On Bi-infinite and Conjugate Post Correspondence Problems

TL;DR

This paper investigates two variants of the Post Correspondence Problem, analyzing their computational complexity and undecidability, with implications for formal language theory and automata.

Contribution

It introduces and analyzes the bi-infinite and conjugate PCP variants, establishing their complexity classes and undecidability results.

Findings

Bi-infinite PCP is in the class Σ₂⁰ of the arithmetical hierarchy.

Conjugate PCP is undecidable, reduced from the word problem for semi-Thue systems.

Provides a formal framework for understanding these PCP variants.

Abstract

We study two modifications of the Post Correspondence Problem (PCP), namely 1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and 2) the conjugate version, where we require the images of a solution for two given morphisms are conjugates of each other. For the bi-infinite PCP we show that it is in the class $\Sigma_2^0$ of the arithmetical hierarchy and for the conjugate PCP we give an undecidability proof by reducing it to the word problem for a special type of semi-Thue systems.