The symmetric Post Correspondence Problem, and errata for the freeness problem for matrix semigroups

TL;DR

This paper introduces the symmetric Post Correspondence Problem (PCP), proves its undecidability, and demonstrates its implications for the freeness problem in 3x3 integer matrix semigroups, clarifying the limits of previous proofs.

Contribution

It defines the symmetric PCP, proves its undecidability, and provides insights into the applicability of existing undecidability proofs for matrix semigroup problems.

Findings

Symmetric PCP is undecidable.

The original proof for matrix semigroups applies to symmetric PCP.

The proof does not extend to PCP in general.

Abstract

We define the symmetric Post Correspondence Problem (PCP) and prove that it is undecidable. As an application we show that the original proof of undecidability of the freeness problem for 3-by-3 integer matrix semigroups works for the symmetric PCP, but not for the PCP in general.