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Register a new userTopological Complexity of omega-Powers : Extended Abstract
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Olivier Finkel
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Olivier Finkel
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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.
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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.
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.
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.
The $omega$-power of a finitary language L over a finite alphabet $Sigma$ is the language of infinite words over $Sigma$ defined by L $infty$ := {w 0 w 1. .. $in$ $Sigma$ $omega$ | $forall$i $in$ $omega$ w i $in$ L}. The $omega$-powers appear very naturally in Theoretical Computer Science in the characterization of several classes of languages of infinite words accepted by various kinds of automata, like B{u}chi automata or B{u}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $omega$-powers.
An abstract machine is a theoretical model designed to perform a rigorous study of computation. Such a model usually consists of configurations, instructions, programs, inputs and outputs for the machine. In this paper we formalize these notions as a very simple algebraic system, called a configuration machine. If an abstract machine is defined as a configuration machine consisting of primitive recursive functions then the functions computed by the machine are always recursive. The theory of configuration machines provides a useful tool to study universal machines.
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