
sub (Code $f) {
sub ( $x) { $f($n($f)($x)) }}}
 
constant $add = sub (Code $n) {
sub (Code $m) {
sub (Code $f) {
sub ( $x) { $m($f)($n($f)($x)) }}}}
 
constant $mult = sub (Code $n) {
sub (Code $m) {
sub (Code $f) {
sub ( $x) { $m($n($f))($x) }}}}
 
constant $power = sub (Code $b) {
sub (Code $e) { $e($b) }}
 
sub to_int (Code $f) {
sub countup (Int $i) { $i + 1 }
return $f(&countup).(0)
}
 
sub from_int (Int $x) {
multi sub countdown ( 0) { $zero }
multi sub countdown (Int $i) { $succ( countdown($i - 1) ) }
return countdown($x);
}
 
constant $three = $succ($succ($succ($zero)));
constant $four = from_int(4);
 
say map &to_int,
$add( $three )( $four ),
$mult( $three )( $four ),
$power( $four )( $three ),
$power( $three )( $four ),
;

Arrow subs without sigils[edit]

Translation of: Julia
my \zero  = -> \f {                 -> \x { x               }}
my \succ = -> \n { -> \f { -> \x { f.(n.(f)(x)) }}}
my \add = -> \n { -> \m { -> \f { -> \x { m.(f)(n.(f)(x)) }}}}
my \mult = -> \n { -> \m { -> \f { -> \x { m.(n.(f))(x) }}}}
my \power = -> \b { -> \e { e.(b) }}
 
my \to_int = -> \f { f.( -> \i { i + 1 } ).(0) }
my \from_int = -> \i { i == 0 ?? zero !! succ.( &?BLOCK(i - 1) ) }
 
my \three = succ.(succ.(succ.(zero)));
my \four = from_int.(4);
 
say map -> \f { to_int.(f) },
add.( three )( four ),
mult.( three )( four ),
power.( four )( three ),
power.( three )( four ),
;
Output:
(7 12 64 81)

Phix[edit]

Translation of: Go
type church(object c)
-- eg {r_add,1,{a,b}}
return sequence(c) and length(c)=3
and integer(c[1]) and integer(c[2])
and sequence(c[3]) and length(c[3])=2
end type
 
function succ(church c)
-- eg {r_add,1,{a,b}} => {r_add,2,{a,b}} aka a+b -> a+b+b
c[2] += 1
return c
end function
 
-- three normal integer-handling routines...
function add(integer n, a, b)
for i=1 to n do
a += b
end for
return a
end function
constant r_add = routine_id("add")
 
function mul(integer n, a, b)
for i=1 to n do
a *= b
end for
return a
end function
constant r_mul = routine_id("mul")
 
function pow(integer n, a, b)
for i=1 to n do
a = power(a,b)
end for
return a
end function
constant r_pow = routine_id("pow")
 
-- ...and three church constructors to match
-- (no maths here, just pure static data)
function addch(church c, d)
church res = {r_add,1,{c,d}}
return res
end function
 
function mulch(church c, d)
church res = {r_mul,1,{c,d}}
return res
end function
 
function powch(church c, d)
church res = {r_pow,1,{c,d}}
return res
end function
 
function tointch(church c)
-- note this is where the bulk of any processing happens
{integer rid, integer n, object x} = c
for i=1 to length(x) do
if church(x[i]) then x[i] = tointch(x[i]) end if
end for
return call_func(rid,n&x)
end function
 
constant church zero = {r_add,0,{0,1}}
 
function inttoch(integer i)
if i=0 then
return zero
else
return succ(inttoch(i-1))
end if
end function
 
church three = succ(succ(succ(zero))),
four = succ(three)
printf(1,"three -> %d\n",tointch(three))
printf(1,"four -> %d\n",tointch(four))
printf(1,"three + four -> %d\n",tointch(addch(three,four)))
printf(1,"three * four -> %d\n",tointch(mulch(three,four)))
printf(1,"three ^ four -> %d\n",tointch(powch(three,four)))
printf(1,"four ^ three -> %d\n",tointch(powch(four,three)))
printf(1,"5 -> five -> %d\n",tointch(inttoch(5)))
Output:
three        -> 3
four         -> 4
three + four -> 7
three * four -> 12
three ^ four -> 81
four ^ three -> 64
5 -> five    -> 5

