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

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 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].
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Contributeur : Olivier Finkel <>
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Dernière modification le : vendredi 4 janvier 2019 - 17:32:32
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Olivier Finkel. Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language. International Journal of Foundations of Computer Science, World Scientific Publishing, 2012, 23 (7), p. 1481-1497. ⟨10.1142/S0129054112400606⟩. ⟨hal-00543309v2⟩

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