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 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.

Upper (n3-n) /6 Frankl, 1982, Pin, 1983

Kljachko,Rystsov,Spivak, 1987

Lower (n-1)2

Cerny 1964

known bounds

Gap

The value (n-1)2 is reached in next cases:

Kari graph

Cerny sequence of graphs (here n=4)

Cerny, Piricka and Rosenauerova

Roman graph

An algorithm of package TESTAS, mostly quadratic, finds new examples with minimal reset word of length (n-1)2

The corresponding reset words of minimal length are:abcacabca, acbaaacba, baab, acba, bacbThe size of the syntactic semigroup is 148, 180, 24, 27 and 27

All automata of minimal reset word of length (n-1)2

for n<12, q=2. n<9, q=3, n<8, q=4

size n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11

q=2

q=3

q=4

@ Cerny automata

@ @ @ @ @ @ @ @ @ @ @

@

@ Known automata

@

@ @ @ @

@ Found by TESTAS

All automata of minimal reset word of length less than (n-1)2

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.

Size n=4 n=5 n=6 n=7 n=8 n=9 q<3 n=10 q<3 n=11 q<3

(n-1)2 9 16 25 36 49 64 81 100

Max of minimal length

q=2 8

q=3 8

q=4 8

q=2 15

q=3 15

q=4 15

23

23

22

32

31

30

44

42

58 74 92

The growing gap between (n-1)2 and Max of minimal length inspires

Conjecture

Algorithms data for the length of synchronizing word

name Roman Kari Cerny6 Cerny10 Cerny17 KMM KMM complete

Cern28 Cern151

Graph size 5 6 6 10 17 28 28 28 151

Semigroup size

1397 17265 2742 >9 X 105 - 22126 >2X 105 - -

Cycle alg for word length

18 27 25 81 256 4 41 729 22500

Eppstein (greedy) alg

17 26 27 97 375 4 51 1202 57190

Minimal length alg

16 25 25 81 256 4 27 729 22500

Word length of FKSR alg

17 26 26 97 375 4 31 1086 57190

Complexity of Eppstein and cycle algorithms is O(n3), algorithm FKSR - O(n4) , semigroup algorithm is subquadratic, minimal length algorithm is not polynomial

Semigroup algorithm

18 28 25 81 256 4 27 729 22500

Properties used in algorithms

An automaton with transition graph G is synchronizing iff G2 has a sink state.

It is a base for a quadratic in the worst case algorithm of synchronizability used in implemented procedures finding synchronizing word

For every state p of synchronizing automaton A there exists a word s of length not greater than |A| such that p not in As.