On Carpi and Alessandro conjecture

The well known open Černý conjecture states that each synchronizing automaton with n states has a synchronizing word of length at most (n−1) . On the other hand, the best known upper bound is cubic of n. Recently, in the paper [1] of Alessandro and Carpi, the authors introduced the new notion of strongly transitivity for automata and conjectured that this property with a help of Extension method allows to get a quadratic upper bound for the length of the shortest synchronizing words. They also confirmed this conjecture for circular automata. We disprove this conjecture and the long-standing Extension conjecture too. We also consider the widely used Extension method and its perspectives. 1 Strongly transitivity and the Černý conjecture Let A = 〈Q,Σ, δ〉 be a complete deterministic finite automaton (DFA), where Q is the state set, Σ is the input alphabet, and δ : Q × Σ → Q is the transition function. The function δ extends uniquely to a function Q × Σ → Q, where Σ stands for the free monoid over Σ ; the latter function is still denoted by δ and λ denotes an empty word. Thus, each word in Σ acts on the set Q via δ . The DFA A is called synchronizing if there exists a word w ∈ Σ whose action resets A , that is, leaves the automaton in one particular state no matter which state in Q it starts at: δ(q, w) = δ(q, w) for all q, q ∈ Q. Any such word w is called a synchronizing word for A . The minimum length of synchronizing words for A is denoted by minsynch(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 survey [12]. Here we discuss one of the main problem in this theory: the Černý conjecture and related problems. In the paper [3] at 1964 Černý conjectured that each synchronizing automaton with n states has a synchronizing word of length at most (n− 1) . He also presented the extremal series of the n-state circular automata with a shortest synchronizing word of length (n − 1) . Thus he proved the lower bound of the conjecture. The conjecture is still open and the best known upper bound for the length of the shortest synchronizing word is n 3 −n 6 . Pin proved this result at 1983 in [8] using combinatorial result of Frankl [5]. Since the lower bound is quadratic and the upper bound is cubic, it is of certain importance to prove a quadratic upper bound. All existing methods for proving the upper bound of minimal length of synchronizing words can be divided to «compress» and «extension» methods. Methods of both types construct a finite ordered collection of words V = (v1, v2, . . . , vm), which concatenation is synchronizing. Let us say that m = |V | is the size of the collection V and LV = maxi |vi| is the length of the collection V . The difference between these types of methods is that the compress collection subsequently compresses the set of states Q to some state p, i.e |Q| > |Q.v1| > |Q.v1v2| > . . . > |Q.v1v2 . . . vm| = |{p}|, while the extension collection subsequently extends some state p to the set of states Q, i.e |{p}| < |p.v 1 | < |p.v 1 v 2 | < . . . < |p.v 1 v 2 . . . v m | = |Q|. Since the size m of the collections can not be more than n− 1, the proof of a quadratic upper bound can be reduced to the proof a linear upper bound for the length of the collection LV . The compress method is used to prove the cubic upper bound n 3 −n 6 in the general case mentioned above. It is also used to prove the Černý conjecture for few «small» classes of automata such as automata with zero, aperiodic automata [13] or interval automata. Since the Černý conjecture is proved for automata with zero, we assume automata is strongly connected in the rest of the paper, otherwise the considered problem can be reduced by using the construction of automaton with zero (see [11] for example). The extension methods seem more productive to prove a quadratic upper bounds. In 1998 Dubuc [4] proved the Černý conjecture for circular automata, i.e. the automata with a letter, which acts as a cyclic substitution. He used an extension method combined with the skilful linear algebra techniques to prove this result. In 2003 Kari [7] proved the Černý conjecture for Eurlian automata using extension method. The quadratic upper bound was also confirmed for the one-cluster automata in the paper [2]. Let us note that it is the largest class of synchronizing automata with proved quadratic upper bound. In 2008 Arturo Carpi and Flavio D’Alessandro introduced the new ideas for constructing the extension collection V of linear length. The ideas are based on the notion of the independent collection (or set) of words. The collection of words W = (w1, w2, . . . , wn) of the n-state automaton A is called independent, if for any two given state s and t there exists an index i such that s.wi = t. The automaton A is called strongly transitive, if it admits some independent collection of words W . It is easy to check, that each synchronizing strongly connected automaton A is strongly transitive. Moreover, if u is synchronizing, then A has an independent collection of length not more than |u| + n − 1. The authors also proved that this bound is tight and if the n-state automaton A is strongly transitive with some independent collection W , then it has a synchronizing word of length not more than (n− 2)(n + LW − 1) + 1. Later they conjectured that each synchronizing automata has an independent collection of linear length. Formally, for some number k > 0 the following kn-Independent-Set conjecture holds true. Conjecture 1 Each strongly connected n-state synchronizing automaton has an independent collection W = (w1, w2, . . . wn) of length less than kn. Since k is a constant, this conjecture implies quadratic upper bound of the minimal length of synchronizing word for all synchronizing automata. If the automaton is circular and a denotes the circular letter, then the independent collection W can be chosen as (λ, a, a, . . . , a). Hence, the 1 ∗ n-Independent-Set conjecture is true for circular automata. This implies the upper bound 2(n − 2)(n − 1) + 1 for this class of automata. Our paper is organized as follows. At first, in the section 2 we consider the Extension Algorithm in the universal form, introduce the knExtension and kn-Balanced conjecture and prove that the last one implies kn-Independent-Set conjecture. After this, in the section 3 we construct a series, which disproves introduced conjectures, in particular, the kn-Independent-Set conjecture of Carpi and Alessandro for each k > 0. Finally, in the section 4 we generalize the disproved conjectures to the «local» form and discuss the perspectives of the extension method.