We prove the following surprising result: there exist a 1-counter B\"uchi automaton and a 2-tape B\"uchi automaton such that the \omega-language of the first and the infinitary rational relation of the second in one model of ZFC are \pi_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an \omega-language accepted by a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by a 2-tape B\"uchi 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 B\"uchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author.

Source : oai:arXiv.org:0910.1268

DOI : 10.2168/LMCS-5(4:4)2009

Volume: Volume 5, Issue 4

Published on: December 21, 2009

Submitted on: January 20, 2009

Keywords: Computer Science - Logic in Computer Science,Computer Science - Computational Complexity,Mathematics - Logic,F.1.1,F.1.3,F.4.1,F.4.3

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