# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.03392