The \v{C}erny conjecture

A word w is called a synchronizing word of deterministic finite automaton (DFA) if w sends all states of the automaton to a unique state. In 1964, Jan Cerny discovered a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n-1)2. The Cerny conjecture claims that it is also the upper bound on the length of such a word for a complete DFA. The problem has motivated great and constantly growing number of investigations and generalizations and together with the Road Coloring problem is considered as a most fascinating old problem in the theory of finite automata.

[1]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[2]  Benjamin Weiss,et al.  SIMILARITY OF AUTOMORPHISMS OF THE TORUS , 1970 .

[3]  Dominique Perrin,et al.  A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata , 2009, Developments in Language Theory.

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

[5]  Avraham Trakhtman,et al.  The Road Coloring and Cerny Conjecture , 2008, Stringology.

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

[7]  Flavio D'Alessandro,et al.  Strongly transitive automata and the Černý conjecture , 2009, Acta Informatica.

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

[9]  Flavio D'Alessandro,et al.  The Synchronization Problem for Strongly Transitive Automata , 2008, Developments in Language Theory.

[10]  T. C. Brown,et al.  Foundations of Linear Algebra. , 1968 .

[11]  Jean Berstel,et al.  Rational series and their languages , 1988, EATCS monographs on theoretical computer science.

[12]  Mikhail V. Volkov,et al.  Synchronizing Automata and the Cerny Conjecture , 2008, LATA.

[13]  Dominique Perrin,et al.  A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata , 2011, Int. J. Found. Comput. Sci..

[14]  Joel Friedman.  On the road coloring problem , 1990 .

[15]  A. N. Trahtman,et al.  Modifying the upper bound on the length of minimal synchronizing word , 2011, FCT 2011.

[16]  Benjamin Steinberg.  The Averaging Trick and the Cerný Conjecture , 2010, Developments in Language Theory.