PHP[edit]

<?php
$zero = function($f) { return function ($x) { return $x; }; };
 
$succ = function($n) {
return function($f) use (&$n) {
return function($x) use (&$n, &$f) {
return $f( ($n($f))($x) );
};
};
};
 
$add = function($n, $m) {
return function($f) use (&$n, &$m) {
return function($x) use (&$f, &$n, &$m) {
return ($m($f))(($n($f))($x));
};
};
};
 
$mult = function($n, $m) {
return function($f) use (&$n, &$m) {
return function($x) use (&$f, &$n, &$m) {
return ($m($n($f)))($x);
};
};
};
 
$power = function($b,$e) {
return $e($b);
};
 
$to_int = function($f) {
$count_up = function($i) { return $i+1; };
return ($f($count_up))(0);
};
 
$from_int = function($x) {
$countdown = function($i) use (&$countdown) {
global $zero, $succ;
if ( $i == 0 ) {
return $zero;
} else {
return $succ($countdown($i-1));
};
};
return $countdown($x);
};
 
$three = $succ($succ($succ($zero)));
$four = $from_int(4);
foreach (array($add($three,$four), $mult($three,$four),
$power($three,$four), $power($four,$three)) as $ch) {
print($to_int($ch));
print("\n");
}
?>
Output:
7
12
81
64

Prolog[edit]

Prolog terms can be used to represent church numerals.

church_zero(z).
 
church_successor(Z, c(Z)).
 
church_add(z, Z, Z).
church_add(c(X), Y, c(Z)) :-
church_add(X, Y, Z).
 
church_multiply(z, _, z).
church_multiply(c(X), Y, R) :-
church_add(Y, S, R),
church_multiply(X, Y, S).
 
% N ^ M
church_power(z, z, z).
church_power(N, c(z), N).
church_power(N, c(c(Z)), R) :-
church_multiply(N, R1, R),
church_power(N, c(Z), R1).
 
int_church(0, z).
int_church(I, c(Z)) :-
int_church(Is, Z),
succ(Is, I).
 
run :-
int_church(3, Three),
church_successor(Three, Four),
church_add(Three, Four, Sum),
church_multiply(Three, Four, Product),
church_power(Four, Three, Power43),
church_power(Three, Four, Power34),
 
int_church(ISum, Sum),
int_church(IProduct, Product),
int_church(IPower43, Power43),
int_church(IPower34, Power34),
 
!,
maplist(format('~w '), [ISum, IProduct, IPower43, IPower34]),
nl.
Output:
7 12 81 64 

Python[edit]

Works with: Python version 3.7
'''Church numerals'''
 
from itertools import repeat
from functools import reduce
 
 
# CHURCH ENCODINGS OF NUMERALS AND OPERATIONS -------------
 
def churchZero():
'''The identity function.
No applications of any supplied f
to its argument.
'''

return lambda f: identity
 
 
def churchSucc(cn):
'''The successor of a given
Church numeral. One additional
application of f. Equivalent to
the arithmetic addition of one.
'''

return lambda f: compose(f)(cn(f))
 
 
def churchAdd(m):
'''The arithmetic sum of two Church numerals.'''
return lambda n: lambda f: compose(m(f))(n(f))
 
 
def churchMult(m):
'''The arithmetic product of two Church numerals.'''
return lambda n: compose(m)(n)
 
 
def churchExp(m):
'''Exponentiation of Church numerals. m^n'''
return lambda n: n(m)
 
 
def churchFromInt(n):
'''The Church numeral equivalent of
a given integer.
'''

return lambda f: (
foldl
(compose)
(identity)
(replicate(n)(f))
)
 
 
# OR, alternatively:
def churchFromInt_(n):
'''The Church numeral equivalent of a given
integer, by explicit recursion.
'''

if 0 == n:
return churchZero()
else:
return churchSucc(churchFromInt(n - 1))
 
 
def intFromChurch(cn):
'''The integer equivalent of a
given Church numeral.
'''

return cn(succ)(0)
 
