2026-08-16T15:58:03Z https://drops.dagstuhl.de/oai

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Lower Bounds on Avoiding Thresholds Ferens, Robert Szykuła, Marek Vorel, Vojtěch avoiding word Černý conjecture rank conjecture reset threshold reset word synchronizing automaton synchronizing word For a DFA, a word avoids a subset of states, if after reading that word the automaton cannot be in any state from the subset regardless of its initial state. A subset that admits an avoiding word is avoidable. The k-avoiding threshold of a DFA is the smallest number such that every avoidable subset of size k can be avoided with a word no longer than that number. We study the problem of determining the maximum possible k-avoiding thresholds. For every fixed k ≥ 1, we show a general construction of strongly connected DFAs with n states and the k-avoiding threshold in Θ(n^k). This meets the known upper bound for k ≥ 3. For k = 1 and k = 2, the known upper bounds are respectively in 𝒪(n²) and in 𝒪(n³). For k = 1, we show that 2n-3 is attainable for every number of states n in the class of strongly connected synchronizing binary DFAs, which is supposed to be the best possible in the class of all DFAs for n ≥ 8. For k = 2, we show that the conjectured solution for k = 1 (an upper bound in 𝒪(n)) also implies a tight upper bound in 𝒪(n²) on 2-avoiding threshold. Finally, we discuss the possibility of using k-avoiding thresholds of synchronizing automata to improve upper bounds on the length of the shortest reset words. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Robert Ferens and Marek Szykuła and Vojtěch Vorel 2021 Is Part Of LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2021.46 urn:nbn:de:0030-drops-144869 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.46 eng https://creativecommons.org/licenses/by/4.0/legalcode