# The Complexity of Infinite Computations In Models of Set Theory

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1 ELM - Équipe de Logique Mathématique

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 $LA$ and the infinitary rational relation $LB$ are ${\bf \Pi} 2^0$-sets, and 2 There is a model $V 2$ of ZFC in which the omega-language $LA$ and the infinitary rational relation $LB$ 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.

Keywords : independence from the axiomatic system ZFC Infinite words omega-languages 1-counter automaton 2-tape automaton two-dimensional words tiling systems Cantor topology topological complexity Borel sets largest effective coanalytic set models of set theory independence from the axiomatic system ZFC.

Author: ** Olivier Finkel - **

Source: https://hal.archives-ouvertes.fr/