Equational Descriptions of Languages

Pin, Jean-Éric

doi:10.1142/s0129054112400497

Cited by 13 publications

(13 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This allows one to characterise the languages of L by a property of their ordered syntactic monoid. Here are three examples of such results, but many more can be found in the literature [23,24].…”

Section: Formal Languagesmentioning

confidence: 99%

Dual Space of a Lattice as the Completion of a Pervin Space

2017

Relational and Algebraic Methods in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Formal Languagesmentioning

confidence: 99%

Dual Space of a Lattice as the Completion of a Pervin Space

2017

Relational and Algebraic Methods in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…The treatment in [25], for which the duality theoretic components have been given above, is the first fully modular treatment and the first to allow the treatment of lattices of recognizable languages without any further properties. We summarize the results and the location of their proofs in the following A full account of the method ensuing from these results will be treated elsewhere, see also [41] and [42]. Some applications have already appeared in the literature, see e.g.…”

Section: Lattices Of Languages Equational Theoriesmentioning

confidence: 99%

Stone duality, topological algebra, and recognition

Gehrke

2016

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognizable subsets of the abstract algebra endowed with certain residuation operations. These results identify a connection between topological algebra as applied in algebra and Stone duality as applied in logic, and show that the notion of recognition originating in computer science is intrinsic to profinite completion in mathematics in general. This connection underlies a number of recent results in automata theory including a generalization of Eilenberg-Reiterman theory for regular languages and a new notion of compact recognition applying beyond the setting of regular languages. The purpose of this paper is to give the underlying duality theoretic result in its general form. Further we illustrate the power of the result by providing a few applications in topological algebra and language theory. In particular, we give a simple proof of the fact that any topological algebra quotient of a profinite algebra which is again based on a Boolean space is in fact a profinite algebra and we derive the conditions dual to the ones of the original Eilenberg theorem in a fully modular manner. We cast our results in the setting of extended Priestley duality for distributive lattices with additional operations as some classes of languages of interest in automata theory fail to be closed under complementation.2010 Mathematics Subject Classification. Primary 06D50,20M35,22A30; keywords: Stone/Priestley duality for lattices with additional operations, topological algebra, automata and recognition, profinite completion.The author acknowledges support from ANR 2010 BLAN 0202 02 FREC.

show abstract

“…It was proved by Almeida [1] to be equal to the topological space underlying the free profinite monoid on A, denoted by A * . We refer to [2,8,9] for more information on this space, but it can be seen as the completion of A * for the profinite metric d defined as follows. A finite monoid M separates two words u and v of A * if there is a monoid morphism ϕ : u,v) , with the usual conventions min ∅ = +∞ and 2 −∞ = 0.…”

Section: Stone Dualitymentioning

confidence: 99%

“…Restricted instances of Result 1 have proved to be very successful long before the result was stated in full generality. It is in particular a powerful tool for characterizing classes of regular languages or for determining the expressive power of various fragments of logic, see the book of Almeida [2] or the survey [9] for more information.…”

mentioning

confidence: 99%

From Ultrafilters on Words to the Expressive Power of a Fragment of Logic

Gehrke

Krebs

2014

Descriptional Complexity of Formal Systems

Self Cite

View full text Add to dashboard Cite

International audienceWe give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving profinite equations characterizing B ∩ Reg via ultrafilter equations satisfied by B. This suffices to establish the decidability of the membership problem for B ∩ Reg.Nous donnons une méthode pour spécifier des équations en ultrafiltres et identifier leurs projections sur l'ensemble des mots profinis. Soit B l'ensemble des langages décrits par des énoncés du premier ordre utilisant des prédicats unaires pour chaque lettre, des prédicats numériques uniformes unaires arbitraires et un prédicat pour la longueur d'un mot. Nous illustrons notre méthode en donnant des équations profinies caractérisant B ∩ Reg à partir d'équations en ultrafiltres satisfaites par B. Cela suffit à établir la décidabilité de l'appartenance à B ∩ Reg

show abstract

Equational Descriptions of Languages

Cited by 13 publications

References 31 publications

Dual Space of a Lattice as the Completion of a Pervin Space

Dual Space of a Lattice as the Completion of a Pervin Space

Stone duality, topological algebra, and recognition

From Ultrafilters on Words to the Expressive Power of a Fragment of Logic

Contact Info

Product

Resources

About