Topological Complexity of Locally Finite omega-Languages

to appear in Archive for Mathematical LogicInternational audienceLocally finite omega-languages were introduced by Ressayre in [Formal Languages defined by the Underlying Structure of their Words, Journal of Symbolic Logic, 53 (4), December 1988, p. 1009-1026]. These languages are defined by local sentences and extend omega-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite omega languages are analytic sets, the class LOC_omega of locally finite omega-languages meets all finite levels of the Borel hierarchy and there exist some locally finite omega-languages which are Borel sets of infinite rank or even analytic but non-Borel sets. This gives partial answers to questions of Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992] and of Duparc, Finkel, and Ressayre [Computer Science and the Fine Structure of Borel Sets, Theoretical Computer Science, Volume 257 (1-2), 2001, p.85-105]

Wadge Degrees of ω\omega-Languages of Petri Nets

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].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326

Borel Ranks and Wadge Degrees of Context Free Omega Languages

We show that, from a topological point of view, considering the Borel and the Wadge hierarchies, 1-counter B\"uchi automata have the same accepting power than Turing machines equipped with a B\"uchi 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 B\"uchi 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]