 
# TEST ----------------------------------------------------
# main :: IO ()
def main():
'Tests'
 
cThree = churchFromInt(3)
cFour = churchFromInt(4)
 
print(list(map(intFromChurch, [
churchAdd(cThree)(cFour),
churchMult(cThree)(cFour),
churchExp(cFour)(cThree),
churchExp(cThree)(cFour),
])))
 
 
# GENERIC FUNCTIONS ---------------------------------------
 
# compose (flip (.)) :: (a -> b) -> (b -> c) -> a -> c
def compose(f):
'''A left to right composition of two
functions f and g'''

return lambda g: lambda x: g(f(x))
 
 
# foldl :: (a -> b -> a) -> a -> [b] -> a
def foldl(f):
'''Left to right reduction of a list,
using the binary operator f, and
starting with an initial value a.
'''

def go(acc, xs):
return reduce(lambda a, x: f(a)(x), xs, acc)
return lambda acc: lambda xs: go(acc, xs)
 
 
# identity :: a -> a
def identity(x):
'''The identity function.'''
return x
 
 
# replicate :: Int -> a -> [a]
def replicate(n):
'''A list of length n in which every
element has the value x.
'''

return lambda x: list(repeat(x, n))
 
 
# succ :: Enum a => a -> a
def succ(x):
'''The successor of a value.
For numeric types, (1 +).
'''

return 1 + x if isinstance(x, int) else (
chr(1 + ord(x))
)
 
 
if __name__ == '__main__':
main()
Output:
[7, 12, 64, 81]

Racket[edit]

#lang racket
 
(define zero (λ (f) (λ (x) x)))
(define zero* (const identity)) ; zero renamed
 
(define one (λ (f) f))
(define one* identity) ; one renamed
 
(define succ (λ (n) (λ (f) (λ (x) (f ((n f) x))))))
(define succ* (λ (n) (λ (f) (λ (x) ((n f) (f x)))))) ; different impl
 
(define add (λ (n) (λ (m) (λ (f) (λ (x) ((m f) ((n f) x)))))))
(define add* (λ (n) (n succ)))
 
(define succ** (add one))
 
(define mult (λ (n) (λ (m) (λ (f) (m (n f))))))
(define mult* (λ (n) (λ (m) ((m (add n)) zero))))
 
(define expt (λ (n) (λ (m) (m n))))
(define expt* (λ (n) (λ (m) ((m (mult n)) one))))
 
(define (nat->church n)
(cond
[(zero? n) zero]
[else (succ (nat->church (sub1 n)))]))
 
(define (church->nat n) ((n add1) 0))
 
(define three (nat->church 3))
(define four (nat->church 4))
 
(church->nat ((add three) four))
(church->nat ((mult three) four))
(church->nat ((expt three) four))
(church->nat ((expt four) three))
Output:
7
12
81
64

Ruby[edit]

Translation of: Perl 6

The traditional methods version uses lambda to declare anonymous functions and calls them with .(); the version with procs all the way down uses proc to declare the anonymous functions and calls them with []. These are stylistic choices and each pair of options is completely interchangeable in the context of this solution.

Traditional methods[edit]

def zero(f)
return lambda {|x| x}
end
Zero = lambda { |f| zero(f) }
 
def succ(n)
return lambda { |f| lambda { |x| f.(n.(f).(x)) } }
end
 
Three = succ(succ(succ(Zero)))
 
def add(n, m)
return lambda { |f| lambda { |x| m.(f).(n.(f).(x)) } }
end
 
def mult(n, m)
return lambda { |f| lambda { |x| m.(n.(f)).(x) } }
end
 
def power(b, e)
return e.(b)
end
 
def int_from_couch(f)
countup = lambda { |i| i+1 }
f.(countup).(0)
end
 
def couch_from_int(x)
countdown = lambda { |i|
case i
when 0 then Zero
else succ(countdown.(i-1))
end
}
countdown.(x)
end
 
Four = couch_from_int(4)
 
puts [ add(Three, Four),
mult(Three, Four),
power(Three, Four),
power(Four, Three) ].map {|f| int_from_couch(f) }
 
Output:
7
12
81
64

Procs all the way down[edit]

