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Classical Descriptive Set Theory

/pdf/comptes-rendus-de-l-academie-des-sciences-48l5a7hs5u.pdf

Comptes Rendus de l'Académie des Sciences.

It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game - with n nodes and m distinct values (aka colours or priorities) - is proven to be in the class of fixed parameter tractable (FPT) problems when parameterised over m. Both results improve known bounds, from runtime nO(√n) to O(nlog(m)+6) and from an XP-algorithm with runtime O(nΘ(m)) for fixed parameter m to an FPT-algorithm with runtime O(n5)+g(m), for some function g depending on m only. As an application it is proven that coloured Muller games with n nodes and m colours can be decided in time O((mm · n)5); it is also shown that this bound cannot be improved to O((2m · n)c), for any c, unless FPT = W[1].

Deciding parity games in quasipolynomial time

Fundamenta Mathematicae私抄 : 退任の辞に代えて

Descriptive Set Theory

https://hal.archives-ouvertes.fr/docs/00/10/36/79/PDF/bor-cf.pdf

Borel hierarchy and omega context free languages

The extension of the Wagner hierarchy to blind counter automata accepting infinite words with a Muller acceptance condition is effective. We determine precisely this hierarchy.

An Effective Extension of the Wagner Hierarchy to Blind Counter Automata

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.

/pdf/topology-and-ambiguity-in-omega-context-free-languages-3b2vdy9c3r.pdf

Topology and Ambiguity in Omega Context Free Languages

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.

https://hal.archives-ouvertes.fr/hal-00349761/document

Highly Undecidable Problems For Infinite Computations

We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20]

Olivier Finkel

Papers

Borel hierarchy and omega context free languages

An Effective Extension of the Wagner Hierarchy to Blind Counter Automata

Topology and Ambiguity in Omega Context Free Languages

Highly Undecidable Problems For Infinite Computations

On the topological complexity of infinitary rational relations