arXiv subjects

A. N. Trahtman

Publications and source records attributed to A. N. Trahtman.

22 records · Page 2Linked to original sources

arXiv2010

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.

cs.DM↗

arXiv2009

The Road Coloring for Mapping on k States(withdrawn)

Let $Γ$ be directed strongly connected finite graph of uniform outdegree (constant outdegree of any vertex) and let some coloring of edges of $Γ$ turn the graph into deterministic complete automaton. Let the word $s$ be a word in the alphabet of colors (considered also as letters) on the edges of $Γ$ and let $Γs$ be a mapping of vertices $Γ$.

cs.DM↗

arXiv2007

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.

cs.DM↗

arXiv2007

An efficient algorithm finds noticeable trends and examples concerning the Černy 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 $n$-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.html

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