Synchronizing Automata and the Černý Conjecture

Volkov, M. V.

doi:10.1007/978-3-540-88282-4_4

Cited by 255 publications

(254 citation statements)

References 28 publications

Supporting

Mentioning

248

Contrasting

Unclassified

Order By: Relevance

“…This conjecture was formulated byČerný in 1964 (Černý 1964), and is considered the most longstanding open problem in the combinatorial theory of finite automata. 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).…”

mentioning

confidence: 99%

“…The standard approach is to construct the power automaton and to compute the shortest path from the whole set state to a singleton (Sandberg 2005;Trahtman 2006;Kudłacik et al 2012;Volkov 2008). Most naturally, the breadth-first-search (BFS) method is used which starts from the set of all states of the given automaton and forms images applying letter transformations until a singleton is reached.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Computing the shortest reset words of synchronizing automata

2013

View full text Add to dashboard Cite

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.

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

Computing the shortest reset words of synchronizing automata

2013

View full text Add to dashboard Cite

show abstract

“…In the study of synchronizing sets, the search for synchronizing words of minimal length in a prefix complete code is tightly related to that of synchronizing words of minimal length for synchronizing complete deterministic automata and the celebratedČerný Conjecture [15] (see also [2,3,4,7,8,9,10,11,12,15,19,20,23] for some results on the problem). In particular, in [3] (see also [4]), Béal and Perrin have proved that a complete synchronizing prefix code X on an alphabet of d letters with n code-words has a synchronizing word of length O(n 2 ).…”

Section: Introductionmentioning

confidence: 99%

On incomplete and synchronizing finite sets

Carpi

D'Alessandro

2017

Theoretical Computer Science

View full text Add to dashboard Cite

This paper situates itself in the theory of variable length codes and of finite automata where the concepts of completeness and synchronization play a central role. In this theoretical setting, we investigate the problem of finding upper bounds to the minimal length of synchronizing words and incompletable words of a finite language X in terms of the length of the words of X. This problem is related to two wellknown conjectures formulated byČerný and Restivo, respectively. In particular, if Restivo's conjecture is true, our main result provides a quadratic bound for the minimal length of a synchronizing pair of any finite synchronizing complete code with respect to the maximal length of its words.

show abstract

“…This conjecture states that any synchronizing automata A with n states admits at least a reset word w with |w| ≤ (n − 1) 2 . For more information on synchronizing automata we refer the reader to the survey by Volkov [12]. In what follows, when there is no ambiguity on the choice of the action δ of the automaton, we use the notation q · u instead of δ(q, u).…”

Section: Introductionmentioning

confidence: 99%

Ideal regular languages and strongly connected synchronizing automata

Reis

Rodaro

2016

Theoretical Computer Science

View full text Add to dashboard Cite

Abstract. We introduce the notion of reset left regular decomposition of an ideal regular language and we prove that the category formed by these decompositions with certain morphisms is equivalent to the category of strongly connected synchronizing automata. We show that each ideal regular language has at least a reset left regular decomposition. As a consequence, each ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal. This notion allows the reformulation ofČerný's conjecture in pure language theoretic terms.

show abstract

Synchronizing Automata and the Černý Conjecture

Cited by 255 publications

References 28 publications

Computing the shortest reset words of synchronizing automata

Computing the shortest reset words of synchronizing automata

On incomplete and synchronizing finite sets

Ideal regular languages and strongly connected synchronizing automata

Contact Info

Product

Resources

About