2022/07/28 by Guillaume Chapuy, Guillem Perarnau, Chapuy, Guillaume +1
Biochemistry, Genetics and Molecular Biology · Computer Science · #Algorithms and Data Compression #Combinatorics (math.CO) #DNA and Biological Computing #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Probability (math.PR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2207.14108
openalex publication_date 2022/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that a uniformly random automaton with n states on a 2-letter alphabet has a synchronizing word of length O(n1/2log 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(nlog3n) due to Nicaud (2016). 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+ε)log2(n), for any ε>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.