Varieties of Formal Languages

Pin, Jean Eric; Miller, Raymond E.

doi:10.1007/978-1-4613-2215-3

Cited by 460 publications

(368 citation statements)

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“…See the book by Pin [11] for a detailed treatment of the matters discussed in this subsection and the next; here we give a brief informal review. A monoid is a set M together with an associative operation for which there is an identity element 1 ∈ M. If A is a finite alphabet, then A * is a monoid with concatenation of words as the multiplication.…”

Section: Finite Monoids and Regular Languagesmentioning

confidence: 99%

An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Krebs

Straubing

2017

ACM Trans. Comput. Logic

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We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that two-sided semidirect products with J as the right-hand factor preserve decidability. , the second author provided an algebraic characterization of the levels of the hierarchy, showing that they correspond to the levels of weakly iterated two-sided semidirect products of the pseudovariety J of finite J -trivial monoids. This still left open the problem of decidability of the hierarchy: effectively determining from a description of a regular language the lowest level of the hierarchy to which the language belongs. This problem was apparently solved in Almeida-Weil [2], from which explicit identities for the iterated product varieties can be extracted. However, an error in that paper called the correctness of these results into question. Here we show that the given identities do indeed characterize these pseudovarieties. In particular, since it is possible to verify effectively whether a given finite monoid satisfies one of these identities, we obtain an effective procedure for exactly determining the alternation depth of a regular language definable in two-variable logic. ACM Subject Classification

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Section: Finite Monoids and Regular Languagesmentioning

confidence: 99%

An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Krebs

Straubing

2017

ACM Trans. Comput. Logic

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“…Good sources for semigroup theory, in particular finite semigroup theory, are [1,16,18,25,40]. Here we introduce some standard notions and terminology.…”

Section: Preliminariesmentioning

confidence: 99%

Representation theory of finite semigroups, semigroup radicals and formal language theory

Almeida

Margolis

Steinberg

et al. 2008

Trans. Amer. Math. Soc.

121

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Abstract. In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; andČerný's conjecture for an important class of automata.

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“…In a regular D-class, any H-class containing an idempotent is a maximal subgroup of the semigroup. Moreover, two regular H-classes contained in a same D-class are isomorphic (as groups), see for instance [20,Chapter 3 Proposition 1.8]. This group is called the characteristic group of the regular D-class.…”

Section: Figurementioning

confidence: 99%

“…In general, the structure of a finite semigroup is determined by the Green's relations (denoted R, L, H, D, J ) [20]. Our invariant is the acyclic directed graph whose nodes the non null regular D-classes of S(X) labeled by their rank and their characteristic group.…”

Section: Introductionmentioning

confidence: 99%