Transitive Closures of Semi-commutation Relations on Regular omega-Languages
Identifieur interne : 004E15 ( Main/Exploration ); précédent : 004E14; suivant : 004E16Transitive Closures of Semi-commutation Relations on Regular omega-Languages
Auteurs : Pierre-Cyrille Héam [France]Source :
English descriptors
Abstract
A semi-commutation $R$ is a relation on a finite alphabet $A$. Given an infinite word $u$ on $A$, we denote by $R(u)=\{xbay\mid x\in A^*,y\in A^\omega \ (a,b)\in R \text{ and } xaby=u\}$ and by $R^*(u)$ the language $\{u\}\cup \cup_{k\geq 1} R^k(u)$. In this paper we prove that if an $\omega$-language $L$ is a finite union of languages of the form $A_0^*a_1A_1^*\ldots a_k A_k^*a_{k+1}A_{k+1}^*$, where the $A_i$'s are subsets of the alphabet and the $a_i$'s are letters, then $R^*(L)$ is a computable regular $\omega$-language accepting a similar decomposition. In addition we prove the same result holds for $\omega$-languages which are finite unions of languages of the form $L_0a_1L_1\ldots a_k L_ka_{k+1}L_{k+1}$, where the $L_i$'s are accepted by diamond automata and the $a_i$'s are letters. These results improve recent works by Bouajjani, Muscholl and Touili on one hand, and by Cécé, Héam and Mainier on the other hand, by extending them to infinite words.
Url:
Affiliations:
- France
- Franche-Comté, Grand Est, Lorraine (région)
- Belfort, Besançon, Nancy
- Institut national polytechnique de Lorraine, Université Nancy 2, Université de Bourgogne Franche-Comté, Université de Franche-Comté, Université de Lorraine, Université de technologie de Belfort-Montbéliard
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Le document en format XML
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<front><div type="abstract" xml:lang="en">A semi-commutation $R$ is a relation on a finite alphabet $A$. Given an infinite word $u$ on $A$, we denote by $R(u)=\{xbay\mid x\in A^*,y\in A^\omega \ (a,b)\in R \text{ and } xaby=u\}$ and by $R^*(u)$ the language $\{u\}\cup \cup_{k\geq 1} R^k(u)$. In this paper we prove that if an $\omega$-language $L$ is a finite union of languages of the form $A_0^*a_1A_1^*\ldots a_k A_k^*a_{k+1}A_{k+1}^*$, where the $A_i$'s are subsets of the alphabet and the $a_i$'s are letters, then $R^*(L)$ is a computable regular $\omega$-language accepting a similar decomposition. In addition we prove the same result holds for $\omega$-languages which are finite unions of languages of the form $L_0a_1L_1\ldots a_k L_ka_{k+1}L_{k+1}$, where the $L_i$'s are accepted by diamond automata and the $a_i$'s are letters. These results improve recent works by Bouajjani, Muscholl and Touili on one hand, and by Cécé, Héam and Mainier on the other hand, by extending them to infinite words.</div>
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