Zero  = proc { |f| proc { |x| x } }
 
Succ = proc { |n| proc { |f| proc { |x| f[n[f][x]] } } }
 
Add = proc { |n, m| proc { |f| proc { |x| m[f][n[f][x]] } } }
 
Mult = proc { |n, m| proc { |f| proc { |x| m[n[f]][x] } } }
 
Power = proc { |b, e| e[b] }
 
ToInt = proc { |f| countup = proc { |i| i+1 }; f[countup][0] }
 
FromInt = proc { |x|
countdown = proc { |i|
case i
when 0 then Zero
else Succ[countdown[i-1]]
end
}
countdown[x]
}
 
Three = Succ[Succ[Succ[Zero]]]
Four = FromInt[4]
 
puts [ Add[Three, Four],
Mult[Three, Four],
Power[Three, Four],
Power[Four, Three] ].map(&ToInt)
Output:
7
12
81
64

Rust[edit]

use std::rc::Rc;
use std::ops::{Add, Mul};
 
#[derive(Clone)]
struct Church<'a, T: 'a> {
runner: Rc<dyn Fn(Rc<dyn Fn(T) -> T + 'a>) -> Rc<dyn Fn(T) -> T + 'a> + 'a>,
}
 
impl<'a, T> Church<'a, T> {
fn zero() -> Self {
Church {
runner: Rc::new(|_f| {
Rc::new(|x| x)
})
}
}
 
fn succ(self) -> Self {
Church {
runner: Rc::new(move |f| {
let g = self.runner.clone();
Rc::new(move |x| f(g(f.clone())(x)))
})
}
}
 
fn run(&self, f: impl Fn(T) -> T + 'a) -> Rc<dyn Fn(T) -> T + 'a> {
(self.runner)(Rc::new(f))
}
 
fn exp(self, rhs: Church<'a, Rc<dyn Fn(T) -> T + 'a>>) -> Self
{
Church {
runner: (rhs.runner)(self.runner)
}
}
}
 
impl<'a, T> Add for Church<'a, T> {
type Output = Church<'a, T>;
 
fn add(self, rhs: Church<'a, T>) -> Church<T> {
Church {
runner: Rc::new(move |f| {
let self_runner = self.runner.clone();
let rhs_runner = rhs.runner.clone();
Rc::new(move |x| (self_runner)(f.clone())((rhs_runner)(f.clone())(x)))
})
}
}
}
 
impl<'a, T> Mul for Church<'a, T> {
type Output = Church<'a, T>;
 
fn mul(self, rhs: Church<'a, T>) -> Church<T> {
Church {
runner: Rc::new(move |f| {
(self.runner)((rhs.runner)(f))
})
}
}
}
 
impl<'a, T> From<i32> for Church<'a, T> {
fn from(n: i32) -> Church<'a, T> {
let mut ret = Church::zero();
for _ in 0..n {
ret = ret.succ();
}
ret
}
}
 
impl<'a> From<&Church<'a, i32>> for i32 {
fn from(c: &Church<'a, i32>) -> i32 {
c.run(|x| x + 1)(0)
}
}
 
fn three<'a, T>() -> Church<'a, T> {
Church::zero().succ().succ().succ()
}
 
fn four<'a, T>() -> Church<'a, T> {
Church::zero().succ().succ().succ().succ()
}
 
fn main() {
println!("three =\t{}", i32::from(&three()));
println!("four =\t{}", i32::from(&four()));
 
println!("three + four =\t{}", i32::from(&(three() + four())));
println!("three * four =\t{}", i32::from(&(three() * four())));
 
println!("three ^ four =\t{}", i32::from(&(three().exp(four()))));
println!("four ^ three =\t{}", i32::from(&(four().exp(three()))));
}
Output:
three =	3
four =	4
three + four =	7
three * four =	12
three ^ four =	81
four ^ three =	64

