Matrix Mortality and the Černý-Pin Conjecture

Almeida, Jorge; Steinberg, Benjamin

doi:10.1007/978-3-642-02737-6_5

Cited by 25 publications

(36 citation statements)

References 30 publications

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“…In this case the stronger result for the variety EDS stated in Theorem 4.2 of [3] holds. However, we do not know whether the condition that E(I i ) is a semilattice for each i = 1, .…”

Section: The Radical Conjecture In Some Casesmentioning

confidence: 89%

“…. k, are semigroups, whence this last theorem implies Theorem 7.3 of [1], but it does not yield the linear bound (n − 1) provided in Theorem 2.6 of [3]. Furthermore, we do not know if Corollary 18 is more general, since it is not known whether the fact that I i \ {0}, for i = 1, .…”

Section: Lemma 14mentioning

confidence: 94%

“…For more information on synchronizing automata we refer the reader to Volkov's survey [21]. In this paper we follow a representation theoretic approach to theČerný conjecture and synchronizing automata initially pursued in [1,3,5,19]. However, our approach has a more ring theoretic flavor making use of the well-known Wedderburn-Artin theory for semisimple rings.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we show how the Wedderburn-Artin Theory may be used to factorize the problem of finding "short" radical words into the sub-problem of finding "short" words that are zero in the projections into the simple components. This approach works for instance in case the 0-minimal ideals in the factor monoids do not contain zero H-classes similarly to [1,3]. In Section 5 we introduce cyclic ideal languages and we characterize them.…”

Section: Introductionmentioning

confidence: 99%

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Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

Almeida¹,

Rodaro²

2014

Developments in Language Theory

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Abstract. We present a ring theoretic approach toČerný's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those inČerný's series. Semisimplicity gives also the advantage of "factorizing" the problem of finding a synchronizing word into the sub-problems of finding "short" words that are zeros into the projection of the simple components in the WedderburnArtin decomposition. In the general case this last problem is related to the search of radical words of length at most (n − 1) 2 where n is the number of states of the automaton. We show that the solution of this "Radical Conjecture" would give an upper bound 2(n−1) 2 for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases andČerný's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

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“…In this case the stronger result for the variety EDS stated in Theorem 4.2 of [3] holds. However, we do not know whether the condition that E(I i ) is a semilattice for each i = 1, .…”

Section: The Radical Conjecture In Some Casesmentioning

confidence: 89%

Section: Lemma 14mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

Almeida¹,

Rodaro²

2014

Developments in Language Theory

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“…A slightly better bound n(7n 2 +6n−16) 48 has been claimed in [14]. Though theČerný conjecture is open in general, it has been confirmed for some restricted classes of synchronizing automata, see [1,6,8,13,16]. For some classes quadratic upper bound is established, see [4,11].…”

Section: Introductionmentioning

confidence: 89%

Synchronizing Automata of Bounded Rank

Gusev

2012

Implementation and Application of Automata

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Abstract. We reduce the problem of synchronization of an n-state automaton with letters of rank at most r < n to the problem of synchronization of an r-state automaton with constraints given by a regular language. Using this technique we construct a series of synchronizing n-state automata in which every letter has rank r < n and whose reset threshold is at least r, such automata are strongly connected.

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Words of Minimum Rank in Deterministic Finite Automata

Kari

Ryzhikov²,

Varonka

2019

Developments in Language Theory

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The rank of a word in a deterministic finite automaton is the size of the image of the whole state set under the mapping defined by this word. We study the length of shortest words of minimum rank in several classes of complete deterministic finite automata, namely, strongly connected and Eulerian automata. A conjecture bounding this length is known as the Rank Conjecture, a generalization of the well known Černý Conjecture. We prove upper bounds on the length of shortest words of minimum rank in automata from the mentioned classes, and provide several families of automata with long words of minimum rank. Some results in this direction are also obtained for automata with rank equal to period (the greatest common divisor of lengths of all cycles) and for circular automata.

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Matrix Mortality and the Černý-Pin Conjecture

Cited by 25 publications

References 30 publications

Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

Synchronizing Automata of Bounded Rank

Words of Minimum Rank in Deterministic Finite Automata

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