Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language

TL;DR

This paper proves that several decision problems related to infinite Post Correspondence Problems, rational relations, and omega-rational functions are highly undecidable, specifically $oldsymbol{ ext{Sigma}}_1^1$-complete or $oldsymbol{ ext{Pi}}_1^1$-complete, highlighting their complexity beyond arithmetical hierarchy.

Contribution

It establishes the $oldsymbol{ ext{Sigma}}_1^1$-completeness of the infinite Post Correspondence Problem in regular omega-languages and the $oldsymbol{ ext{Pi}}_1^1$-completeness of related problems about omega-rational functions, providing exact complexities.

Findings

Infinite Post Correspondence Problem in regular omega-languages is $oldsymbol{ ext{Sigma}}_1^1$-complete.

Deciding disjointness of infinitary rational relations is $oldsymbol{ ext{Pi}}_1^1$-complete.

Determining the existence of a point of continuity for omega-rational functions is $oldsymbol{ ext{Sigma}}_1^1$-complete, and checking if the continuity set is omega-regular is $oldsymbol{ ext{Pi}}_1^1$-complete.

Abstract

It was noticed by Harel in [Har86] that "one can define $\Sigma_1^1$-complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular $\omega$-language is $\Sigma_1^1$-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is $\Pi_1^1$-complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about $\omega$-rational functions realized by finite state B\"uchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given $\omega$-rational function is continuous, while we show here that it is $\Sigma_1^1$-complete to determine whether a given $\omega$-rational function has at least one point of continuity. Next we prove that it is $\Pi_1^1$-complete to determine whether the continuity set of a given $\omega$-rational function is $\omega$-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].