Citation Context

...e uniform distribution on E” therefore means that the probability that P holds tends to 1 as n tends to infinity, when for each n we consider the uniform distribution on En. The reader is referred to =-=[5]-=- for more information on combinatorial probabilistic models. Random mappings and random p-mappings. A random mapping of size 3The size is often clear in the context (number of nodes in a tree, ...) an...

Citation Context

...1). Then the word uv synchronizes x1,. . . ,xi+1 and has length at most ` · i. The result follows by induction. Random mappings and random p-mappings have been studied intensively in the literature =-=[7, 4, 10]-=-, using probabilistic techniques or methods from analytic combinatorics. In this section, we only recall basic properties of the typical number of cyclic points and of the typical height of a random p...

Citation Context

...tures in automata theory. Though established for important subclasses of automata, Černý’s conjecture remains open in the general case. The best known upper bound, established in the early eighties =-=[12, 6]-=-, is 16 (n 3−n). For a more detailed account on Černý’s conjecture, we refer the interested reader to Volkov’s article [14]. Probabilistic Černý conjecture. Considering Černý’s conjecture from a...

Citation Context

...1). Then the word uv synchronizes x1,. . . ,xi+1 and has length at most ` · i. The result follows by induction. Random mappings and random p-mappings have been studied intensively in the literature =-=[7, 4, 10]-=-, using probabilistic techniques or methods from analytic combinatorics. In this section, we only recall basic properties of the typical number of cyclic points and of the typical height of a random p...

Citation Context

...tures in automata theory. Though established for important subclasses of automata, Černý’s conjecture remains open in the general case. The best known upper bound, established in the early eighties =-=[12, 6]-=-, is 16 (n 3−n). For a more detailed account on Černý’s conjecture, we refer the interested reader to Volkov’s article [14]. Probabilistic Černý conjecture. Considering Černý’s conjecture from a...

Citation Context

...rd for that automaton. The remainder of this section is devoted to a more detailed proof of Theorem 4. For the presentation, we will follow an idea used by Karp in his article on random direct graphs =-=[8]-=-: we start from an automaton with no transition, then add new random transitions during at each step of the construction, progressively improving the synchronization. From now on, we fix a real > 0 ...

Citation Context

...ess of the starting position. This notion, first formalized by Černý in the sixties, arises naturally in automata theory and its extensions, and plays an important role in several application areas =-=[14]-=-. Perhaps one of the reasons synchronizing automata are still intensively studied in theoretical computer science is a beautiful question asked by Černý [13] back in 1964: “Does every synchronizing ...

Citation Context

...ortant role in several application areas [14]. Perhaps one of the reasons synchronizing automata are still intensively studied in theoretical computer science is a beautiful question asked by Černý =-=[13]-=- back in 1964: “Does every synchronizing n-state automaton admits a synchronizing word of length at most (n − 1)2?” The bound of (n− 1)2, as shown by Černý, is best possible. This question, known as...

Citation Context

...ore precise statements4, but we will only need the following results in the sequel5. 4For instance, limit distributions of some parameters [5] or even a notion of continuous limit for random mappings =-=[1]-=-. 5The bound are not tight, we choose them for readability. 4 Lemma 2 Let > 0 be a fixed real number. For n large enough, the probability that a random p-mapping of size n has more than n 1 2+ cycl...

Citation Context

...utomata are synchronized by a short synchronizing word, of length sublinear in the number of states. Note that simulating the second question is nontrivial, as finding the shortest reset word is hard =-=[11]-=-; the best experimental results we are aware of were obtained by Kisielewicz, Kowalski, and Szykula [9]. Our results. In this paper we give a positive answer to Question 2 when the automaton is chosen...

Citation Context

...e that simulating the second question is nontrivial, as finding the shortest reset word is hard [11]; the best experimental results we are aware of were obtained by Kisielewicz, Kowalski, and Szykula =-=[9]-=-. Our results. In this paper we give a positive answer to Question 2 when the automaton is chosen uniformly among deterministic and complete n-state automata on an alphabet with at least two letters. ...

Citation Context

...conjecture, we refer the interested reader to Volkov’s article [14]. Probabilistic Černý conjecture. Considering Černý’s conjecture from a probabilistic point of view is natural (see for instance =-=[3]-=-), and leads to the following questions: ∗This work is supported by the French National Agency (ANR) through ANR-10-LABX-58 and through ANR-2010-BLAN-0204. 1 ar X iv :1 40 4. 69 62 v1s[ cs .FL ]s28sA ...

Citation Context

...1)2 with high probability? Here, with high probability means “with probability that tends to 1 as n goes to infinity”. Berlinkov recently made a breakthrough by giving a positive answer to Question 1 =-=[2]-=-: he proved that the probability that a random automaton is not synchronizing is in O(n− 12 |A|), for an alphabet A with at least two letters. Question 2 can be simulated and experimental evidence sug...

Citation Context

...1). Then the word uv synchronizes x1,. . . ,xi+1 and has length at most ` · i. The result follows by induction. Random mappings and random p-mappings have been studied intensively in the literature =-=[7, 4, 10]-=-, using probabilistic techniques or methods from analytic combinatorics. In this section, we only recall basic properties of the typical number of cyclic points and of the typical height of a random p...