Syntactic Semigroups

Pin, Jean-Éric

doi:10.1007/978-3-642-59136-5_10

Cited by 179 publications

(170 citation statements)

References 129 publications

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“…The set of all transformations of Q induced in D by non-empty words is the transition semigroup of D. This semigroup is a subsemigroup of T Q . If D is minimal, its transition semigroup is isomorphic to the syntactic semigroup of the language L(D) [14,15]. A language is regular if and only if its syntactic semigroup is finite.…”

Section: Terminology and Notationmentioning

confidence: 99%

“…It has been suggested that other measures of complexity may also be useful [2], in particular, the syntactic complexity of a regular language which is the cardinality of its syntactic semigroup [15]. This is the same as the cardinality of the transition semigroup of a minimal DFA accepting the language, and it is this latter representation that we use here.…”

Section: Introductionmentioning

confidence: 99%

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Large Aperiodic Semigroups

Brzozowski

Szykuła

2014

Implementation and Application of Automata

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Abstract. The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of a minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having n left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with n states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For n 4 there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that 2 n − 1 is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.

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Section: Terminology and Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Large Aperiodic Semigroups

Brzozowski

Szykuła

2014

Implementation and Application of Automata

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“…× M k . A profound introduction on recognizability of languages by monoids (or semigroups) and syntactic equivalency is given in [13]. One of the well-known pumping lemmas for regular languages is:…”

Section: Regular Languages and Finite Monoidsmentioning

confidence: 99%

On the maximality of languages with combined types of code properties

Kari

Konstantinidis

Kopecki

2014

Theoretical Computer Science

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We consider the decision problem of whether or not a given regular language is maximal with respect to certain combined types of code properties. In the recent past, there have been a few formal methods for defining code properties, such as trajectory-based codes and transducer-based codes, that allow one to decide the maximality problem, including the case where maximality is tested with respect to combined properties within these formal methods. The property of "decoding delay 1" is not known to be definable with these methods, but it is known that the maximality of this property alone is decidable for regular languages. Here, we consider the problem of deciding maximality of a regular language with respect to decoding delay 1 and a transducer-based property, such as suffix-free, overlap-free, and error-detection properties.

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“…Let L be a language at level 1 2 of the STH. Languages at level 1 2 are the polynomial closure of languages at level 0 [13]. The polynomial closure of level 0 results in languages of the form Σ * a 1 Σ * .…”

Section: Comparing Classes Of Regular Languages and Spider Diagramsmentioning

confidence: 99%

“…This paper builds on our previous work [4,5] and provides a proof that star-free regular languages are definable in spider diagrams of order, when augmented with a product operator. Star-free languages may be described by regular expressions without the use of the Kleene star, a fact from which the name of the language class derives [13]. For example, the language a * over the alphabet Σ = {a, b} is star free as it may be written as the star-free expression ∅b∅ i.e.…”

Section: Introductionmentioning

confidence: 99%

Spider Diagrams of Order and a Hierarchy of Star-Free Regular Languages

Delaney

Taylor

Thompson

Diagrammatic Representation and Inference

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Abstract. The spider diagram logic forms a fragment of constraint diagram logic and is designed to be primarily used as a diagrammatic software specification tool. Our interest is in using the logical basis of spider diagrams and the existing known equivalences between certain logics, formal language theory classes and some automata to inform the development of diagrammatic logic. Such developments could have many advantages, one of which would be aiding software engineers who are familiar with formal languages and automata to more intuitively understand diagrammatic logics. In this paper we consider relationships between spider diagrams of order (an extension of spider diagrams) and the star-free subset of regular languages. We extend the concept of the language of a spider diagram to encompass languages over arbitrary alphabets. Furthermore, the product of spider diagrams is introduced. This operator is the diagrammatic analogue of language concatenation. We establish that star-free languages are definable by spider diagrams of order equipped with the product operator and, based on this relationship, spider diagrams of order are as expressive as first order monadic logic of order.

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Syntactic Semigroups

Cited by 179 publications

References 129 publications

Large Aperiodic Semigroups

Large Aperiodic Semigroups

On the maximality of languages with combined types of code properties

Spider Diagrams of Order and a Hierarchy of Star-Free Regular Languages

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