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					                    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
              q              graph Γ

                     ά
                             ά
Deterministic                                            Complete – for any vertex
                p                                        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 k-
 synchronizing 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:
(n3-n)/6           The gap exists
                                     (n-1)2
Frankl, 1982, Pin, almost 30 years
1983                                 Cerny, 1964
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   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
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=7     n=8 q<3   n=9 q<3   n=10 q<3

(n-1)2       16         25    36     49        64        81
Max of       q=2 15     23    32     44         58        74
minimal      q=3 15     23    31     <=44
length       q=4 15     22    30

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

     90%
     80%
     70%
     60%
     50%
     40%
     30%
     20%
     10%                                                         n=10,q=2
      0%                                                         n=7,q=3
           n - 2n   2n -3n   3n -4n   4n -5n    5n -6n   6n-7n
                                                   |             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: Letevery 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

				
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