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Register a new userOn the Length of the Wadge Hierarchy of Omega Context Free Languages
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Added by
Olivier Finkel
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Olivier Finkel
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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 the 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.
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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].
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.
We prove that $omega$-languages of (non-deterministic) Petri nets and $omega$-languages of (non-deterministic) Turing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of $omega$-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of $omega$-languages of (non-deterministic) Turing machines which also form the class of effective analytic sets. In particular, for each non-null recursive ordinal $alpha < omega_1^{{rm CK}} $ there exist some ${bf Sigma}^0_alpha$-complete and some ${bf Pi}^0_alpha$-complete $omega$-languages of Petri nets, and the supremum of the set of Borel ranks of $omega$-languages of Petri nets is the ordinal $gamma_2^1$, which is strictly greater than the first non-recursive ordinal $omega_1^{{rm CK}}$. We also prove that there are some ${bf Sigma}_1^1$-complete, hence non-Borel, $omega$-languages of Petri nets, and that it is consistent with ZFC that there exist some $omega$-languages of Petri nets which are neither Borel nor ${bf Sigma}_1^1$-complete. This answers the question of the topological complexity of $omega$-languages of (non-deterministic) Petri nets which was left open in [DFR14,FS14].
We study the links between the topological complexity of an omega context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel omega context free languages which are recognized by Buchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation of taking the omega power of a finitary context free language. We prove also that taking the adherence or the delta-limit of a finitary language preserves neither unambiguity nor inherent ambiguity. On the other side we show that methods used in the study of omega context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by Buchi 2-tape automata and we get first results in that direction.
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-deterministic Petri nets are topologically more complex than those accepted by deterministic Petri nets.
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