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تسجيل مستخدم جديدTopological Complexity of Context-Free omega-Languages: A Survey
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Olivier Finkel
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We survey recent results on the topological complexity of context-free omega-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free omega-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of ambiguity, and the special case of omega-powers.
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This is an extended abstract presenting new results on the topological complexity of omega-powers (which are included in a paper Classical and effective descriptive complexities of omega-powers available from arXiv:0708.4176) and reflecting also some
open questions which were discussed during the Dagstuhl seminar on Topological and Game-Theoretic Aspects of Infinite Computations 29.06.08 - 04.07.08.
We show that, from a topological point of view, considering the Borel and the Wadge hierarchies, 1-counter Buchi automata have the same accepting power than Turing machines equipped with a Buchi acceptance condition. In particular, for every non null
recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free languages accepted by 1-counter Buchi automata, and the supremum of the set of Borel ranks of context free omega languages is the ordinal gamma^1_2 which is strictly greater than the first non recursive ordinal. This very surprising result gives answers to questions of H. Lescow and W. Thomas [Logical Specifications of Infinite Computations, In:A Decade of Concurrency, LNCS 803, Springer, 1994, p. 583-621].
An {omega}-language is a set of infinite words over a finite alphabet X. We consider the class of recursive {omega}-languages, i.e. the class of {omega}-languages accepted by Turing machines with a Buchi acceptance condition, which is also the class
{Sigma}11 of (effective) analytic subsets of X{omega} for some finite alphabet X. We investigate here the notion of ambiguity for recursive {omega}-languages with regard to acceptance by Buchi Turing machines. We first present in detail essentials on the literature on {omega}-languages accepted by Turing Machines. Then we give a complete and broad view on the notion of ambiguity and unambiguity of Buchi Turing machines and of the {omega}-languages they accept. To obtain our new results, we make use of results and methods of effective descriptive set theory.
We prove in this paper that the length of the Wadge hierarchy of omega context free languages is greater than the Cantor ordinal epsilon_omega, which is the omega-th fixed point of the ordinal exponentiation of base omega. The same result holds for t
he conciliating Wadge hierarchy, defined by J. Duparc, of infinitary context free languages, studied by D. Beauquier. We show also that there exist some omega context free languages which are Sigma^0_omega-complete Borel sets, improving previous results on omega context free languages and the Borel hierarchy.
We show that there are $Sigma_3^0$-complete languages of infinite words accepted by non-deterministic Petri nets with Buchi acceptance condition, or equivalently by Buchi blind counter automata. This shows that omega-languages accepted by non-determi
nistic Petri nets are topologically more complex than those accepted by deterministic Petri nets.
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