An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture

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.

[1]  Ján Cerný,et al.  On directable automata , 1971, Kybernetika (Praha).

[2]  Avraham Trakhtman.  Optimal Estimation on the Order of Local Testability of Finite Automata , 2000, Theor. Comput. Sci..

[3]  J. Pin.  On two Combinatorial Problems Arising from Automata Theory , 1983 .

[4]  David Eppstein,et al.  Reset Sequences for Monotonic Automata , 1990, SIAM J. Comput..

[5]  Zvi Kohavi,et al.  Establishing Certain Bounds Concerning Finite Automata , 1973, J. Comput. Syst. Sci..

[6]  Carl Pixley,et al.  Minimum Length Synchronizing Sequences of Finite State Machine , 1993, 30th ACM/IEEE Design Automation Conference.

[7]  Jarkko Kari,et al.  A Counter Example to a Conjecture Concerning Synchronizing Words in Finite Automata , 2001, Bull. EATCS.

[8]  P. FRANKL,et al.  An Extremal Problem for two Families of Sets , 1982, Eur. J. Comb..

[9]  Sam M. Kim,et al.  A Polynomial Time Algorithm for the Local Testability Problem of Deterministic Finite Automata , 1989, IEEE Trans. Computers.

[10]  Alessandra Cherubini,et al.  An Inverse Automata Algorithm for Recognizing 2-Collapsing Words , 2002, Developments in Language Theory.

[11]  Arto Salomaa,et al.  Generation of Constants and Synchronization of Finite Automata , 2002, J. Univers. Comput. Sci..

[12]  David Lee,et al.  Principles and methods of testing finite state machines-a survey , 1996, Proc. IEEE.

[13]  Balas K. Natarajan.  An algorithmic approach to the automated design of parts orienters , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).