Ambiguity of ω-Languages of Turing Machines

An ω-language is a set of infinite words over a finite alphabet X . We consider the class of recursive ω-languages, i.e. the class of ω-languages accepted by Turing machines with a Büchi acceptance condition, which is also the class Σ 1 of (effective) analytic subsets of Xω for some finite alphabet X . We investigate the notion of ambiguity for recursive ωlanguages with regard to acceptance by Büchi Turing machines. We first show that the class of unambiguous recursive ω-languages is the class ∆ 1 of hyperarithmetical sets. We obtain also that the∆ 1 -subsets ofXω are the subsets ofXω which are accepted by strictly recursive unambiguous finitely branching Büchi transition systems; this provides an effective analogue to a theorem of Arnold on Büchi transition systems [Arn83]. Moreover, using some effective descriptive set theory, we prove that recursive ω-languages satisfy the following dichotomy property. A recursive ω-language L ⊆ Xω is either unambiguous or has a great degree of ambiguity in the following sense: for every Büchi Turing machine T accepting L, there exist infinitely many ω-words which have 20 accepting runs by T . We also show that if L ⊆ Xω is accepted by a Büchi Turing machine T and L is an analytic but non Borel set, then the set of ω-words, which have 20 accepting runs by T , has cardinality 20 . In that case we say that the recursive ω-language L has the maximum degree of ambiguity. We prove that it is Π 2 -complete to determine whether a given recursive ω-language is unambiguous and that it is Σ 2 -complete to determine whether a given recursive ω-language has the maximum degree of ambiguity. Moreover, using some results of set theory, we prove that it is consistent with the axiomatic system ZFC that there exists a recursive ω-language in the Borel class Π 2 , hence of low Borel rank, which has also this maximum degree of ambiguity. 1998 ACM Subject Classification: F.1.1 Models of Computation; F.4.1 Mathematical Logic.