Short synchronizing words for random automata
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 w of such length, and a state v0, such that w sends all states to v0. Prior to this work, the...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/448799 |
| Acceso en línea: | https://hdl.handle.net/2117/448799 https://dx.doi.org/10.1145/3736722 |
| Access Level: | acceso abierto |
| Palabra clave: | Mathematics of computing Random graphs Theory of computation Formal languages and automata theory Classificació AMS::05 Combinatorics::05C Graph theory Classificació AMS::60 Probability theory and stochastic processes::60C05 Combinatorial probability Classificació AMS::68 Computer science::68Q Theory of computing Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria |
| Sumario: | 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 w of such length, and a state v0, such that w 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 [26]. 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 + e)log2(n), for any e > 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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