Some problems in automata theory which depend on the models of set theory

Équipe de Logique Mathématique, Institut de Mathématiques de Jussieu, CNRS et Université Paris 7, France
finkel@logique.jussieu.fr

Received: 16 April 2010
Accepted: 4 July 2011

Abstract

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 ω-language accepted by a Büchi 1-counter automaton . We prove the following surprising result: there exists a 1-counter Büchi automaton such that the cardinality of the complement of the ω-language is not determined by ZFC: (1) There is a model V₁ of ZFC in which is countable. (2) There is a model V₂ of ZFC in which has cardinal 2^ℵ₀. (3) There is a model V₃ of ZFC in which has cardinal ℵ₁ with ℵ₀ < ℵ₁ < 2^ℵ₀.

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 ω-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 ω-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 ω-language (respectively, infinitary rational relation) is countable is in Σ¹₃\(Π¹₂ ∪ Σ¹₂). This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).