Slowly Synchronizing Automata and Digraphs

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. 1 Background and overview 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. Let Σ stand for the collection of all finite words over the alphabet Σ, including the empty word. The function δ extends to a function Q × Σ → Q (still denoted by δ) in the following natural way: for every q ∈ Q and w ∈ Σ, we set δ(q, w) = q if w is empty 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 called synchronizing if the action of some word w ∈ Σ resets A , that is, leaves the automaton in one particular state no matter at which state in Q it is applied: 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 length of A . Synchronizing automata serve as transparent and natural models of error-resistant systems in many applications (coding theory, robotics, testing of reactive systems) and also reveal interesting connections with symbolic dynamics and other parts of mathematics. For a brief introduction to the theory of synchronizing automata we refer the reader to the recent surveys [15,22]. Here we focus on the so-called Černý conjecture that constitutes a major open problem in this area. In 1964 Černý [5] constructed for each n > 1 a synchronizing automaton Cn with n states whose reset length is (n− 1) . Soon after that ⋆ Supported by the Russian Foundation for Basic Research, grants 09-01-12142 and 10-01-00524, and by the Federal Education Agency of Russia, grant 2.1.1/3537. he conjectured that these automata represent the worst possible case, that is, every synchronizing automaton with n states can be reset by a word of length (n−1). This simply looking conjecture resists researchers’ efforts for more than 40 years. Even though the conjecture has been confirmed for various restricted classes of synchronizing automata (cf., e.g., [9,6,11,19,20,2,23]), no upper bound of magnitude O(n) for the reset length of n-state synchronizing automata is known in general. The best upper bound achieved so far is n 3 −n 6 , see [13]. One of the difficulties that one encounters when approaching the Černý conjecture is that there are only very few extreme automata, that is, n-state synchronizing automata with reset length (n−1). In fact, the Černý series Cn is the only known infinite series of extreme automata. Besides that, only a few isolated examples of such automata have been found, see [22] for a complete list. Moreover, even slowly synchronizing automata, that is, automata with reset length close to the Černý bound are very rare. This empirical observation is supported also by probabilistic arguments. For instance, the probability that a composition of 2n random self-maps of a set of size n is a constant map tends to 1 as n goes to infinity [10]. In terms of automata, this result means that the reset length of a random automaton with n states and at least 2n input letters does not exceed 2n. For further results of the same flavor see [16]. Thus, there is no hope to find new examples of slowly synchronizing automata by a lucky chance or via a random sampling experiment. We therefore have designed and performed a set of exhaustive search experiments. Our experiments are briefly described in Section 5 while the main body of the paper is devoted to a theoretical analysis of their outcome. We concentrate on two principal issues. In Section 3 we discuss a similarity between the distribution of reset lengths of synchronizing automata and the distribution of exponents of primitive digraphs. Section 4 presents a few series of slowly synchronizing automata. Most of these series have been expanded from new examples discovered in the course of our experiments. In our opinion, the proof technique is also of interest; in fact, we provide a transparent and uniform approach to all sufficiently large slowly synchronizing automata with 2 input letters, both new and already known ones. 2 Preliminaries We start with recalling two elementary and well-known number-theoretic results. Lemma 1 ([14, Theorem 1.0.1]). If k1, . . . , km are positive integers whose greatest common divisor is equal to 1, then there exists an integer N such that every integer larger than N is expressible as a non-negative integer combination of k1, . . . , km. The question of how the least N with the property stated in Lemma 1 depends on the integers k1, . . . , km is known as the diophantine Frobenius problem and in general is highly non-trivial, see [14]. There is, however, a simple special case which we will need in Section 4. Lemma 2 ([14, Theorem 2.1.1]). If k1, k2 are relatively prime positive integers, then k1k2 − k1 − k2 is the largest integer that is not expressible as a non-negative integer combination of k1 and k2. A directed graph (digraph) is a pair D = 〈V,E〉 where V is a finite set and E ⊆ V × V . We refer to elements of V and E as vertices and edges. Observe that our definition allows loops but excludes multiple edges. If v, v ∈ V and e = (v, v) ∈ E, the edge e is said to be outgoing for v. We assume the reader’s acquaintance with basic notions of the theory of directed graphs such as (directed) path, cycle, isomorphism etc. Given a DFA A = 〈Q,Σ, δ〉, its underlying digraph D(A ) hasQ as the vertex set and (q, q) ∈ Q×Q is an edge of D(A ) if and only if q = δ(q, a) for some a ∈ Σ. It is easy to see that a digraph D is isomorphic to the underlying digraph of some DFA if and only if each vertex of D has at least one outgoing edge. In the sequel, we always consider only digraphs satisfying this property. Every DFA A such that D ∼= D(A ) is called a coloring ofD. Thus, every coloring ofD is defined by assigning non-empty sets of labels (colors) from some alphabet Σ to edges of D such that the label sets assigned to the outgoing edges of each vertex form a partition of Σ. Fig. 1 shows a digraph and two of its colorings by Σ = {a, b}.