Synchronizing random automata on a 4-letter alphabet

Zaks, Yu. I.; Skvortsov, Evgeny S.

doi:10.1007/s10958-013-1396-4

Cited by 4 publications

(3 citation statements)

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“…automata taken uniformly at random among the n 2n automata with n states, when n goes to infinity). This question was considered by several authors at least since the 2010's [SZ10,ZS12]. Cameron [Cam13] conjectured that a random automaton is synchronizable with probability tending to 1 when n goes to infinity (i.e., with high probability, or w.h.p.).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Short Synchronizing Words for Random Automata

Chapuy¹,

Perarnau²

2022

Preprint

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We prove that a uniformly random automaton with n states on a 2-letter alphabet has a synchronizing word of length O(n 1/2 log n) with high probability (w.h.p.). That is to say, w.h.p. there exists a word ω of such length, and a state v0, such that ω sends all states to v0. Prior to this work, the best upper bound was the quasilinear bound O(n log 3 n) due to Nicaud [Nic19]. The correct scaling exponent had been subject to various estimates by other authors between 0.5 and 0.56 based on numerical simulations, and our result confirms that the smallest one indeed gives a valid upper bound (with a log factor).Our proof introduces the concept of w-trees, for a word w, that is, automata in which the w-transitions induce a (loop-rooted) tree. We prove a strong structure result that says that, w.h.p., a random automaton on n states is a w-tree for some word w of length at most (1 + ) log 2 (n), for any > 0. The existence of the (random) word w is proved by the probabilistic method. This structure result is key to proving that a short synchronizing word exists.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Short Synchronizing Words for Random Automata

Chapuy¹,

Perarnau²

2022

Preprint

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“…-Skvortsov and Zaks [10] obtained some results for large alphabets (where the number of letters grows with n); -Berlinkov [1] proved that the probability that a random automaton is not synchronizing is in Ø(n −k/2 ), where k is the number of letters, for any k ≥ 2 (this bound is tight for k = 2); -Nicaud [5] proved that with high probability a random automaton admits a reset word of length Ø(n log 3 n), for k ≥ 2 (but with less precise error terms than in [1]).…”

Section: Introductionmentioning

confidence: 99%

Synchronizing Random Almost-Group Automata

Berlinkov

Nicaud²

2018

Implementation and Application of Automata

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In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on n − 1 states, and the others as permutations. We prove that this small change is enough for automata to become synchronizing with high probability.More precisely, we establish that the probability that a strongly connected almost-group automaton is not synchronizing is 2 k−1 −1 n 2(k−1) (1 + o(1)), for a k-letter alphabet.

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“…Skvortsov and Zaks have shown that a random automaton with sufficiently large number of letters is synchronizing with high probability [9]. Later on, they proved that a random 4-letter automaton is synchronizing with a positive probability that is independent of the number of states [13]. The last step in this direction seems to be done by Berlinkov [2].…”

Section: Introductionmentioning

confidence: 99%