Classical and Effective Descriptive Complexities of omega-Powers

Final Version, published in A.P.A.L. This paper is an extended version of a conference paper which appeared in the Proceedings of the 16th EACSL Annual Conference on Computer Science and Logic, CSL 07. Part of the results in this paper have been also presented at the International Conference Computability in Europe, CiE 07, Siena, Italy, June 2007.International audienceWe prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel set as the range of a closed subset of the Baire space by a continuous bijection. This leads us to prove closure properties for the classes Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical hierarchy 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 by the second author in [Omega-Powers and Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p. 1210-1232]

Classical and Effective Descriptive Complexities of omega-Powers

We prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel set as the range of a closed subset of the Baire space by a continuous bijection. This leads us to prove closure properties for the classes Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical hierarchy 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 by the second author in [Omega-Powers and Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p. 1210-1232].Comment: Final Version, published in A.P.A.L. This paper is an extended version of a conference paper which appeared in the Proceedings of the 16th EACSL Annual Conference on Computer Science and Logic, CSL 07. Part of the results in this paper have been also presented at the International Conference Computability in Europe, CiE 07, Siena, Italy, June 200

Topological Complexity of omega-Powers : Extended Abstract

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

Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

Polishness of some topologies related to word or tree automata

We prove that the B\"uchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies are not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the proceedings of the 26th EACSL Annual Conference on Computer Science and Logic, CSL 2017. The main addition with regard to the conference paper consists in the study of the B\"uchi topology and of the Muller topology in the case of a space of trees, which now forms Section

On Recognizable Languages of Infinite Pictures

An erratum is added at the end of the paper: The supremum of the set of Borel ranks of Büchi recognizable languages of infinite pictures is not the first non recursive ordinal $\omega_1^{CK}$ but an ordinal $\gamma^1_2$ which is strictly greater than the ordinal $\omega_1^{CK}$. This follows from a result proved by Kechris, Marker and Sami (JSL 1989).International audienceIn a recent paper, Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for infinite words. The authors asked for comparing the tiling system acceptance with an acceptance of pictures row by row using an automaton model over ordinal words of length $\omega^2$. We give in this paper a solution to this problem, showing that all languages of infinite pictures which are accepted row by row by Büchi or Choueka automata reading words of length $\omega^2$ are Büchi recognized by a finite tiling system, but the converse is not true. We give also the answer to two other questions which were raised by Altenbernd, Thomas and Wöhrle, showing that it is undecidable whether a Büchi recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable)

Modelling local autonomy and dependence through cooperative relations

Purpose Given the spread of multi‐level governance tools, interaction between local and regional governments has become an important mechanism for service delivery and the implementation of public policies. The paper will empirically test a model of cooperative relations affecting local governments and thus having impact on local autonomy and dependence. Design/methodology/approach This article takes previous typologies as a starting point to theoretically build and empirically test a model of cooperative relations based on two selected indicators: the degree of autonomy-dependence and the degree of rigidity-flexibility of the cooperative system. In a second step, the authors stimulate the model numerically to match real data on it coming from South‐European local governments to assess the functioning of the theory in a concrete space and time. Findings The combination of the aforementioned concepts, creates a four-option theoretical model that describes four possible situations where cooperative intergovernmental relations can be empirically placed. Originality/value This article points out the need to incorporate empirical studies to trace the characteristics and evolution of the cooperative relations between local governments and upper tiers. This is particularly relevant if we are referring to mechanisms that can vary through time. In the current big and open data era, this empirical process will become easier and more affordable. In this context, local government studies benefit from particular features that improve the operation of this kind of analysis: large 'N' configuration (a large number of units to be included) and a reasonable equivalence in concepts and bodies that allows comparability