Getting around the Halting Problem

TL;DR

This paper explores theoretical limits of computation related to the halting problem, presenting conditions under which certain programs behave in paradoxical ways, with implications for logic and formal systems.

Contribution

It introduces specific program constructions that circumvent the halting problem, extending understanding of computability and incompleteness.

Findings

Existence of programs with paradoxical halting behavior

Implications for the liar paradox and G"odel incompleteness

Theoretical framework for non-halting program conditions

Abstract

There are numbers k and s and a URM program A(n,m) satisfying the following conditions. 1. If A(n,m) halts, then Cn(m) diverges. 2. For all n, C_k(n) = A(n,n) and C_s(n) = C_k(s). 3. A(k,s) halts and for all n, A(s,n) diverges. Here C_n(_) is a program with index n in some exhaustive enumeration of all possible programs. This has implications for solving the liar paradox and for generalization of G\"odel incompleteness theorem to formal systems other than PA.