How to resolve the algorithm Church numerals step by step in the Java programming language

Published on 12 May 2024 09:40 PM
#Java

How to resolve the algorithm Church numerals step by step in the Java programming language

Table of Contents

Problem Statement

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.

Arithmetic operations on natural numbers can be similarly represented as functions on Church numerals. In your language define:

You should:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Church numerals step by step in the Java programming language

Churh encoding is a way to represent numbers using functions. In this code, a Church number is represented as a function that takes a function f and an argument x and returns the result of applying f to x.

The zero function simply returns the identity function, which returns its argument unchanged. The next function takes a Church number n and returns a function that applies f to n.apply(f).apply(x). This is equivalent to incrementing n by 1.

The plus function takes two Church numbers a and b and returns a function that applies f to b.apply(f).apply(a.apply(f).apply(x)). This is equivalent to adding a and b.

The pow function takes a Church number m and returns a function that applies m to n. This is equivalent to raising m to the power of n.

The mult function takes two Church numbers a and b and returns a function that applies f to b.apply(a.apply(f)).apply(x). This is equivalent to multiplying a and b.

The toChurchNum function takes an integer n and returns the corresponding Church number. The toInt function takes a Church number c and returns the corresponding integer.

The main function tests the Church number implementation by creating some Church numbers and performing some arithmetic operations on them. The results are printed to the console.

Output:

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

Source code in the java programming language

package lvijay;

import java.util.concurrent.atomic.AtomicInteger;
import java.util.function.Function;

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
    }
}

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