An efficient algorithm finds noticeable trends and examples concerning the \v{C}erny conjecture

TL;DR

This paper presents an efficient algorithm that identifies notable trends and new examples related to the Cerny conjecture, which concerns the maximum length of shortest synchronizing words in finite automata.

Contribution

The authors developed a quadratic-time algorithm that discovers new automata examples and reveals significant patterns in the minimal synchronizing word lengths.

Findings

Discovered new automata with minimal synchronizing words of length (n-1)^2.

Identified remarkable trends in the length of minimal synchronizing words.

Validated the quadratic complexity of the algorithm for most automata tested.

Abstract

A word w is called synchronizing (recurrent, reset, directed) word of a deterministic finite automaton (DFA) if w sends all states of the automaton on a unique state. Jan Cerny had found in 1964 a sequence of n-state complete DFA with shortest synchronizing word of length (n-1)^2. He had conjectured that it is an upper bound for the length of the shortest synchronizing word for any $n$-state complete DFA. The examples of DFA with shortest synchronizing word of length (n-1)^2 are relatively rare. To the Cerny sequence were added in all examples of Cerny, Piricka and Rosenauerova (1971), of Kari (2001) and of Roman (2004). By help of a program based on some effective algorithms, a wide class of automata of size less than 11 was checked. The order of the algorithm finding synchronizing word is quadratic for overwhelming majority of known to date automata. Some new examples of n-state DFA with minimal synchronizing word of length (n-1)^2 were discovered. The program recognized some remarkable trends concerning the length of the minimal synchronizing word. http://www.cs.biu.ac.il/~trakht/Testas.html