Swift[edit]

func succ<A, B, C>(_ n: @escaping (@escaping (A) -> B) -> (C) -> A) -> (@escaping (A) -> B) -> (C) -> B {
return {f in
return {x in
return f(n(f)(x))
}
}
}
 
func zero<A, B>(_ a: A) -> (B) -> B {
return {b in
return b
}
}
 
func three<A>(_ f: @escaping (A) -> A) -> (A) -> A {
return {x in
return succ(succ(succ(zero)))(f)(x)
}
}
 
func four<A>(_ f: @escaping (A) -> A) -> (A) -> A {
return {x in
return succ(succ(succ(succ(zero))))(f)(x)
}
}
 
func add<A, B, C>(_ m: @escaping (B) -> (A) -> C) -> (@escaping (B) -> (C) -> A) -> (B) -> (C) -> C {
return {n in
return {f in
return {x in
return m(f)(n(f)(x))
}
}
}
}
 
func mult<A, B, C>(_ m: @escaping (A) -> B) -> (@escaping (C) -> A) -> (C) -> B {
return {n in
return {f in
return m(n(f))
}
}
}
 
func exp<A, B, C>(_ m: A) -> (@escaping (A) -> (B) -> (C) -> C) -> (B) -> (C) -> C {
return {n in
return {f in
return {x in
return n(m)(f)(x)
}
}
}
}
 
func church<A>(_ x: Int) -> (@escaping (A) -> A) -> (A) -> A {
guard x != 0 else { return zero }
 
return {f in
return {a in
return f(church(x - 1)(f)(a))
}
}
}
 
func unchurch<A>(_ f: (@escaping (Int) -> Int) -> (Int) -> A) -> A {
return f({i in
return i + 1
})(0)
}
 
let a = unchurch(add(three)(four))
let b = unchurch(mult(three)(four))
// We can even compose operations
let c = unchurch(exp(mult(four)(church(1)))(three))
let d = unchurch(exp(mult(three)(church(1)))(four))
 
print(a, b, c, d)
Output:
7 12 64 81

zkl[edit]

class Church{  // kinda heavy, just an int + fcn churchAdd(ca,cb) would also work
fcn init(N){ var n=N; } // Church Zero is Church(0)
fcn toInt(f,x){ do(n){ x=f(x) } x } // c(3)(f,x) --> f(f(f(x)))
fcn succ{ self(n+1) }
fcn __opAdd(c){ self(n+c.n) }
fcn __opMul(c){ self(n*c.n) }
fcn pow(c) { self(n.pow(c.n)) }
fcn toString{ String("Church(",n,")") }
}
c3,c4 := Church(3),c3.succ();
f,x := Op("+",1),0;
println("f=",f,", x=",x);
println("%s+%s=%d".fmt(c3,c4, (c3+c4).toInt(f,x) ));
println("%s*%s=%d".fmt(c3,c4, (c3*c4).toInt(f,x) ));
println("%s^%s=%d".fmt(c4,c3, (c4.pow(c3)).toInt(f,x) ));
println("%s^%s=%d".fmt(c3,c4, (c3.pow(c4)).toInt(f,x) ));
println();
T(c3+c4,c3*c4,c4.pow(c3),c3.pow(c4)).apply("toInt",f,x).println();
Output:
f=Op(+1), x=0
Church(3)+Church(4)=7
Church(3)*Church(4)=12
Church(4)^Church(3)=64
Church(3)^Church(4)=81

L(7,12,64,81)

OK, that was the easy sleazy cheat around way to do it. The wad of nested functions way is as follows:

fcn churchZero{ return(fcn(x){ x }) } // or fcn churchZero{ self.fcn.idFcn }
fcn churchSucc(c){ return('wrap(f){ return('wrap(x){ f(c(f)(x)) }) }) }
fcn churchAdd(c1,c2){ return('wrap(f){ return('wrap(x){ c1(f)(c2(f)(x)) }) }) }
fcn churchMul(c1,c2){ return('wrap(f){ c1(c2(f)) }) }
fcn churchPow(c1,c2){ return('wrap(f){ c2(c1)(f) }) }
fcn churchToInt(c,f,x){ c(f)(x) }
fcn churchFromInt(n){ c:=churchZero; do(n){ c=churchSucc(c) } c }
//fcn churchFromInt(n){ (0).reduce(n,churchSucc,churchZero) } // what ever
c3,c4 := churchFromInt(3),churchSucc(c3);
f,x  := Op("+",1),0; // x>=0, ie natural number
T(c3,c4,churchAdd(c3,c4),churchMul(c3,c4),churchPow(c4,c3),churchPow(c3,c4))
.apply(churchToInt,f,x).println();
Output:
L(3,4,7,12,64,81)