In computer science, more precisely, in the theory ofdeterministic finite automata, a synchronizing word or reset sequence is a word in the input alphabet of the DFA that sends any state of the DFA to one and the same state. That is, if an ensemble of copies of the DFA are each started in different states, and all of the copies process the synchronizing word at the same speed, they will all end up reaching the same state as each other, at the same time as each other. Not every DFA has a synchronizing word; for instance, a DFA with two states, one for words of even length and one for words of odd length, can never be synchronized.
Existence
Given a DFA, the problem of determining if it has a synchronizing word can be solved in polynomial time using a theorem due to Ján Černý. A simple approach considers the power set of states of the DFA, and builds a directed graph where nodes belong to the power set, and a directed edge describes the action of the transition function. A path from the node of all states to a singleton state shows the existence of a synchronizing word. This algorithm is exponential in the number of states. A polynomial algorithm results however, due to a theorem of Černý that exploits the substructure of the problem, and shows that a synchronizing word exists if and only if every pair of states has a synchronizing word.
Length
The problem of estimating the length of synchronizing words has a long history and was posed independently by several authors, but it is commonly known as the Černý conjecture. In 1964 Ján Černý conjectured that 2 is the upper bound for the length of the shortest synchronizing word for any n-state complete DFA. If this is true, it would be tight: in his 1964 paper, Černý exhibited a class of automata for which the shortest reset words have this length. The best upper bound known is /6, far from the lower bound. For n-state DFAs over a k-letter input alphabet, an algorithm by David Eppstein finds a synchronizing word of length at most 11n3/48 + O, and runs in time complexity O. This algorithm does not always find the shortest possible synchronizing word for a given automaton; as Eppstein also shows, the problem of finding the shortest synchronizing word is NP-complete. However, for a special class of automata in which all state transitions preserve the cyclic order of the states, he describes a different algorithm with time O that always finds the shortest synchronizing word, proves that these automata always have a synchronizing word of length at most 2, and exhibits examples of automata with this special form whose shortest synchronizing word has length exactly 2.
A transformation semigroup is synchronizing if it contains an element ofrank 1, that is, an element whose image is of cardinality 1. A DFA corresponds to a transformation semigroup with a distinguished generator set.
