A new characterization of computable functions

TL;DR

This paper introduces algorithms that, for any computable function, generate systems of equations within E_n that are consistent over integers or non-negative integers, with solutions encoding the function's value.

Contribution

It provides a novel method to characterize computable functions through systems of simple equations, linking computability with algebraic systems.

Findings

Algorithms exist to generate consistent systems encoding computable functions.

These systems are constructed within the framework of E_n equations.

Solutions to these systems reflect the values of the original functions.

Abstract

Let E_n={x_i=1, x_i+x_j=x_k, x_i*x_j=x_k: i,j,k \in {1,...,n}}. We prove: (1) there is an algorithm that for every computable function f:N-->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a system S \subseteq E_n such that S is consistent over the integers and each integer tuple (x_1,...,x_n) that solves S satisfies x_1=f(n), (2) there is an algorithm that for every computable function f:N-->N returns a positive integer w(f), for which a second algorithm accepts on the input f and any integer n>=w(f), and returns a system S \subseteq E_n such that S is consistent over N and each tuple (x_1,...,x_n) of non-negative integers that solves S satisfies x_1=f(n).