On the Complexity of the Bi-infinite Post Correspondence Problem

2026-06-09

AI Summary

This study investigates the decision complexity of the doubly infinite Post Correspondence Problem (ZPCP). By constructing a reduction chain from the non-halting problem for Turing machines through the infinite PCP and its s-shift variant, and integrating tools from recursion theory, formal language theory, semi-Thue systems, and shift-invariance analysis, the authors precisely locate ZPCP within the second level of the arithmetical hierarchy: it belongs to Σ₂⁰ but lies outside Π₁⁰ ∪ Σ₁⁰. The work further establishes the undecidability of the infinite PCP under injective morphisms and demonstrates that several related variants are Π₁⁰-complete. These results definitively resolve the long-standing open question regarding the exact arithmetical complexity of ZPCP, thereby filling a significant gap in the theoretical understanding of this fundamental problem.

Abstract

In the bi-infinite Post Correspondence Problem ($\Z$PCP), it is asked whether the same bi-infinite word can be constructed correspondingly from a given finite set of pairs of words. In this article, we study its complexity with respect to the arithmetical hierarchy and prove that it is in $\Si^0_2 \setminus (Π^0_1 \cup \Si^0_1)$ and, therefore, at the level 2 of the arithmetical hierarchy. For the proof, we present a sequence of reductions starting from the nonhalting of the Turing machine all the way to $\Z$PCP via infinite PCP, an $s$-shift infinite PCP and $s$-shift $\Z$PCP for all natural numbers $s$. In the process, we prove that the infinite PCP is undecidable for injective morphisms, and that the infinite injective PCP, $s$-shift infinite PCP, $s$-shift $\Z$PCP and the non-termination problem for (deterministic and reversible) semi-Thue systems are all $Π^0_1$-complete.

Problem

• bi-infinite Post Correspondence Problem
• arithmetical hierarchy
• computational complexity
• undecidability

Innovation

• bi-infinite Post Correspondence Problem
• arithmetical hierarchy
• undecidability
• injective morphisms
• reductions