44 research outputs found
An Algorithm for Road Coloring
A coloring of edges of a finite directed graph turns the graph into
finite-state automaton. The synchronizing word of a deterministic automaton is
a word in the alphabet of colors (considered as letters) of its edges that maps
the automaton to a single state. A coloring of edges of a directed graph of
uniform outdegree (constant outdegree of any vertex) is synchronizing if the
coloring turns the graph into a deterministic finite automaton possessing a
synchronizing word. The road coloring problem is the problem of synchronizing
coloring of a directed finite strongly connected graph of uniform outdegree if
the greatest common divisor of the lengths of all its cycles is one. The
problem posed in 1970 had evoked a noticeable interest among the specialists in
the theory of graphs, automata, codes, symbolic dynamics as well as among the
wide mathematical community. A polynomial time algorithm of complexity
in the most worst case and quadratic in majority of studied cases for the road
coloring of the considered graph is presented below. The work is based on
recent positive solution of the road coloring problem. The algorithm was
implemented in the package TESTASComment: 10 page
Matrix approach to synchronizing automata
A word of letters on edges of underlying graph of deterministic
finite automaton (DFA) is called synchronizing if sends all states of the
automaton to a unique state. J. \v{C}erny discovered in 1964 a sequence of
-state complete DFA possessing a minimal synchronizing word of length
. The hypothesis, well known today as \v{C}erny conjecture, claims
that is a precise upper bound on the length of such a word over
alphabet of letters on edges of for every complete -state
DFA. The hypothesis was formulated distinctly in 1966 by Starke. A special
classes of matrices induced by words in the alphabet of labels on edges of the
underlying graph of DFA are used for the study of synchronizing automata.Comment: 19-pages.3 figures An error removed. arXiv admin note: text overlap
with arXiv:1405.2435, arXiv:1202.462
The road coloring problem
The synchronizing word of deterministic automaton is a word in the alphabet
of colors (considered as letters) of its edges that maps the automaton to a
single state. A coloring of edges of a directed graph is synchronizing if the
coloring turns the graph into deterministic finite automaton possessing a
synchronizing word.
The road coloring problem is a problem of synchronizing coloring of directed
finite strongly connected graph with constant outdegree of all its vertices if
the greatest common divisor of lengths of all its cycles is one. The problem
was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked a noticeable
interest among the specialists in theory of graphs, deterministic automata and
symbolic dynamics. The problem is described even in "Wikipedia" - the popular
Internet Encyclopedia. The positive solution of the road coloring problem is
presented.Comment: 9 pages, correct typo
An efficient algorithm finds noticeable trends and examples concerning the \v{C}erny conjecture
A word w is called synchronizing (recurrent, reset, directed) word of a
deterministic finite automaton (DFA) if w sends all states of the automaton on
a unique state. Jan Cerny had found in 1964 a sequence of n-state complete DFA
with shortest synchronizing word of length (n-1)^2. He had conjectured that it
is an upper bound for the length of the shortest synchronizing word for any
-state complete DFA.
The examples of DFA with shortest synchronizing word of length (n-1)^2 are
relatively rare. To the Cerny sequence were added in all examples of Cerny,
Piricka and Rosenauerova (1971), of Kari (2001) and of Roman (2004).
By help of a program based on some effective algorithms, a wide class of
automata of size less than 11 was checked. The order of the algorithm finding
synchronizing word is quadratic for overwhelming majority of known to date
automata. Some new examples of n-state DFA with minimal synchronizing word of
length (n-1)^2 were discovered. The program recognized some remarkable trends
concerning the length of the minimal synchronizing word.
http://www.cs.biu.ac.il/~trakht/Testas.htmlComment: MFCS06. LNCS 4162, 10 page
The length of a minimal synchronizing word and the \v{C}erny conjecture
A word w of letters on edges of underlying graph Gamma of deterministic
finite automaton (DFA) is called the synchronizing word if w sends all states
of the automaton to a unique state. J. \v{C}erny discovered in 1964 a sequence
of n-state complete DFA possessing a minimal synchronizing word of length
(n-1)^2. The hypothesis, well known today as the \v{C}erny conjecture, claims
that it is also precise upper bound on the length of such a word for a complete
DFA. This simple-looking conjecture is arguably the most fascinating and
longstanding open problem in the combinatorial theory of finite automata.
