The ω-inequality problem for concatenation hierarchies of star-free languages

Almeida, Jorge; Klíma, Ondřej; Kunc, Michal

doi:10.1515/forum-2016-0028

Forum Mathematicum

2017

DOI: 10.1515/forum-2016-0028

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The ω-inequality problem for concatenation hierarchies of star-free languages

Jorge Almeida

Ondřej Klíma

Michal Kunc

Abstract: The problem considered in this paper is whether an inequality of ω-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing-Thérien hierarchy.

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Cited by 2 publications

References 30 publications

(21 reference statements)

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Pro-aperiodic monoids via saturated models

Gool

Steinberg

2019

Isr. J. Math.

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We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for ω-terms that avoids using McCammond's normal forms, as well as new proofs and extensions of other structural results concerning free pro-aperiodic monoids.

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Pro-aperiodic monoids via saturated models

Gool

Steinberg

2019

Isr. J. Math.

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The omega-reducibility of pseudovarieties of ordered monoids representing low levels of concatenation hierarchies

Volaříková

2024

Int. J. Algebra Comput.

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We deal with the question of the [Formula: see text]-reducibility of pseudovarieties of ordered monoids corresponding to levels of concatenation hierarchies of regular languages. A pseudovariety of ordered monoids [Formula: see text] is called [Formula: see text]-reducible if, given a finite ordered monoid M, for every inequality of pseudowords that is valid in [Formula: see text], there exists an inequality of [Formula: see text]-words that is also valid in [Formula: see text] and has the same “imprint” in M. Place and Zeitoun have recently proven the decidability of the membership problem for levels [Formula: see text], 1, [Formula: see text] and [Formula: see text] of concatenation hierarchies with level 0 being a finite Boolean algebra of regular languages closed under quotients. The solutions of these membership problems have been found by considering a more general problem of separation of regular languages and its further generalization — a problem of covering. Following the results of Place and Zeitoun, we prove that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels [Formula: see text] and [Formula: see text] are [Formula: see text]-reducible. As a corollary of these results, we obtain that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels [Formula: see text] and [Formula: see text] are definable by [Formula: see text]-inequalities. Furthermore, in the special case of the Straubing–Thérien hierarchy, using a characterization theorem for level 2 by Place and Zeitoun, we obtain that the level 2 is definable by [Formula: see text]-identities.

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The ω-inequality problem for concatenation hierarchies of star-free languages

Abstract: The problem considered in this paper is whether an inequality of ω-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing-Thérien hierarchy.

Cited by 2 publications

References 30 publications

Pro-aperiodic monoids via saturated models

Pro-aperiodic monoids via saturated models

The omega-reducibility of pseudovarieties of ordered monoids representing low levels of concatenation hierarchies

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