The Cerny conjecture for automata respecting intervals of a directed graph

Thě Cern´y's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n−1) 2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors. In this paper we consider finite (deterministic complete) automata A = Q, Σ, δ with the state set Q, the input alphabet Σ, and the transition function δ : Q × Σ → Q. The transition function defines the action of the letters in Σ on Q, which, in this paper, is denoted simply by concatenation: δ(q, a) = qa. The action extends in a natural way to the words in Σ * , and we use the same notation qw = δ(q, w). Accordingly, we write Qw = {qw : q ∈ Q}. The automaton A is called synchronizing if there exists a word w ∈ Σ * such that |Qw| = 1 (in other words, w resets A sending all the states into one particular state). Such a word w is called synchronizing (or a reset word) for A. The problem of synchronization is very natural and its various aspects are considered in the literature (see e.g. [6, 8, 13] for general information and further references). The most famous is the following conjecture due toČern´y. Conjecture (Jaň Cern´y 1964, [4]) If a deterministic finite n-state automaton A = Q, Σ, δ is synchronizing, then it has a reset word of length ≤ (n − 1) 2. This conjecture is considered as one of the most longstanding open problems in the theory of finite automata. The consequent research includes verifying the conjecture for various classes of automata, establishing bounds for the length of reset words, investigating natural algorithmic and complexity questions, and many other related problems. For more detailed discussion we refer the reader to the most recent survey [13] by Volkov. Here, we mention only the most important results proving the conjecture in special cases. In 1978, Pin [9] proved the conjecture for circular automata with a prime number of states (an automaton is circular if it has a letter acting as a cyclic permutation of all the states). In 1990, Eppstein [6] proved the conjecture for orientable au-tomata (preserving a given cyclic ordering of all the states). In 1998, Dubuc [5], completing his earlier …