An attempt to prove the \v{C}erny conjecture is wrong.Comment: lemmas 11 is wrong. The conjecture is not proved. arXiv admin note:
substantial text overlap with arXiv:1202.462
The Visualization of the Road Coloring Algorithm in the package TESTAS
A synchronizing word of a deterministic automaton is a word in the alphabet
of colors of its edges that maps the automaton to a single state. A coloring of
edges of a directed graph is synchronizing if the coloring turns the graph into
a deterministic finite automaton possessing a synchronizing word.
The road coloring problem is the problem of synchronizing coloring of a
directed finite strongly connected graph with constant outdegree of all its
vertices if the greatest common divisor of the lengths of all its cycles is
one. A polynomial time algorithm of the road coloring has been based on recent
positive solution of this old famous problem.
One can use our new visualization program for demonstration of the algorithm
as well as for visualization of the transition graph of any finite automaton.
The visual image presents some structure properties of the transition graph.
This help tool is linear in the size of the automaton.Comment: 12 page
Reducing the time complexity of testing for local threshold testability
A locally threshold testable language L is a language with the property that
for some non negative integers k and l and for some word u from L, a word v
belongs to L if and only if
(1) the prefixes [suffixes] of length k-1 of words u and v coincide,
(2) the numbers of occurrences of every factor of length k in both words u
and v are either the same or greater than l-1.
A deterministic finite automaton is called locally threshold testable if the
automaton accepts a locally threshold testable language for some l and k.
New necessary and sufficient conditions for a deterministic finite automaton
to be locally threshold testable are found. On the basis of these conditions,
we modify the algorithm to verify local threshold testability of the automaton
and to reduce the time complexity of the algorithm. The algorithm is
implemented as a part of the package TESTAS.
\texttt{http://www.cs.biu.ac.il/trakht/Testas.html}.Comment: 11 pages, 4 figure
Polynomial time algorithm for left [right] local testability
A right [left] locally testable language S is a language with the property
that for some non negative integer k two words u and v in alphabet S are equal
in the semi group if (1) the prefix and suffix of the words of length k
coincide, (2) the set of segments of length k of the words as well as 3) the
order of the first appearance of these segments in prefixes [suffixes]
coincide. We present necessary and sufficient condition for graph [semi group]
to be transition graph [semi group] of the deterministic finite automaton that
accepts right [left] locally testable language and necessary and sufficient
condition for transition graph of the deterministic finite automaton with
locally idempotent semi group. We introduced polynomial time algorithms for the
right [left] local testable problem for transition semi group and transition
graph of the deterministic finite automaton based on these conditions.
Polynomial time algorithm verifies transition graph of automaton with locally
idempotent transition semi group.Comment: 10 page
Precise estimation on the order of local testability of deterministic finite automaton
A locally testable language L is a language with the property that for some
non negative integer k, called the order or the level of local testable,
whether or not a word u in the language L depends on (1) the prefix and the
suffix of the word u of length k-1 and (2) the set of intermediate partial
strings of length k of the word u. For given k the language is called
k-testable. We give necessary and sufficient conditions for the language of an
automaton to be k-testable in the terms of the length of paths of a related
graph. Some estimations of the upper and of the lower bound of testable order
follow from these results. We improve the upper bound on the testable order of
locally testable deterministic finite automaton with n states to n(n-2)+1 This
bound is the best possible. We give an answer on the following conjecture of
Kim, McNaughton and Mac-CLoskey for deterministic finite locally testable
automaton with n states: \Is the local testable order of no greater than n in
power 1.5 when the alphabet size is two?" Our answer is negative. In the case
of size two the situation is the same as in general case.Comment: 15 page
Cerny-Starke conjecture from the sixties of XX century
A word of letters on edges of underlying graph of deterministic
finite automaton (DFA) is called synchronizing if sends all states of the
automaton to a unique state. J. \v{C}erny discovered in 1964 a sequence of
-state complete DFA possessing a minimal synchronizing word of length
. The hypothesis, mostly known today as \v{C}erny conjecture, claims
that is a precise upper bound on the length of such a word over
alphabet of letters on edges of for every complete -state
DFA. The hypothesis was formulated in 1966 by Starke. Algebra with nonstandard
operation over special class of matrices induced by words in the alphabet of
labels on edges is used to prove the conjecture. The proof is based on the
connection between length of words and dimension of the space generated by
solution of matrix equation for synchronizing word , as
well as on relation between ranks of and . Important role below
placed the notion of pseudo inverseL matrix, sometimes reversible.Comment: 18 pages, 9 lemmas, graphs, matrices. 4 examples arXiv admin note:
substantial text overlap with arXiv:1904.07694, arXiv:1202.4626; text overlap
with arXiv:1405.243