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Slowly synchronizing automata with fixed alphabet size [article]

Michiel de Bondt, Henk Don, Hans Zantema
2017 arXiv   pre-print
Furthermore, we give constructions of 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.  ... 
arXiv:1609.06853v5 fatcat:fsu3nmkapncsldrgpmbzuuw2vq

Primitive digraphs with large exponents and slowly synchronizing automata

D. S. Ananichev, M. V. Volkov, V. V. Gusev
2013 Journal of Mathematical Sciences  
All these 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.  ... 
doi:10.1007/s10958-013-1392-8 fatcat:vz6twmlz4ndrvabobdyg6xphaa

Slowly synchronizing automata with zero and incomplete sets [article]

E. V. Pribavkina
2009 arXiv   pre-print
Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic 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.  ... 
arXiv:0907.4576v1 fatcat:5ehyld5mgvhlnfkqu7rho4dumy

On the synchronization of planar automata [article]

J. Andres Montoya, Christian Nolasco
2016 arXiv   pre-print
Planar 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.  ... 
arXiv:1612.04462v1 fatcat:sezrkso3urfulal2ukhbwa6luq

A Fast Algorithm Finding the Shortest Reset Words [chapter]

Andrzej Kisielewicz, Jakub Kowalski, Marek Szykuła
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] .  ... 
doi:10.1007/978-3-642-38768-5_18 fatcat:zdses6z64ngoplkcgtbwg7q2by

Reset Complexity of Ideal Languages Over a Binary Alphabet

Marina Maslennikova
2019 International Journal of Foundations of Computer Science  
We compare the reset complexity and the state complexity for languages related to 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.  ... 
doi:10.1142/s0129054119400343 fatcat:ldw2uwuyebcnbn3zwwgdd72vwm

Slowly Synchronizing Automata and Digraphs [chapter]

Dmitry Ananichev, Vladimir Gusev, Mikhail Volkov
2010 Lecture Notes in Computer Science  
These 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.  ... 
doi:10.1007/978-3-642-15155-2_7 fatcat:73yaaviz2jhutojfizqji5jv24

Effective synchronizing algorithms

R. Kudłacik, A. Roman, H. Wagner
2012 Expert systems with applications  
The main problem in this approach is to find the shortest possible sequence which 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).  ... 
doi:10.1016/j.eswa.2012.04.079 fatcat:o5eiggqrabd6ncid66b56zbyty

Synchronizing random automata

Evgeny Skvortsov, Yulia Zaks
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 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?  ... 
doi:10.46298/dmtcs.514 fatcat:lber3dkjlfcy7i4ce7gu2yzfhm

Synchronizing Automata of Bounded Rank [chapter]

Vladimir V. Gusev
2012 Lecture Notes in Computer Science  
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 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.  ... 
doi:10.1007/978-3-642-31606-7_15 fatcat:rdvmslklunbp3doq7pwf56s2k4

Sync-Maximal Permutation Groups Equal Primitive Permutation Groups [article]

Stefan Hoffmann
2021 arXiv   pre-print
Hence, it is natural to investigate 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.  ... 
arXiv:2111.13527v1 fatcat:tx75rsdsrnghvano3cd5yrmdau

On Randomized Generation of Slowly Synchronizing Automata

Costanza Catalano, Raphaël M. Jungers, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
We present a constructive randomized procedure to generate 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  ... 
doi:10.4230/lipics.mfcs.2018.48 dblp:conf/mfcs/CatalanoJ18 fatcat:du76kamufvfyzdlrfh7ncogrcu

Careful synchronization of partial deterministic finite automata [article]

Hanan Shabana, Mikhail Volkov
2020 arXiv   pre-print
Our experiments demonstrate that this approach gives satisfactory results for 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  ... 
arXiv:2002.01045v2 fatcat:olexdmnsnfgoldgcdrsn4bxtia

State Complexity of the Set of Synchronizing Words for Circular Automata and Automata over Binary Alphabets [article]

Stefan Hoffmann
2020 arXiv   pre-print
Most 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.  ... 
arXiv:2011.14404v1 fatcat:4azyecvvyjaunhxusmp65f2lyy

A series of slowly synchronizing automata with a zero state over a small alphabet

P.V. Martugin
2008 Information and Computation  
the shortest reset word for n -state 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.  ... 
doi:10.1016/j.ic.2008.03.020 fatcat:eh3jv6qcuncnxbpetomngttzwu
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