871 research outputs found
Varieties of Cost Functions.
Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring
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
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