How to resolve the algorithm Y combinator step by step in the Mathematica / Wolfram Language programming language

Published on 22 June 2024 08:30 PM
#Wolfram

How to resolve the algorithm Y combinator step by step in the Mathematica / Wolfram Language programming language

Table of Contents

Problem Statement

In strict functional programming and the lambda calculus, functions (lambda expressions) don't have state and are only allowed to refer to arguments of enclosing functions. This rules out the usual definition of a recursive function wherein a function is associated with the state of a variable and this variable's state is used in the body of the function. The   Y combinator   is itself a stateless function that, when applied to another stateless function, returns a recursive version of the function. The Y combinator is the simplest of the class of such functions, called fixed-point combinators.

Define the stateless   Y combinator   and use it to compute factorials and Fibonacci numbers from other stateless functions or lambda expressions.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Y combinator step by step in the Mathematica / Wolfram Language programming language

The provided Wolfram code defines two functions, factorial and fibonacci, using the Y function as a higher-order function. Here's an explanation of the code:

  1. Y Function:

    • The Y function takes a function f as input and returns a new function that applies f twice to its argument.
    • #[#] &[Function[g, f[g[g][##] &]]: This part of the Y function creates a new function that takes a single argument #. It applies the function f to the result of applying the function g three times to #.

  2. factorial Function:

    • The factorial function is defined as Y[Function[f, If[# < 1, 1, # f[# - 1]] &]].
    • This means that the factorial function is created by applying the Y function to a function that calculates the factorial of a number.
    • The inner function takes a number # and checks if it is less than 1. If it is, it returns 1 (the base case for factorial). Otherwise, it multiplies # by the factorial of # - 1.

  3. fibonacci Function:

    • The fibonacci function is defined as Y[Function[f, If[# < 2, #, f[# - 1] + f[# - 2]] &]].
    • This means that the fibonacci function is created by applying the Y function to a function that calculates the Fibonacci number for a given index.
    • The inner function takes an index # and checks if it is less than 2. If it is, it returns # (the base cases for Fibonacci). Otherwise, it adds the Fibonacci numbers of # - 1 and # - 2 to get the Fibonacci number for #.

In summary, the code defines functions to calculate the factorial of a number and the Fibonacci number for a given index using the fixed-point combinator Y that allows defining recursive functions in a functional programming style.

Source code in the wolfram programming language

Y = Function[f, #[#] &[Function[g, f[g[g][##] &]]]];
factorial = Y[Function[f, If[# < 1, 1, # f[# - 1]] &]];
fibonacci = Y[Function[f, If[# < 2, #, f[# - 1] + f[# - 2]] &]];

You may also check:How to resolve the algorithm Fast Fourier transform step by step in the BBC BASIC programming language
You may also check:How to resolve the algorithm First-class functions step by step in the Mathematica / Wolfram Language programming language
You may also check:How to resolve the algorithm Factors of an integer step by step in the HicEst programming language
You may also check:How to resolve the algorithm Empty program step by step in the Avail programming language
You may also check:How to resolve the algorithm Self-describing numbers step by step in the ALGOL 68 programming language