Semisimple Synchronizing Automata and the Wedderburn-Artin Theory
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | https://hdl.handle.net/10216/107488 |
Resumo: | 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. |
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Semisimple Synchronizing Automata and the Wedderburn-Artin TheoryWe 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.20162016-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/10216/107488eng0129-054110.1142/s0129051116400037Almeida, JRodaro, Einfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-29T15:16:08Zoai:repositorio-aberto.up.pt:10216/107488Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T00:19:23.975548Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory |
title |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory |
spellingShingle |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory Almeida, J |
title_short |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory |
title_full |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory |
title_fullStr |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory |
title_full_unstemmed |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory |
title_sort |
Semisimple Synchronizing Automata and the Wedderburn-Artin Theory |
author |
Almeida, J |
author_facet |
Almeida, J Rodaro, E |
author_role |
author |
author2 |
Rodaro, E |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Almeida, J Rodaro, E |
description |
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. |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016 2016-01-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://hdl.handle.net/10216/107488 |
url |
https://hdl.handle.net/10216/107488 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0129-0541 10.1142/s0129051116400037 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
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RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799136112049913856 |