An Efficient Algorithm Finds Noticeable Trends and Examples Concerning the Černy Conjecture

Abstract

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. JanČerny 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 theČerny 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.

Computing the shortest reset words of synchronizing automata

“…The average computation time is about 100 up to 1000 times faster than the time of Trahtman's program TESTAS (Trahtman 2003(Trahtman , 2006 for binary automata with 50 states. The reduction to SAT used in (Skvortsov and Tipikin 2011) seemed to be the fastest recently known algorithm and the reported average time for 50 states automata is 2.7 seconds, and for 100 states automata is 70 seconds.…”

“…So far, the conjecture has been proved only for some special classes of automata and a general cubic upper bound (n 3 − n)/6 has been established (see Volkov (2008) for an excellent survey of the results). Using computers the conjecture has been verified for small automata with 2 letters and n ≤ 11 states (Kisielewicz and Szykuła 2013) (and with k ≤ 4 letters and n ≤ 7 states (Trahtman 2006); see also (Ananichev et al 2010(Ananichev et al , 2012 for n = 9 states). It is known that, in general, the problem is computationally hard, since it involves an NP-hard decision problem.…”

“…In computing reset words, either exponential algorithms finding the shortest reset words (Kudłacik et al 2012;Rho and Yu 1993;Sandberg 2005;Skvortsov and Tipikin 2011;Trahtman 2006) or polynomial heuristics finding relatively the shortest reset words (Gerbush and Heeringa 2011;Kudłacik et al 2012;Podolak et al 2012;Roman 2009a,b;Trahtman 2006) are widely used. The standard approach is to construct the power automaton and to compute the shortest path from the whole set state to a singleton (Sandberg 2005;Trahtman 2006;Kudłacik et al 2012;Volkov 2008).…”

“…The standard approach is to construct the power automaton and to compute the shortest path from the whole set state to a singleton (Sandberg 2005;Trahtman 2006;Kudłacik et al 2012;Volkov 2008). Most naturally, the breadth-first-search (BFS) method is used which starts from the set of all states of the given automaton and forms images applying letter transformations until a singleton is reached.…”

Synchronization of Some DFA

“…The best known upper bound is now equal to (n 3 − n)/6 [3,5,7]. By now, this simple looking conjecture with rich and intriguing story of investigations [4,7,10,12] is one of the most longstanding open problems in the theory of finite automata. The existence of some non-trivial subgroup in the transition semigroup of the automaton is essential in many investigations ofČerny conjecture [2,7,8].…”