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Slowly synchronizing automata with fixed alphabet size
[article]

2017
*
arXiv
*
pre-print

Furthermore, we give constructions of

arXiv:1609.06853v5
fatcat:fsu3nmkapncsldrgpmbzuuw2vq
*automata**with*any number of states, and 3, 4, or 5 symbols, which*synchronize**slowly*, namely in n^2 - 3n + O(1) steps. ... In this paper, we investigate the role of the*alphabet**size*. For each possible*alphabet**size*, we count DFAs on n < 6 states which*synchronize*in (n-1)^2 - e steps, for all e < 2 n/2 . ... Moreover, we also investigate bounds on*synchronization*lengths for*fixed**alphabet**size*. ...##
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Primitive digraphs with large exponents and slowly synchronizing automata

2013
*
Journal of Mathematical Sciences
*

All these

doi:10.1007/s10958-013-1392-8
fatcat:vz6twmlz4ndrvabobdyg6xphaa
*automata*are tightly related to primitive digraphs*with*large exponent. ... 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. ... letters, then a bound of the same magnitude (but probably*with*a worse constant) exists also for the reset threshold of*synchronizing*n-*automata**with*any*fixed**size*of the input*alphabet*. ...##
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Slowly synchronizing automata with zero and incomplete sets
[article]

2009
*
arXiv
*
pre-print

Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic

arXiv:0907.4576v1
fatcat:5ehyld5mgvhlnfkqu7rho4dumy
*automata**with*zero whose shortest*synchronizing*word has length n^2/4+n/2-1. ... Thus a natural question is to determine the maximum length c m (n) of shortest reset words for n-state*synchronizing**automata**with*zero over a*fixed*m-lettered input*alphabet*as a function of n. ... An essential feature of the example in Fig. 1 is that the input*alphabet**size*grows*with*the number of states. ...##
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On the synchronization of planar automata
[article]

2016
*
arXiv
*
pre-print

Planar

arXiv:1612.04462v1
fatcat:sezrkso3urfulal2ukhbwa6luq
*automata*seems to be representative of the*synchronizing*behavior of deterministic finite state*automata*. ... This evidence amounts to show that the class of planar*automata*is representative of the algorithmic hardness of*synchronization*... However, it seems that all the sequences of*slowly**synchronizing**automata*can be obtained this way: By locally perturbing a sequence of*slowly**synchronizing*planar*automata*. ...##
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A Fast Algorithm Finding the Shortest Reset Words
[chapter]

2013
*
Lecture Notes in Computer Science
*

*With*our algorithm we are able to consider much larger sample of

*automata*

*with*up to n=300 states. ... In this paper we present a new fast algorithm finding minimal reset words for finite

*synchronizing*

*automata*. The problem is know to be computationally hard, and our algorithm is exponential. ... Curiously, it works in polynomial time for known

*slowly*

*synchronizing*

*automata*series [1] . ...

##
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Reset Complexity of Ideal Languages Over a Binary Alphabet

2019
*
International Journal of Foundations of Computer Science
*

We compare the reset complexity and the state complexity for languages related to

doi:10.1142/s0129054119400343
fatcat:ldw2uwuyebcnbn3zwwgdd72vwm
*slowly**synchronizing**automata*. ... We prove PSPACE-completeness of checking whether a given ideal language serves as the language of reset words for some automaton*with*at most four states over a binary*alphabet*. ... State Complexity of Languages Related to*Slowly**Synchronizing**Automata*In this section, we study series of "*slowly*"*synchronizing**automata*. ...##
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Slowly Synchronizing Automata and Digraphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

These

doi:10.1007/978-3-642-15155-2_7
fatcat:73yaaviz2jhutojfizqji5jv24
*automata*are closely related to primitive digraphs*with*large exponent. ... 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. ... Observe that in general underlying digraphs of*slowly**synchronizing**automata*may admit colorings*with*rather short reset words. ...##
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Effective synchronizing algorithms

2012
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Expert systems with applications
*

The main problem in this approach is to find the shortest possible sequence which

doi:10.1016/j.eswa.2012.04.079
fatcat:o5eiggqrabd6ncid66b56zbyty
*synchronizes*the automaton being a model of the system under test. This can be done*with*a*synchronizing*algorithm. ... In this paper we analyze the*synchronizing*algorithms described in the literature, both exact (*with*exponential runtime) and greedy (polynomial). ... After these simple optimizations, all*automata*of*size*up to 26 (or more) should be handled easily (space complexity becomes a bigger problem in case of*slowly*-*synchronizing**automata*). ...##
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Synchronizing random automata

