On the Interplay Between Babai and Černý’s Conjectures

Abstract

Abstract. Motivated by the Babai conjecture and theČerný conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upperbounded by 2n 2 − 6n + 5 and can attain the value. In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter Background and OverviewCompletely reachable automata, i.e., deterministic finite automata in which every non-empty subset of the state set occurs as the image of the whole state set under the action of a suitable input word, appeared in the study of descriptional complexity of formal languages [26] and in relation to theČerný conjecture [13]. In [6] an emphasis has been made on automata in this class with minimal transition monoid size. In the present paper we focus on automata being in a sense the extreme opposites of those studied in [6], namely, on automata of maximal transition monoid size. In other words, we consider automata with full transition monoid, i.e., transition monoid equal to the full monoid of transformations of the state set; clearly, automata with this property are completely reachable. There are several reasons justifying special attention to automata with full transition monoid. First, as observed in [6], the membership problem for this class of automata is decidable in polynomial time (of the size of the input automaton) while ⋆ Vladimir Gusev and Mikhail V.

A Characterization of Completely Reachable Automata

“…The following example, also taken from [1], demonstrates that the condition of Theorem 1 is not necessary. a [2] , a [3] a [3] a [1,2] a [1,2] a [1] a [2] , a [3] a [2] , a [1] , a [1,2] 1 2 3 a [1] a [2] a [3] Fig. 1.…”

“…If all SCCs of Γ 1 (A ) are singletons, we also set Γ (A ) := Γ 1 (A ) and the process stops with FAILURE. Except for these two extreme cases, we extend the a [1] a [2] a [3] {1,2} a [1,2] graph Γ 1 (A ) as follows. Let Q 2 be the collection of the vertex sets of all at least 2-element SCCs of the graph Γ 1 (A ) and let W 2 (A ) stand for the set of all words of defect 2 with respect to A .…”

Automata specialities: Cerny automata

“…One can useČerný automata to illustrate those concepts. The interested reader can consult the master degree thesis of the first author [4].In this paper we focussed on their decomposability, but we could focus on several other interesting properties (see for instance [12] and [15]) becauseČerný automata are real peculiarities.…”

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

“…Indeed on the one hand, the best upper bound known on the reset threshold of any synchronizing n-state automaton is cubic in n [25,12,33], while on the other hand automata having quadratic reset threshold, called extremal automata, are very difficult to find and few of them are known (see e.g. [14,9,29,21]). Some of these families have been found by Ananichev et.…”