Dynamical Aspects of Automata and Semigroup Theories
Wien, 2010
Upper bound on the length of reset word
Trahtman A.N.
Complete deterministic directed finite automaton with transition
graph Γ
q
ά
Deterministic
p
ά
Complete – for any vertex
outgoing edges of all colors
from given alphabet
For edge q → p suppose p= q ά
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.
Synchronizing and K-synchronizing graph
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 ksynchronizing word of automaton with transition graph Γ
and the automaton is called k-synchronizing.
K-synchronizing coloring of directed graph Γ turns the
graph into k-synchronizing automaton.
Černy conjecture
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.
Upper bound:
• Lower bound:
The gap exists
(n3-n)/6
2
(n-1)
Frankl, 1982, Pin, almost 30 years
Cerny, 1964
1983
Kljachko, Rystsov,
Spivak, 1987
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
q=2
n=3
@
@
n=4
n=5
@
@
@
q=3
@@ @@
@
q=4
@ Cerny automata
@ Known automata
@ Found by TESTAS
n=6
n=7
n=8
n=9
n=10
@
@
@
@
@
@
Some improvement of the known upper
bound (no changed from 1982)
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
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
Size
n=5
n=6
(n-1)2
Max of
minimal
length
16
25
q=2 15
23
23
22
q=3 15
q=4 15
n=7
n=8 q<3
n=9 q<3
n=10 q<3
36
49
64
81
32
31
30
44
<=44
58
74
The growing gap between (n-1)2 and Max of 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
mentioned above, 3 of size 3, 3 of size 4, one of
size 5 and one of size 6.
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
81.01% 16.2%
1.82% 0.8%
0.05%
6n - 7n
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
three cases: n=10, q=2; n=7,q=3 and n=7,q=4
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
n=10,q=2
n - 2n
2n -3n
3n -4n
4n -5n
5n -6n
|
6n-7n
n=7,q=3
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 interest
and 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.
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 -
Theorem: (Beal, Perrin)
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
An arbitrary graph Γ
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.
Then Γ has K- synchronizing coloring.
Finite directed graph of uniform outdegree is either
K- synchronizing or has no sink SCC
The package TESTAS finds k- synchronizing road coloring for a
graph having a subgraph with sink SCC of uniform outdegree.
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
Visualization
of a directed labeled graph
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.
Among the important visual properties of a graph
structure one can mention paths, cycles, strongly
connected components (SCC), cliques, bunches,
reachable states etc.
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.
.
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