Church Numerals

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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}

Fōrmulæ[edit]

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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 = (<*>) . (<$>) (.)
 
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]

JavaScript[edit]

(() => {
'use strict';
 
const main = () => {
 
const churchZero = f => x => x;
 
const churchSucc = n => f => x => f(n(f)(x));
 
const churchAdd = m => n => f => x => n(f)(m(f)(x));
 
const churchMult = m => n => f => x => n(m(f))(x);
 
const churchExp = m => n => n(m);
 
const intFromChurch = n => n(succ)(0);
 
const churchFromInt = n =>
f => foldl(composeR, id, replicate(n, f));
 
// Or, recursively ...
// const churchFromInt = x => {
// const go = i =>
// 0 === i ? (
// churchZero
// ) : churchSucc(go(i - 1));
// return go(x);
// };
 
// TEST -------------------------------------------
const [cThree, cFour] = map(churchFromInt, [3, 4]);
 
return map(
intFromChurch, [
churchAdd(cThree)(cFour),
churchMult(cThree)(cFour),
churchExp(cFour)(cThree),
churchExp(cThree)(cFour),
]
);
};
 
// GENERIC FUNCTIONS ------------------------------
 
// composeR (>>>) :: (a -> b) -> (b -> c) -> a -> c
const composeR = (f, g) => x => f(g(x));
 
// foldl :: (a -> b -> a) -> a -> [b] -> a
const foldl = (f, a, xs) => xs.reduce(f, a);
 
// id :: a -> a
const id = x => x;
 
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.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;
 
// MAIN ---------------------
return 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

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

Python[edit]

import functools
import itertools
 
# CHURCH ENCODINGS ---------------------------------
 
 
def churchZero():
return lambda f: id
 
 
def churchSucc(cn):
return lambda f: lambda x: f(cn(f)(x))
 
 
def churchAdd(m):
return lambda n: lambda f: lambda x: n(f)(m(f)(x))
 
 
def churchMult(m):
return lambda n: lambda f: lambda x: n(m(f))(x)
 
 
def churchExp(m):
return lambda n: n(m)
 
 
def churchFromInt(n):
return lambda f: (
foldl
(composeR)
(id)
(replicate(n)(f))
)
 
# OR, recursively:
# def churchFromInt(n):
# if 0 == n:
# return churchZero()
# else:
# return churchSucc(churchFromInt(n - 1))
 
 
def intFromChurch(cn):
return cn(succ)(0)
 
 
# GENERIC FUNCTIONS -------------------------------
 
# composeR (>>>) :: (a -> b) -> (b -> c) -> a -> c
def composeR(f):
return lambda g: lambda x: f(g(x))
 
 
# foldl :: (a -> b -> a) -> a -> [b] -> a
def foldl(f):
return lambda a: lambda xs: (
functools.reduce(uncurry(f), xs, a)
)
 
 
# id :: a -> a
def id(x):
return x
 
 
# replicate :: Int -> a -> [a]
def replicate(n):
return lambda x: itertools.repeat(x, n)
 
 
# succ :: Int -> Int
def succ(x):
return 1 + x
 
 
# uncurry :: (a -> b -> c) -> ((a, b) -> c)
def uncurry(f):
def g(x, y):
return f(x)(y)
return g
 
 
# MAIN -------------------------------------------
def main():
cThree = churchFromInt(3)
cFour = churchFromInt(4)
 
print (list(map(intFromChurch, [
churchAdd(cThree)(cFour),
churchMult(cThree)(cFour),
churchExp(cFour)(cThree),
churchExp(cThree)(cFour),
])))
 
 
main()
Output:
[7, 12, 64, 81]

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)