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Algorithms and trends for synchronizing automata

Algorithms and trends for synchronizing automata. A word w is called synchronizing ( magic, recurrent, reset , directable ) word of an automaton if w sends all states of the automaton on a n unique state.

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Algorithms and trends for synchronizing automata

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  1. Algorithms and trends for synchronizing automata • A word w is called synchronizing (magic, recurrent, reset, directable) word of an automatonif w sends all states of the automaton on an unique state. Jan Cerny had found in 1964 n-state complete DFA with shortest synchronizing word of length (n-1)2. The Cerny conjecture states that it is an upper bound for the length of the shortest synchronizing word for any n-state automaton. known bounds Upper (n3-n)/6 Frankl, 1982, Pin, 1983 Kljachko,Rystsov,Spivak, 1987 Lower (n-1)2 Cerny 1964 Gap

  2. The value(n-1)2is reached in next cases: Cerny sequence of graphs (here n=4) Kari graph Cerny, Piricka and Rosenauerova Roman graph

  3. An algorithm of package TESTAS, mostly quadratic,finds new exampleswith minimal reset word of length(n-1)2 The corresponding reset words of minimal length are: abcacabca, acbaaacba, baab, acba, bacb The size of the syntactic semigroup is 148, 180, 24, 27 and 27

  4. All automata of minimal reset word of length (n-1)2for n<12, q=2. n<9, q=3, n<8, q=4 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ Cerny automata @ Found by TESTAS @ Known automata

  5. All automata of minimal reset word of lengthless than (n-1)2 The growing gap between (n-1)2 and Maxof minimal length inspires Conjecture The set of n-state complete DFA (n>2) with minimal reset word of length (n-1)2 contains only the sequence of Cerny and the eight automata, three of size 3, three of size 4, one of size 5 and one of size 6.

  6. Algorithms data for the length of synchronizing word Complexity of Eppstein and cycle algorithms is O(n3), algorithm FKSR - O(n4) , semigroup algorithm is subquadratic, minimal length algorithm is not polynomial

  7. Properties used in algorithms An automaton with transition graph Gis synchronizing iff G2 has a sink state. It is a base foraquadratic in the worst casealgorithm of synchronizability used in implemented procedures finding synchronizing word For every state p of synchronizing automaton A there exists a word sof length not greater than |A| such that pnot in As.

  8. THANKS !

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