Three computational models and its equivalence

TL;DR

This paper revisits the classical equivalence of three foundational models of computation—Turing machines, recursive functions, and Lambda Calculus—by providing a clear, modern proof that emphasizes mathematical rigor.

Contribution

It offers a comprehensive, accessible proof of the equivalence of the three main models of computation, filling a gap in existing literature.

Findings

Confirmed the equivalence of Turing machines, recursive functions, and Lambda Calculus

Provided a modern, detailed proof accessible to contemporary readers

Strengthened the foundational understanding of computability theory

Abstract

The study of computability has its origin in Hilbert's conference of 1900, where an adjacent question, to the ones he asked, is to give a precise description of the notion of algorithm. In the search for a good definition arose three independent theories: Turing and the Turing machines, G\"odel and the recursive functions, Church and the Lambda Calculus. Later there were established by Kleene that the classic models of computation are equivalent. This fact is widely accepted by many textbooks and the proof is omitted since the proof is tedious and unreadable. We intend to fill this gap presenting the proof in a modern way, without forgetting the mathematical details.