Paper Info
Title
Semisimple Synchronizing Automata And The Wedderburn-Artin Theory
Abstract
We present a ring theoretic approach to Cerny's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in Cerny's series. Semisimplicity gives also the advantage of "factorizing" the problem of finding a synchronizing word into the sub-problems of finding "short" words that are zeros into the projection of the simple components in the Wedderburn-Artin decomposition. In the general case this last problem is related to the search of radical words of length at most (n-1)(2) where n is the number of states of the automaton. We show that the solution of this "Radical Conjecture" would give an upper bound 2(n-1)(2) for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases and Cerny's conjecture for the class of strongly semisirnple synchronizing automata. 'These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.
Year
DOI
Venue
2014
10.1142/S0129054116400037
INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
Keywords
Field
DocType
Synchronizing automaton, Cerny's conjecture, Wedderburn-Artin theory, radical word, ideal regular language
Discrete mathematics,Radical of an ideal,Automata theory,Synchronizing automaton,Computer science,Synchronizing,Automaton,Synchronizing word,Regular language,Conjecture
Conference
Volume
Issue
ISSN
27
2
0129-0541
Citations 
PageRank 
References 
0
0.34
7
Authors
2
Name
Order
Citations
PageRank
J. Almeida16115.24
Emanuele Rodaro25515.63