Some Problems in Automata Theory Which Depend on the Models of Set Theory - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue RAIRO - Theoretical Informatics and Applications (RAIRO: ITA) Année : 2011

Some Problems in Automata Theory Which Depend on the Models of Set Theory

Olivier Finkel
  • Fonction : Auteur
  • PersonId : 834645

Résumé

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an omega-language $L(A)$ accepted by a Büchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter Büchi automaton $A$ such that the cardinality of the complement $L(A)^-$ of the omega-language $L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which $L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A)^-$ has cardinal $2^{\aleph_0}$. (3). There is a model $V_3$ of ZFC in which $L(A)^-$ has cardinal $\aleph_1$ with $\aleph_0<\aleph_1<2^{\aleph_0}$. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton $B$. As a corollary, this proves that the Continuum Hypothesis may be not satisfied for complements of 1-counter omega-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter omega-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter omega-language (respectively, infinitary rational relation) is countable is in $\Sigma_3^1 \setminus (\Pi_2^1 \cup \Sigma_2^1)$. This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).
Fichier principal
Vignette du fichier
Automata-theory-and-set-theory.pdf (123.81 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00614658 , version 1 (14-08-2011)

Identifiants

Citer

Olivier Finkel. Some Problems in Automata Theory Which Depend on the Models of Set Theory. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), 2011, 45 (4), pp.383 - 397. ⟨10.1051/ita/2011113⟩. ⟨hal-00614658⟩
131 Consultations
240 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More