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- W3100488283 abstract "We present a few classes of synchronizing automata exhibiting certain extremal properties with regard to synchronization. The first is a series of automata with subsets whose shortest extending words are of length $varTheta(n^2)$, where $n$ is the number of states of the automaton. This disproves a conjecture that every subset in a strongly connected synchronizing automaton is $cn$-extendable, for some constant $c$, and in particular, shows that the cubic upper bound on the length of the shortest reset words cannot be improved generally by means of the extension method. A detailed analysis shows that the automata in the series have subsets that require words as long as $n^2/4+O(n)$ in order to be extended by at least one element. We also discuss possible relaxations of the conjecture, and propose the image-extension conjecture, which would lead to a quadratic upper bound on the length of the shortest reset words. In this regard we present another class of automata, which turn out to be counterexamples to a key claim in a recent attempt to improve the Pin-Frankl bound for reset words. Finally, we present two new series of slowly irreducibly synchronizing automata over a ternary alphabet, whose lengths of the shortest reset words are $n^2-3n+3$ and $n^2-3n+2$, respectively. These are the first examples of such series of automata for alphabets of size larger than two." @default.
- W3100488283 created "2020-11-23" @default.
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- W3100488283 date "2015-01-01" @default.
- W3100488283 modified "2023-09-25" @default.
- W3100488283 title "Synchronizing Automata with Extremal Properties" @default.
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- W3100488283 doi "https://doi.org/10.1007/978-3-662-48057-1_26" @default.
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