Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language Olivier Finkel To cite this version: 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 Com- puter Science, World Scientific Publishing, 2012, 23 (7), p. 1481-1497. 10.1142/S0129054112400606. hal-00543309v2 HAL Id: hal-00543309 https://hal.archives-ouvertes.fr/hal-00543309v2 Submitted on 28 Jul 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular ω-Language Olivier Finkel Equipe de Logique Math´ematique CNRS et Universit´eParis Diderot Paris 7 UFR de Math´ematiques case 7012, site Chevaleret, 75205 Paris Cedex 13, France. fi
[email protected] Abstract 1 It was noticed by Harel in [Har86] that “one can define Σ1-complete versions of the well-known Post Correspondence Problem”. We first give a complete proof of this result, showing that the infinite Post 1 Correspondence Problem in a regular ω-language is Σ1-complete, hence located beyond the arithmetical hierarchy and highly undecidable.