Deterministic Concatenation Product of Languages Recognized by Finite Idempotent Monoids

Branco, Mário J. J.

doi:10.1007/s00233-006-0669-3

Cited by 3 publications

(3 citation statements)

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“…Some specialized topics require even more sophisticated algebraic tools, like the kernel category of a morphism. This is the case for instance for the bideterministic product [9][10][11] or for the marked product of two languages [4]. Another topic that we did not mention at all, but which is highly interesting, is the extension of these results to infinite words or even to words over ordinals or linear orders.…”

Section: Other Variations Recent Advancesmentioning

confidence: 97%

Theme and Variations on the Concatenation Product

2011

Algebraic Informatics

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The concatenation product is one of the most important operations on regular languages. Its study requires sophisticated tools from algebra, finite model theory and profinite topology. This paper surveys research advances on this topic over the last fifty years. The concatenation product plays a key role in two of the most important results of automata theory: Kleene's theorem on regular languages [23] and Schützenberger's theorem on star-free languages [60]. This survey article surveys the most important results and tools related to the concatenation product, including connections with algebra, profinite topology and finite model theory. The paper is organised as follows: Section 1 presents some useful algebraic tools for the study of the concatenation product. Section 2 introduces the main definitions on the product and its variants. The classical results are summarized in Section 3. Sections 4 and 5 are devoted to the study of two algebraic tools: Schützenberger products and relational morphisms. Closure properties form the topic of Section 6. Hierarchies and their connection with finite model theory are presented in Sections 7 and 8. Finally, new directions are suggested in Section 9. 1 The instruments This section is a brief reminder on the algebraic notions needed to study the concatenation product: semigroups and semirings, syntactic ordered monoids, free profinite monoids, equations and identities, varieties and relational morphisms. More information can be found in [1-3, 18, 35, 42, 45]. 1.1 Semigroups and semirings If S is a semigroup, the set P(S) of subsets of S is also a semiring, with union as addition and multiplication defined, for every X, Y ∈ P(S), by XY = {xy | x ∈ X, y ∈ Y } In this semiring, the empty set is the zero and for this reason, is denoted by 0. It is also convenient to denote simply by x a singleton {x}. If k is a semiring, we denote by M n (k) be the semiring of square matrices of size n with entries in k.

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Section: Other Variations Recent Advancesmentioning

confidence: 97%

Theme and Variations on the Concatenation Product

2011

Algebraic Informatics

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“…We consider in this section two variants of the product introduced by Schützenberger in [15]: unambiguous and deterministic products. These products were also studied in [2,3,4,5,9,11,12,13].…”

Section: Some Variants Of the Productmentioning

confidence: 99%

An Explicit Formula for the Intersection of Two Polynomials of Regular Languages

2013

Lecture Notes in Computer Science

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Let L be a set of regular languages of A *. An L-polynomial is a finite union of products of the form L0a1L1 • • • anLn, where each ai is a letter of A and each Li is a language of L. We give an explicit formula for computing the intersection of two L-polynomials. Contrary to Arfi's formula (1991) for the same purpose, our formula does not use complementation and only requires union, intersection and quotients. Our result also implies that if L is closed under union, intersection and quotient, then its polynomial closure, its unambiguous polynomial closure and its left [right] deterministic polynomial closure are closed under the same operations.

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“…In this paper we refine this by showing that the basis may be chosen to consist solely of pseudoidentities between finite products of regular pseudowords, whenever V is a pseudovariety in the interval [Sl, DS] that is closed under bideterministic product; motivated by this result, we call a finite product of regular pseudowords a multiregular pseudoword. Conversely, we give a proof that every pseudovariety of semigroups that has a basis of pseudoidentities between multiregular pseudowords is closed under bideterministic product; one may say that this converse is already hidden in the paper [31], where pseudovarieties closed under bideterministic product were first introduced, but note that neither in [31] nor in the sequels [15,22,16,17] the profinite approach is explicitly present. In view of these results, one may argue that closure under bideterministic product is a relatively mild condition to impose upon a pseudovariety.…”

Section: Introductionmentioning

confidence: 99%

Bases for pseudovarieties closed under bideterministic product

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2019

Algebra Univers.

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We show that if V is a semigroup pseudovariety containing the finite semilattices and contained in DS, then it has a basis of pseudoidentities between finite products of regular pseudowords if, and only if, the corresponding variety of languages is closed under bideterministic product. The key to this equivalence is a weak generalization of the existence and uniqueness of J-reduced factorizations. This equational approach is used to address the locality of some pseudovarieties. In particular, it is shown that DH ∩ ECom is local, for any group pseudovariety H.

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Deterministic Concatenation Product of Languages Recognized by Finite Idempotent Monoids

Cited by 3 publications

References 12 publications

Theme and Variations on the Concatenation Product

Theme and Variations on the Concatenation Product

An Explicit Formula for the Intersection of Two Polynomials of Regular Languages

Bases for pseudovarieties closed under bideterministic product

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