Hilbert’s Tenth Problem, Word Problems

Abstract

A recursive function f:Nⁿ → N has been defined as one for which there is a Turing machine, T _f say, which computes f(x ₁,..., x _n ) for all (x ₁,..., x _n ) ∈ Nⁿ. Accordingly, in order to show that a particular function g:Nⁿ → N is recursive, we must construct a Turing machine which computes g. This is a tiresome process, even for functions of relatively simple form, and consequently it is natural to seek an alternative characterisation of recursive functions that will facilitate their recognition.