4,549 research outputs found
On Varieties of Ordered Automata
The Eilenberg correspondence relates varieties of regular languages to
pseudovarieties of finite monoids. Various modifications of this correspondence
have been found with more general classes of regular languages on one hand and
classes of more complex algebraic structures on the other hand. It is also
possible to consider classes of automata instead of algebraic structures as a
natural counterpart of classes of languages. Here we deal with the
correspondence relating positive -varieties of languages to
positive -varieties of ordered automata and we present various
specific instances of this correspondence. These bring certain well-known
results from a new perspective and also some new observations. Moreover,
complexity aspects of the membership problem are discussed both in the
particular examples and in a general setting
On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)
Eilenberg correspondence, based on the concept of syntactic monoids, relates
varieties of regular languages with pseudovarieties of finite monoids. Various
modifications of this correspondence related more general classes of regular
languages with classes of more complex algebraic objects. Such generalized
varieties also have natural counterparts formed by classes of finite automata
equipped with a certain additional algebraic structure. In this survey, we
overview several variants of such varieties of enriched automata.Comment: In Proceedings AFL 2014, arXiv:1405.527
A Fibrational Approach to Automata Theory
For predual categories C and D we establish isomorphisms between opfibrations
representing local varieties of languages in C, local pseudovarieties of
D-monoids, and finitely generated profinite D-monoids. The global sections of
these opfibrations are shown to correspond to varieties of languages in C,
pseudovarieties of D-monoids, and profinite equational theories of D-monoids,
respectively. As an application, we obtain a new proof of Eilenberg's variety
theorem along with several related results, covering varieties of languages and
their coalgebraic modifications, Straubing's C-varieties, fully invariant local
varieties, etc., within a single framework
Varieties of Languages in a Category
Eilenberg's variety theorem, a centerpiece of algebraic automata theory,
establishes a bijective correspondence between varieties of languages and
pseudovarieties of monoids. In the present paper this result is generalized to
an abstract pair of algebraic categories: we introduce varieties of languages
in a category C, and prove that they correspond to pseudovarieties of monoids
in a closed monoidal category D, provided that C and D are dual on the level of
finite objects. By suitable choices of these categories our result uniformly
covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer,
respectively, and yields new Eilenberg-type correspondences
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
Commutative positive varieties of languages
We study the commutative positive varieties of languages closed under various
operations: shuffle, renaming and product over one-letter alphabets
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