Extended Temporal Logic on Finite Words and Wreath Product of Monoids with Distinguished Generators

Ésik, Zoltán

doi:10.7146/brics.v9i47.21762

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…For this reason, there are generalizations in various directions that relax one or more of the requirements on the classes of languages to which the theory applies, e.g. [40,43,44,45,21,55,36].…”

Section: 4mentioning

confidence: 99%

Stone duality, topological algebra, and recognition

Gehrke

2016

Journal of Pure and Applied Algebra

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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.

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Section: 4mentioning

confidence: 99%

Stone duality, topological algebra, and recognition

Gehrke

2016

Journal of Pure and Applied Algebra

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“…In this section we mention a connection between the Linear Temporal Logic (LTL) and the concept of the literal idempotency. The expressive power of certain variants of the temporal logic were successfully characterized applying algebraic methods; in particular, the concept of the syntactic monoid was used in [2,3,12]. In the center of our interest is the expressive power of LTL formulas which do not use the "next" operator.…”

Section: Linear Temporal Logic Without the Operator "Next"mentioning

confidence: 99%

On Varieties of Literally Idempotent Languages

Klíma

Polák

2008

RAIRO-Theor. Inf. Appl.

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A language L ⊆ A * is literally idempotent in case that ua 2 v ∈ L if and only if uav ∈ L, for each u, v ∈ A * , a ∈ A. Varieties of literally idempotent languages result naturally by taking all literally idempotent languages in a classical (positive) variety or by considering a certain closure operator on classes of languages. We initiate the systematic study of such varieties. Various classes of literally idempotent languages can be characterized using syntactic methods. A starting example is the class of all finite unions of B * 1 B * 2. .. B * k where B1,. .. , B k are subsets of a given alphabet A.

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“…Closure under inverses of morphisms can be relaxed by requiring only closure under inverses of length-preserving, length-increasing or length-decreasing morphisms, leading to the notions of lp-variety, li-variety and ld-variety, respectively. These notions were introduced independently by Esik [27] and Straubing [85].…”

Section: Varieties Of Languages and Varieties Of Finite Monoidsmentioning

confidence: 99%

“…Let us just mention for the record that a generalization of Eilenberg's variety theorem also holds for lp-varieties, li-varieties and ld-varieties. See [27,10,49,65,66,85] for more details.…”

Section: Is a Variety Of Languages Moreover The Correspondence V → mentioning

confidence: 99%

Open Problems About Regular Languages, 35 Years Later

2017

The Role of Theory in Computer Science

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In December 1979, Janusz A. Brzozowski presented a selection of six open problems about regular languages and mentioned two other problems in the conclusion of his article. These problems have been the source of some of the greatest breakthroughs in automata theory over the past 35 years. This survey article summarizes the state of the art on these questions and the hopes for the next 35 years. Thirty-five years ago, at a conference in Santa Barbara in December 1979, Janusz A. Brzozowski [8] presented a selection of six open problems about regular languages and mentioned two other topics in the conclusion of his article. These six open problems were, in order, star height, restricted star height, group complexity, star removal, regularity of non-counting classes and optimality of prefix codes. The two other topics were the limitedness problem and the dotdepth hierarchy. These problems proved to be very influential in the development of automata theory and were the source of critical breakthroughs. The aim of this paper is to survey these results, to describe their impact on current research and to outline some hopes for the next thirty-five years. Due to the lack of space, the dot-depth hierarchy is treated in a separate article [61].

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Extended Temporal Logic on Finite Words and Wreath Product of Monoids with Distinguished Generators

Abstract: Abstract. We associate a modal operator with each language belonging to a given class of regular languages and use the (reverse) wreath product of monoids with distinguished generators to characterize the expressive power of the resulting logic.

Cited by 5 publications

References 11 publications

Stone duality, topological algebra, and recognition

Stone duality, topological algebra, and recognition

On Varieties of Literally Idempotent Languages

Open Problems About Regular Languages, 35 Years Later

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