Slowly Synchronizing Automata and Digraphs

Ananichev, D. S.; Gusev, Vladimir V.; Volkov, M. V.

doi:10.1007/978-3-642-15155-2_7

Cited by 46 publications

(95 citation statements)

References 21 publications

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“…As a concrete demonstration of our modified approach, we exhibit a series of slowly synchronizing Eulerian automata whose reset threshold is twice as large as the reset threshold of automata that can be obtained by a direct application of techniques from [1]. We believe that the method suggested in this paper can find a number of other applications and its further development may shed a new light on the properties of synchronizing automata.…”

Section: Background and Overviewmentioning

confidence: 92%

“…Our first attempt was following an approach from [1]. In that paper, several examples of slowly synchronizing automata, which had been discovered in the course of a massive computational experiment, have been related to known examples of primitive graphs with large exponent from [5] and then have been expanded to infinite series.…”

Section: Background and Overviewmentioning

confidence: 99%

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Lower Bounds for the Length of Reset Words in Eulerian Automata

Gusev

2011

Lecture Notes in Computer Science

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Section: Background and Overviewmentioning

confidence: 92%

Section: Background and Overviewmentioning

confidence: 99%

Lower Bounds for the Length of Reset Words in Eulerian Automata

Gusev

2011

Lecture Notes in Computer Science

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“…So far, the conjecture has been proved only for some special classes of automata and a general cubic upper bound (n 3 − n)/6 has been established (see Volkov (2008) for an excellent survey of the results). Using computers the conjecture has been verified for small automata with 2 letters and n ≤ 11 states (Kisielewicz and Szykuła 2013) (and with k ≤ 4 letters and n ≤ 7 states (Trahtman 2006); see also (Ananichev et al 2010(Ananichev et al , 2012 for n = 9 states). It is known that, in general, the problem is computationally hard, since it involves an NP-hard decision problem.…”

mentioning

confidence: 93%

“…also (Berlinkov 2010) and (Martyugin 2009(Martyugin , 2011]. On the other hand, there are several theoretical and experimental results showing that most automata are synchronizing (Berlinkov 2013) and most of them have relatively short reset words (Ananichev et al 2010;Skvortsov and Tipikin 2011).…”

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confidence: 99%

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Computing the shortest reset words of synchronizing automata

2013

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In this paper we give the details of our new algorithm for finding minimal reset words of finite synchronizing automata. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional breadth-first-search and radix (Patricia) tries to store and compare subsets. A good performance is due to a number of heuristics we apply and describe here in a suitable detail. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with up to n = 350 states and up to k = 10 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word ≈ 2.5 √ n − 5 for binary automata and show that the error of this estimate is sufficiently small. Experiments for automata with more than two input letters show certain trends with the same general pattern.

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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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Slowly Synchronizing Automata and Digraphs

Abstract: We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent.

Cited by 46 publications

References 21 publications

Lower Bounds for the Length of Reset Words in Eulerian Automata

Lower Bounds for the Length of Reset Words in Eulerian Automata

Computing the shortest reset words of synchronizing automata

Words of Minimum Rank in Deterministic Finite Automata

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