(Generalized) Post Correspondence Problem and semi-Thue systems

TL;DR

This paper provides a clear, detailed proof connecting the undecidability of semi-Thue system word problems to the Post Correspondence Problem, refining bounds on their computational complexity.

Contribution

It offers a simplified proof of Claus's result, establishing new undecidability bounds for PCP and GPCP using the Generalized Post Correspondence Problem.

Findings

Proves that undecidability of ACCESSIBILITY(k) implies GPCP(k+2) undecidable.

Shows GPCP(k) undecidable implies PCP(k+2) undecidable.

Refines the bounds for PCP and GPCP undecidability based on semi-Thue systems.

Abstract

Let PCP(k) denote the Post Correspondence Problem for k input pairs of strings. Let ACCESSIBILITY(k) denote the the word problem for k-rule semi-Thue systems. In 1980, Claus showed that if ACCESSIBILITY(k) is undecidable then PCP(k + 4) is also undecidable. The aim of the paper is to present a clean, detailed proof of the statement. We proceed in two steps, using the Generalized Post Correspondence Problem as an auxiliary. First, we prove that if ACCESSIBILITY(k) is undecidable then GPCP(k + 2) is also undecidable. Then, we prove that if GPCP(k) is undecidable then PCP(k + 2) is also undecidable. (The latter result has also been shown by Harju and Karhumaki.) To date, the sharpest undecidability bounds for both PCP and GPCP have been deduced from Claus's result: since Matiyasevich and Senizergues showed that ACCESSIBILITY(3) is undecidable, GPCP(5) and PCP(7) are undecidable.