Primitive digraphs with large exponents and slowly synchronizing automata

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 the results of certain computational experiments in synchronization of finite automata. Recall that 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 the input alphabet respectively, and δ : Q × Σ → Q is a totally defined function called the transition function. As usual, Σ stands for the collection of all finite words over the alphabet Σ, including the empty word 1. The function δ extends to a function Q× Σ → Q (still denoted by δ) as follows: for every q ∈ Q and w ∈ Σ, we set δ(q, w) = q if w = 1 and δ(q, w) = δ(δ(q, v), a) if w = va for some v ∈ Σ and a ∈ Σ. Thus, via δ, every word w ∈ Σ acts on the set Q. A DFA A = 〈Q,Σ, δ〉 is said to be synchronizing if some word w ∈ Σ brings all states to one particular state: δ(q, w) = δ(q, w) for all q, q ∈ Q. Any such word w is said to be a reset word for the DFA. The minimum length of reset words for A is called the reset threshold of A . Synchronizing automata serve as transparent and natural models of errorresistant systems in many applied areas (system and protocol testing, information coding, robotics). At the same time, synchronizing automata surprisingly arise in some parts of pure mathematics (symbolic dynamics, theory of substitution systems and others). Basics of the theory of synchronizing automata as well as its diverse connections and applications are discussed, for instance, in [20, 27]. Here we focus on only one aspect of the theory, namely, on the question of how the reset threshold of a DFA depends on the state number. For brevity, a DFA with n states will be referred to as an n-automaton. In 1964 Černý [9] constructed a series of synchronizing n-automata with reset threshold (n − 1). Soon after that he conjectured that these automata represent the worst possible case with respect to synchronization speed, i.e. that ∗A preliminary version of a part of the results of this paper was published in [3].