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We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. All these automata are tightly related to primitive digraphs with large exponent. 1 Background and the structure of the paper This paper has arisen from our attempts to find a theoretical explanation for(More)
We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent. A complete deterministic finite automaton (DFA) is a triple A = Q, Σ, δ, where Q and Σ are finite sets called the state set and(More)
We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, q > p. We describe reset thresholds of the colorings of primitive digraphs with exactly one cycle of length p and one cycle of length q. Also, we study reset thresholds of the colorings(More)
For each odd n ≥ 5 we present a synchronizing Eulerian automaton with n states for which the minimum length of reset words is equal to n 2 −3n+4 2. We also discuss various connections between the reset threshold of a synchronizing automaton and a sequence of reachability properties in its underlying graph. A complete deterministic finite automaton A is(More)
A finite set S of words over the alphabet Σ is called non-complete if Fact(S *) = Σ *. A word w ∈ Σ * \ Fact(S *) is said to be uncompletable. We present a series of non-complete sets S k whose minimal uncompletable words have length 5k 2 − 17k + 13, where k ≥ 4 is the maximal length of words in S k. This is an infinite series of counterexamples to(More)
A set of nonnegative matrices M = {M 1 , M 2 ,. .. , M k } is called primitive if there exist indices im is positive (i.e. has all its entries > 0). The length of the shortest such product is called the exponent of M. The concept of primitive sets of matrices comes up in a number of problems within control theory, non-homogeneous Markov chains, automata(More)
Motivated by the Babai conjecture and thě Cern´y conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upper-bounded by 2n 2 − 6n + 5 and can attain the value n(n−1) 2. In addition, we(More)