2010
*
Discrete Mathematics & Theoretical Computer Science
*

In this paper we study a random automaton that is sampled uniformly at random from the set of all

doi:10.46298/dmtcs.514
fatcat:lber3dkjlfcy7i4ce7gu2yzfhm
*automata**with*n states and m(n) letters. ... We show that for m(n) > 18 ln n any random automaton is*synchronizing**with*high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton*with*high probability satisfies the. ... • What*size*of an*alphabet*implies that almost all*automata**with*the*alphabet*of this*size*are*synchronizing*and comply*with*theČerný conjecture? ...##
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Synchronizing Automata of Bounded Rank
[chapter]

2012
*
Lecture Notes in Computer Science
*

We reduce the problem of

doi:10.1007/978-3-642-31606-7_15
fatcat:rdvmslklunbp3doq7pwf56s2k4
*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 2 − r − 1 Moreover, if r > n 2 , such*automata*...*Synchronizing**automata**with*a letter of deficiency 2 were considered in [3] while in [2] a series of*slowly**synchronizing**automata*in which all letters deficiency 2 was reported. ...##
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Sync-Maximal Permutation Groups Equal Primitive Permutation Groups
[article]

2021
*
arXiv
*
pre-print

Hence, it is natural to investigate

arXiv:2111.13527v1
fatcat:tx75rsdsrnghvano3cd5yrmdau
*synchronizing**automata*extremal*with*this property, i.e., such that the minimal deterministic automaton for the set of*synchronizing*words has 2^n - n states. ... The*size*of a recognizing automaton for the set of*synchronizing*words is linked to computational problems related to*synchronization*and to the length of*synchronizing*words. ... Unfortunately, due to space, I could not discuss all of them, in particular the connections to decoders and probabilistic investigations on the length of*synchronizing*words. ...##
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On Randomized Generation of Slowly Synchronizing Automata

2018
*
International Symposium on Mathematical Foundations of Computer Science
*

We present a constructive randomized procedure to generate

doi:10.4230/lipics.mfcs.2018.48
dblp:conf/mfcs/CatalanoJ18
fatcat:du76kamufvfyzdlrfh7ncogrcu
*synchronizing**automata*of that kind*with*(potentially) large*alphabet**size*based on recent results on primitive sets of matrices. ... Motivated by the randomized generation of*slowly**synchronizing**automata*, we study*automata*made of permutation letters and a merging letter of rank n − 1. ... Almost all the families of*slowly**synchronizing**automata*listed above are closely related to the Černý automaton C(n) = {a, b}, where a is the cycle over n vertices and b the letter that*fixes*all the ...##
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Careful synchronization of partial deterministic finite automata
[article]

2020
*
arXiv
*
pre-print

Our experiments demonstrate that this approach gives satisfactory results for

arXiv:2002.01045v2
fatcat:olexdmnsnfgoldgcdrsn4bxtia
*automata**with*up to 100 states even if very modest computational resources are used. ... We compare our results*with*the ones obtained by the first author for exact*synchronization*, which is another version of*synchronization*studied in the literature, and draw some theoretical conclusions ... Benchmarks and*slowly**synchronizing**automata*Besides experimenting*with*random PFAs, we have tested our approach on certain provably '*slowly**synchronizing*'*automata*, that is, the ones*with*the minimum ...##
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State Complexity of the Set of Synchronizing Words for Circular Automata and Automata over Binary Alphabets
[article]

2020
*
arXiv
*
pre-print

Most

arXiv:2011.14404v1
fatcat:4azyecvvyjaunhxusmp65f2lyy
*slowly**synchronizing**automata*over binary*alphabets*are circular, i.e., containing a letter permuting the states in a single cycle, and their set of*synchronizing*words has maximal state complexity ... We derive that over a binary*alphabet*every completely reachable automaton must be circular, a consequence of a structural result stating that completely reachable*automata*over strictly less letters than ... These properties are also shared by a wealth of different*slowly**synchronizing**automata*[1, 2, 12, 13] . Our criteria apply to all the*automata*mentioned in this previous work. ...##
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A series of slowly synchronizing automata with a zero state over a small alphabet

2008
*
Information and Computation
*

the shortest reset word for n -state

doi:10.1016/j.ic.2008.03.020
fatcat:eh3jv6qcuncnxbpetomngttzwu
*synchronizing*0 -*automata*over a*fixed*input*alphabet*. ... An essential feature of the example in Fig. 1 is that the input*alphabet**size*grows*with*the number of states. ...
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