We prove that, for each countable ordinal ξ≥1, there exist some -complete ω-powers, and some -complete ω-powers, extending previous works on the topological complexity of ω-powers [O. Finkel, Topological properties of omega context free languages, Theoretical Computer Science 262 669–697; O. Finkel, Borel hierarchy and omega context free languages, Theoretical Computer Science 290 1385–1405; O. Finkel, An omega-power of a finitary language which is a borel set of infinite rank, Fundamenta informaticae 62 333–342; D. Lecomte, Sur les ensembles de phrases (...) infinies constructibles a partir d’un dictionnaire sur un alphabet fini, Séminaire d’Initiation a l’Analyse, 1, année 2001–2002; D. Lecomte, Omega-powers and descriptive set theory, Journal of Symbolic Logic 70 1210–1232; J. Duparc, O. Finkel, An ω-Power of a Context-Free Language Which Is Borel Above , in: S. Bold, B. Löwe, T. Räsch, J. van Benthem , in the Proceedings of the international conference foundations of the formal sciences V : Infinite Games, November 26th to 29th, 2004, Bonn, Germany, in: Studies in Logic, vol. 11, College Publications at King’s College, 2007, pp. 109–122]. We prove effective versions of these results; in particular, for each recursive ordinal there exist some recursive sets A2<ω such that , where and denote classes of the hyperarithmetical hierarchy. To do this, we prove effective versions of a result by Kuratowski, describing a set as the range of a closed subset of the Baire space ωω by a continuous bijection. This leads us to prove closure properties for the pointclasses in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some sets of dictionaries considered in [D. Lecomte, Omega-powers and descriptive set theory, Journal of Symbolic Logic 70 1210–1232]. (shrink)

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by realtime 1-counter Büchi automata is equivalent to the determinacy of analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge ofω-languages accepted by 1-counter Büchi automata is equivalent to the analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automatonand a (...) Büchi automatonsuch that: There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge gameW,L()); There exists a model of ZFC in which the Wadge gameW,L()) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge gameW,L()). (shrink)

Locally finite omega languages were introduced by Ressayre [Formal languages defined by the underlying structure of their words. J Symb Log 53(4):1009–1026, 1988]. These languages are defined by local sentences and extend ω-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite ω-languages are analytic sets, the class LOC ω of locally finite ω-languages meets all finite levels of the Borel hierarchy and there exist some locally finite ω-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 et al. [Computer science and the fine structure of Borel sets. Theor Comput Sci 257(1–2):85–105, 2001]. (shrink)

We consider ω n -automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ω n for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω 2 -automatic (resp. ω n -automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not (...) determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ω n -automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a ${\mathrm{\Sigma }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ nor a ${\mathrm{\Pi }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ . We obtain that there exist infinitely many ω n -automatic, hence also ω-tree-automatic, atomless boolean algebras $\mathcal{B}_{n}$ , n ≥ 1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14]. (shrink)

Local sentences and the formal languages they define were introduced by Ressayre in. We prove that locally finite ω‐languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω‐languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non‐null recursive ordinal there exist some ‐complete and some ‐complete locally finite ω‐languages, and the supremum of the set of Borel ranks (...) of locally finite ω‐languages is the ordinal, which is strictly greater than the first non‐recursive ordinal. This gives an answer to the question of the topological complexity of locally finite ω‐languages, which was asked by Simonnet and also by Duparc, Finkel, and Ressayre in. Moreover we show that the topological complexity of a locally finite ω‐language defined by a local sentence φ may depend on the models of the Zermelo‐Fraenkel axiomatic system. Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω‐languages are ‐complete, hence highly undecidable. (shrink)

Local sentences were introduced by Ressayre in [6] who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact strength was left open in [4]. Here we solve this problem, using a combinatorial result of J. H. Schmerl [7]. In fact, we show that the stretching (...) principles are equivalent to the existence of n -Mahlo cardinals for appropriate integers n. This is done by proving first that for every integer n, there is a local sentence φn having well ordered models of order type τ, for every infinite ordinal τ > ω which is not an n -Mahlo cardinal. (shrink)

A dictionary is a set of finite words over some finite alphabet X. The omega-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [Omega-powers and descriptive set theory, JSL 2005] the complexity of the set of dictionaries whose associated omega-powers have a given complexity. In particular, he considered the sets $W({bfSi}^0_{k})$ (respectively, $W({bfPi}^0_{k})$, $W({bfDelta}_1^1)$) of dictionaries $V subseteq 2^star$ whose omega-powers are ${bfSi}^0_{k}$-sets (respectively, ${bfPi}^0_{k}$-sets, Borel sets). In (...) this paper we first establish a new relation between the sets $W({bfSigma}^0_{2})$ and $W({bfDelta}_1^1)$, showing that the set $W({bfDelta}_1^1)$ is ``more complex' than the set $W({bfSigma}^0_{2})$. As an application we improve the lower bound on the complexity of $W({bfDelta}_1^1)$ given by Lecomte. Then we prove that, for every integer $kgeq 2$, (respectively, $kgeq 3$) the set of dictionaries $W({bfPi}^0_{k+1})$ (respectively, $W({bfSi}^0_{k+1})$) is ``more complex' than the set of dictionaries $W({bfPi}^0_{k})$ (respectively, $W({bfSi}^0_{k})$). (shrink)

We prove that, for any natural number \, we can find a finite alphabet \ and a finitary language L over \ accepted by a one-counter automaton, such that the \-power $$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$is \-complete. We prove a similar result for the class \.

We prove that, for any natural number n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}, we can find a finite alphabet Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} and a finitary language L over Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} accepted by a one-counter automaton, such that the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-power L∞:={w0w1…∈Σω∣∀i∈ωwi∈L}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (...) L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$\end{document}is Πn0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Pi }}^0_n$$\end{document}-complete. We prove a similar result for the class Σn0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}^0_n$$\end{document}. (shrink)