Reset Word Of Length Research Articles

A bound for the length of the shortest reset words for semisimple synchronizing automata via the packing number

We show that if a semisimple synchronizing automaton with $n$ states has a minimal reachable non-unary subset of cardinality $r\ge 2$, then there is a reset word of length at most $(n-1)D(2,r,n)$, where $D(2,r,n)$ is the $2$-packing number for families of $r$-subsets of $[1,n]$.

Between primitive and 2-transitive: Synchronization and its friends

An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture , one of the oldest and most famous problems in automata theory, according to which a synchronizing n -state automaton has a reset word of length at most (n-1)^2 .The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set \Omega is said to synchronize a map f if the monoid \langle G,f\rangle generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.

Synchronizing random automata

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications Conjecture that any synchronizing automaton with n states has a reset word of length (n - 1)(2) was made by. Cerny in 1964. Notwithstanding the numerous attempts made by various researchers this conjecture hasn't been definitively proven yet. In this paper we study a random automaton that is sampled uniformly at random from the set of all automata with n states and m(n) letters. We show that for m(n) > 18 ln n any random automaton is synchronizing with high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton with high probability satisfies the. Cerny conjecture.

Notable trends concerning the synchronization of graphs and automata

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). He had conjectured that it is an upper bound for the length of the shortest synchronizing word for any n-state complete DFA. This simply looking conjecture is now one of the most longstanding open problems in the theory of finite automata. Moreover, the examples of DFA with shortest synchronizing word of length (n − 1) 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). An effective program for search of automata with minimal reset word of relatively great length has studied all automata of size n ≤ 10 for q = 2 (q size of alphabet) and of size n ≤ 7 for q ≤ 4. There are no contradictory examples for the Cerny conjecture in this class of automata. Moreover, the program does not find new examples of DFA with reset word of length (n−1)2 for automata with n > 4 as well as for q > 3. New examples of DFA with shortest synchronizing word of length (n − 1) were found only for n = 3, 4 and for q = 2, 3. And what is more, the examples with minimal length of reset word disappear even for values near the Cerny bound (n− 1) with growth of the size of the automaton as well as of the size of the alphabet. The gap between (n − 1) and the nearest of the minimal lengths of reset word appears for n = 6. The following table displays this interesting trend for the length of minimal reset words less than (n− 1).