Complete deterministic directed finite automaton with transition graph Γ

For a set of states Q and mapping ά consider a map Qά and Qs for s=ά1ά2… άi . Γs presents a map of Γ.

Deterministic

Complete – for any vertex outgoing edges of all colors from given alphabet

ά

q

ά

For edge q → p suppose p= q ά

Upper bound on the length of reset word Trahtman A.N.

Dynamical Aspects of Automata and Semigroup Theories Wien, 2010

p

K-synchronizing coloring of directed graph Γ turns the graph into k-synchronizing automaton.

If for some word s |Γs|=1 then s is synchronizing word of automaton with transition graph Γ and the automaton is called synchronizing.

If for some word s |Γs|=k and k is minimal then s is k-synchronizing word of automaton with transition graph Γ and the automaton is called k-synchronizing.

Synchronizing and K-synchronizing graph

Černy conjecture

• Lower bound: (n-1)2

Cerny, 1964

The gap exists almost 30 years

• Upper bound: (n3-n)/6 Frankl, 1982, Pin, 1983 Kljachko, Rystsov, Spivak, 1987

Jan Černy found in 1964 n-state complete DFA with shortest synchronizing word of length (n-1)2.

Conjecture: (n-1)2 is an upper bound for the length of the shortest synchronizing word for any n-state automaton.

The conjecture holds in a lot of private cases. Some interesting corollaries follow from the study of small DFA

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

for n<11, q=2 and n<8, q<5

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

q=2

q=3

q=4

@ Cerny automata

@ @ @ @ @ @ @ @ @ @

@

@ Known automata

@

@ @ @ @

@ Found by TESTAS

The upper bound on the length of the minimal synchronizing word of n-state automaton is not greater than

(n3-n) 7/ (6x8) + n2 /2

Some improvement of the known upper bound (no changed from 1982)

A small modification of the old upper bound

makes the coefficient 7/8

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 mentioned above, 3 of size 3, 3 of size 4, one of size 5 and one of size 6.

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

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

Max of minimal length

q=2 15

q=3 15

q=4 15

23

23

22

32

31

30

44

<=44

58 74

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

Conjecture

Synchronization algorithms of TESTAS 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 for to check synchronizability. The algorithm is used in procedures of the package TESTAS finding synchronizing word.

The procedures are based on semigroup approach (almost quadratic algorithm) on Eppstein algorithm, O(n3) and its generalizations, on the ideas of works of Kljachko, Rystsov, Spivak and Frankl. O(n4)

A minimal length synchronizing word is found by non-polynomial algorithm

Distribution of the length of synchronizing word

Lengths (near minimal) are found by an algorithm based on the package.

The algorithm consistently sifts non-synchronizing graphs, graphs with very short reset word and a part of isomorphic graphs. The minimal length is found for graphs with very long reset words.

All remaining graphs of 10 vertices over 2 letters

n - 2n 2n – 3n 3n - 4n 4n – 5n 5n - 6n 6n - 7n

81.01% 16.2% 1.82% 0.8% 0.05% 0.006%

Maximal value of the length found by the algorithm – 93

The length found by minimal length algorithm – 81 (Err < 13%)

Distribution of synchronizing automata of size n, size of alphabet q,

according to the length of reset word

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

n - 2n 2n -3n 3n -4n 4n -5n 5n -6n |

6n-7n

n=10,q=2

n=7,q=3

n=7,q=4

three cases: n=10, q=2; n=7,q=3 and n=7,q=4

The maximal number of graphs has its length of reset word

near n+1

Road coloring problem Adler, Goodwyn, Weiss, 1970 1.directed finite strongly connected graph 2.constant outdegree of all its vertices 3. the greatest common divisor of lengths of all its cycles is one.

Has such graph a synchronizing coloring?

The problem awaked an unusual interestand not only among the mathematicians

Theorem: Let every vertex of strongly connected directed finite graph Γ have the same number of outgoing edges (uniform outdegree).

Then Γ has synchronizing coloring if and only if the greatest common divisor of lengths of all its cycles is one.

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……...

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…….….

Road coloring for mapping on k states 1.directed finite strongly connected graph

2.constant outdegree of all its vertices The problem also depends only on sink (minimal) strongly connected component with constant outdegree - for to be complete and deterministic

The problem was solved by Beal, Perrin, A quadratic algorithm for road coloring, arXiv:0803.0726v6, 2008, see also Budzban, Feinsilver, The Generalized Road Coloring Problem and periodic digraphs arXiv:0903.0192, 2009 -

Directed finite strongly connected graph with constant outdegree of all its vertices has K- synchronizing coloring

if and only if

the greatest common divisor of lengths of all its cycles is K

Theorem: (Beal, Perrin)

An arbitrary graph Γ

Then Γ has K- synchronizing coloring.

Let a finite directed graph Γ have a sink component Γ1. Suppose that by removing some edges of Γ1 one obtains strongly connected directed graph Γ2 of uniform outdegree. Let K be the gcd of lengths of the cycles of Γ2.

The package TESTAS finds k- synchronizing road coloring for a graph having a subgraph with sink SCC of uniform outdegree.

Finite directed graph of uniform outdegree is eitherK- synchronizing or has no sink SCC

Algorithms for Road Coloring

The known algorithms are based on the proof of the Road Coloring Conjecture.

The cubic algorithm (quadratic in most cases) of Trahtman is implemented in the package TESTAS

Beal and Perrin declared creation of a quadratic algorithm

The coloring for K-synchronizing and for arbitrary automaton is also implemented in the package TESTAS

The visualization used the cyclic layout ( the vertices are at the periphery of a circle). The visibility of inner structure of a digraph with labels on the edges is our goal.

It is clear that the curve edges (used, for instance, in some packages) hinder to recognize the cycles and paths. Therefore, we use only direct and, hopefully, short edges. .

Among the important visual properties of a graph structure one can mention paths, cycles, strongly connected components (SCC), cliques, bunches,

reachable states etc.

Visualization of a directed labeled graph

Linear Visualization Algorithm

The first step is the selection of the strongly connected components (SCC). A linear algorithm is used in order to find them.

Our modification of the cyclic layout considered two levels of circles, the first level consists of SCC , the second level presents the whole graph with SCC at the periphery of the circle.

So strongly connected components can be easily recognized. The pictorial diagram demonstrates the graph structure.

Any deterministic finite automaton is accepted