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

Olivier Finkel (1)

1 ELM - Équipe de Logique Mathématique

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 Büchi 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].

Mots clés: points of continuity; Decision problems; infinite Post Correspondence Problem; analytical hierarchy; high undecidability; infinitary rational relations; omega rational functions; topology; points of continuity.

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]; Complexité [cs.CC]

Olivier Finkel. Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language. 2010. ⟨hal-00543309v1⟩