# The Complexity of Infinite Computations In Models of Set Theory

Abstract : We prove the following surprising result: there exist a 1-counter Büchi automaton A and a 2-tape Büchi automaton B such that : (1) There is a model $V_1$ of ZFC in which the omega-language $L(A)$ and the infinitary rational relation $L(B)$ are ${\bf \Pi}_2^0$-sets, and (2) There is a model $V_2$ of ZFC in which the omega-language $L(A)$ and the infinitary rational relation $L(B)$ are analytic but non Borel sets. This shows that the topological complexity of an omega-language accepted by a 1-counter Büchi automaton or of an infinitary rational relation accepted by a 2-tape Büchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by Büchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied in previous papers [Fin09a, Fin09b].
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https://hal.archives-ouvertes.fr/hal-00422538
Contributeur : Olivier Finkel <>
Soumis le : lundi 19 octobre 2009 - 17:42:11
Dernière modification le : vendredi 4 janvier 2019 - 17:32:32
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 17:37:32

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• HAL Id : hal-00422538, version 3
• ARXIV : 0910.1268

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Olivier Finkel. The Complexity of Infinite Computations In Models of Set Theory. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2009, 5 (4:4), pp.1-19. ⟨hal-00422538v3⟩

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