Synchronizing word

This drawing represents a DFA with eight states and two input symbols, red and blue. The word blue-red-red-blue-red-red-blue-red-red is a synchronizing word that sends all states to the yellow state; the word blue-blue-red-blue-blue-red-blue-blue-red is another synchronizing word that sends all states to the green state.

In computer science, more precisely, in the theory of deterministic finite automata (DFA), a synchronizing word or reset sequence is a word in the input alphabet of the DFA which sends any state of the DFA to one and the same state.^[1] 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.

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 Jan Černý conjectured that (n − 1)² is the upper bound for the length of the shortest synchronizing word for any n-state complete DFA (a DFA with complete state transition graph).^[1][2] If this is true, it would be tight: in his 1964 paper, Černý exhibited a class of automata (indexed by the number n of states) for which the shortest reset words have this length. The best upper bound known is n(7n²+6n-16)/48, far from the lower bound. ^[3]. 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(n²), and runs in time complexity O(n³+kn²). 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(kn²) that always finds the shortest synchronizing word, proves that these automata always have a synchronizing word of length at most (n − 1)² (the bound given in Černý's conjecture), and exhibits examples of automata with this special form whose shortest synchronizing word has length exactly (n − 1)².^[4]

The road coloring problem is the problem of labeling the edges of a regular directed graph with the symbols of a k-letter input alphabet (where k is the outdegree of each vertex) in order to form a synchronizable DFA. It was conjectured in 1970 by Benjamin Weiss and Roy Adler that any strongly connected and aperiodic regular digraph can be labeled in this way; their conjecture was proven in 2007 by Avraham Trahtman.^[5][6]

References

1. ^ ^{a b} Avraham Trakhtman: Synchronizing automata, algorithms, Cerny Conjecture. Accessed May 15, 2010.
2. ^ Černý, J. (1964), "Poznámka k homogénnym eksperimentom s konecnými automatami", Matematicko-fyzikálny Časopis Slovenskej Akadémie Vied 14: 208–216, http://dml.cz/bitstream/handle/10338.dmlcz/126647/MathSlov_14-1964-3_2.pdf  (in Slovak).
3. ^ A.N. Trahtman. Modifying the Upper Bound on the Length of Minimal Synchronizing Word. Lect. Notes in Comp. Sci, 6914(2011) Springer, 173-180
4. ^ Eppstein, David (1990), "Reset Sequences for Monotonic Automata", SIAM Journal on Computing 19 (3): 500–510, doi:10.1137/0219033, http://www.ics.uci.edu/~eppstein/pubs/Epp-SJC-90.pdf. 
5. ^ Adler, R. L.; Weiss, B. (1970), "Similarity of automorphisms of the torus", Memoires of the American Mathematical Society 98 .
6. ^ Trahtman, Avraham (2007). "The road coloring problem". arXiv:0709.0099. .

Further reading

• Rystsov, I. C. (2004), "Černý's conjecture: retrospects and prospects", Proc. Worksh. Synchronizing Automata, Turku (WSA 2004) .
• Jürgensen, H. (2008). "Synchronization". Information and Computation 206 (9–10): 1033–1044. doi:10.1016/j.ic.2008.03.005.  edit
• Volkov, Mikhail V. (2008), "Synchronizing Automata and the Černý Conjecture", Proc. 2nd Int'l. Conf. Language and Automata Theory and Applications (LATA 2008), LNCS, 5196, Springer-Verlag, pp. 11–27, http://csseminar.kadm.usu.ru/tarragona_volkov2008.pdf