Slowly synchronizing automata with zero and incomplete sets

arXiv:0907.4576v1 [cs.FL] 27 Jul 2009 Slowly Synchronizing Automata with Zero and Incomplete Sets Elena V. Pribavkina Ural State University, 620083 Ekaterinburg, Russia elena.pribavkina@usu.ru Abstract. Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic automata with zero whose shortest synchronizing word has length n2 4 + n 2 −1. KEYWORDS: synchronizing automata, shortest synchronizing words, automata with zero, incomplete sets, incompletable words. 1 Introduction Recall that a deterministic finite automaton A = ⟨Q, A, δ⟩is defined by specify- ing a finite state set Q, an input alphabet A, and a transition function δ : Q×A → Q. The function δ naturally extends to the free monoid A∗; this extension is still denoted by δ. An automaton A = ⟨Q, A, δ⟩is said to be synchronizing (or reset) if there is a synchronizing (reset) word, that is a word w ∈A∗which takes all the states of A to a particular one: δ(q, w) = δ(q′, w) for all q, q′ ∈Q. Reset automata turn out to have various applications in different fields such as model-based testing of reactive systems, robotics, dna-computing, symbolic dynamics. In view of the applications an important question is about the length of the shortest reset word for a given synchronizing automaton. This issue has been widely studied over the past forty years, especially in connection with the famous ˇCern´y conjecture [1] which states that any n-state synchronizing automa- ton possesses a synchronizing word of length at most (n −1)2. This conjecture has been proved for a large number of classes of synchronizing automata, nev- ertheless in general it remains one of the most longstanding open problems in automata theory. For more details see the surveys [5,9,10]. It is known (see for example [11, Proposition 3]) that the proof of the ˇCern´y’s conjecture reduces to proving it in two particular cases: for reset automata whose underlying graph is strongly-connected, i. e. each state is reachable from any other one, and for reset automata with zero, i. e. with a particular state 0 such that δ(0, a) = 0 for any a ∈A. Thus obtaining bounds for the maximal length of shortest synchronizing words for the class of reset automata with zero is a rather natural and interesting problem. It is clear that any synchronizing automaton with zero possesses a unique zero state, and any synchronizing word brings the automaton in the zero state. A rather simple argument shows that the length of a synchronizing word for a given n-state reset automaton with zero is at most n(n−1) 2 , see e. g. [8]. This bound is tight, since for each n there is an n-state reset automaton with zero and n −1 input letters whose shortest reset word has length n(n−1) 2 (Fig. 1). 0 1 2 3 · · · n-2 n-1 a1 a2 a2 a3 a3 an−1 an−1 A A \ {a1, a2} A \ {a2, a3} A \ {a3, a4} A \ {an−2, an−1} A \ {an−1} Fig. 1. An n-state reset automaton with zero over an n −1-lettered alphabet whose shortest reset word is of length n(n−1) 2 . An essential feature of the example in Fig. 1 is that the input alphabet size grows with the number of states. This contrasts with the aforementioned examples due to ˇCern´y [1] in which the alphabet does not depend on the state number. Thus a natural question is to determine the maximum length cm(n) of shortest reset words for n-state synchronizing automata with zero over a fixed m-lettered input alphabet as a function of n. Recently in [4] with the help of computer experiments P. Martjugin found a series of n-state automata with zero over a binary alphabet whose shortest reset words have length n2 4 + o(n2). The main result of [4] can be stated as follows: Theorem 1. For each integer n ≥8 there exists a synchronizing n-state au- tomaton with zero over a binary alphabet whose shortest synchronizing word has length l n2+6n−16 4 m . Note that the construction from [4] is not trivial and beside that, it can not be extended on larger alphabets. Let us explain what we mean by such an extension. We say that a synchronizing automaton A = ⟨Q, A, δ⟩is proper, if each letter of the alphabet A appears in each word synchronizing this automaton. Putting this another way, each letter is essential for synchronization of such an automaton. Naturally, it is the class of proper automata for which the problem of estimation of the function cm(n) should be considered, but adding new letters to the automata from Theorem 1 violates this property. The main result of the present paper is the following Theorem 2. Let A be an alphabet with |A| ≥2. For any integer k > |A| there is a proper synchronizing automaton with zero and n = 2k states over the alphabet A whose shortest synchronizing word has length n2 4 + n 2 −1. Our construction leads to the same growth rate of the length of the shortest reset word as the construction from [4], but essentially differs from it since there are no limitations on the number of input letters. Another important feature is that is was found not by

…(Full text truncated)…

This content is AI-processed based on ArXiv data.