49 research outputs found
Separation Property for wB- and wS-regular Languages
In this paper we show that {\omega}B- and {\omega}S-regular languages satisfy
the following separation-type theorem If L1,L2 are disjoint languages of
{\omega}-words both recognised by {\omega}B- (resp. {\omega}S)-automata then
there exists an {\omega}-regular language Lsep that contains L1, and whose
complement contains L2. In particular, if a language and its complement are
recognised by {\omega}B- (resp. {\omega}S)-automata then the language is
{\omega}-regular. The result is especially interesting because, as shown by
Boja\'nczyk and Colcombet, {\omega}B-regular languages are complements of
{\omega}S-regular languages. Therefore, the above theorem shows that these are
two mutually dual classes that both have the separation property. Usually (e.g.
in descriptive set theory or recursion theory) exactly one class from a pair C,
Cc has the separation property. The proof technique reduces the separation
property for {\omega}-word languages to profinite languages using Ramsey's
theorem and topological methods. After that reduction, the analysis of the
separation property in the profinite monoid is relatively simple. The whole
construction is technically not complicated, moreover it seems to be quite
extensible. The paper uses a framework for the analysis of B- and S-regular
languages in the context of the profinite monoid that was proposed by
Toru\'nczyk
Equational theories of profinite structures
In this paper we consider a general way of constructing profinite struc-
tures based on a given framework - a countable family of objects and a
countable family of recognisers (e.g. formulas). The main theorem states:
A subset of a family of recognisable sets is a lattice if and only if it is
definable by a family of profinite equations.
This result extends Theorem 5.2 from [GGEP08] expressed only for finite words
and morphisms to finite monoids. One of the applications of our theorem is the
situation where objects are finite relational structures and recognisers are
first order sentences. In that setting a simple characterisation of lattices of
first order formulas arise
Index problems for game automata
For a given regular language of infinite trees, one can ask about the minimal
number of priorities needed to recognize this language with a
non-deterministic, alternating, or weak alternating parity automaton. These
questions are known as, respectively, the non-deterministic, alternating, and
weak Rabin-Mostowski index problems. Whether they can be answered effectively
is a long-standing open problem, solved so far only for languages recognizable
by deterministic automata (the alternating variant trivializes).
We investigate a wider class of regular languages, recognizable by so-called
game automata, which can be seen as the closure of deterministic ones under
complementation and composition. Game automata are known to recognize languages
arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is,
the alternating index problem does not trivialize any more.
Our main contribution is that all three index problems are decidable for
languages recognizable by game automata. Additionally, we show that it is
decidable whether a given regular language can be recognized by a game
automaton
Deciding the topological complexity of Büchi languages *
International audienceWe study the topological complexity of languages of Büchi automata on infinite binary trees. We show that such a language is either Borel and WMSO-definable, or Σ 1 1-complete and not WMSO-definable; moreover it can be algorithmically decided which of the two cases holds. The proof relies on a direct reduction to deciding the winner in a finite game with a regular winning condition
On Determinisation of Good-for-Games Automata
International audienceIn this work we study Good-For-Games (GFG) automata over ω-words: non-deterministic automata where the non-determinism can be resolved by a strategy depending only on the prefix of the ω-word read so far. These automata retain some advantages of determinism: they can be composed with games and trees in a sound way, and inclusion LpAq Ě LpBq can be reduced to a parity game over A ˆ B if A is GFG. Therefore, they could be used to some advantage in verification, for instance as solutions to the synthesis problem. The main results of this work answer the question whether parity GFG automata actually present an improvement in terms of state-complexity (the number of states) compared to the deterministic ones. We show that a frontier lies between the Büchi condition, where GFG automata can be determinised with only quadratic blow-up in state-complexity; and the co-Büchi condition, where GFG automata can be exponentially smaller than any deterministic automaton for the same language. We also study the complexity of deciding whether a given automaton is GFG