Partial Conway and Iteration Semiring-Semimodule Pairs

A Conway semiring is a semiring S equipped with a unary operation *:S → S, called “star”, satisfying the sum star and product star identities. A Conway semiring-semimodule pair consists of a Conway semiring S and a left S-semimodule V together with a function ω: S → V, called “omega power”, subject...

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Bibliographic Details
Published inAlgebraic Foundations in Computer Science Vol. 7020; pp. 56 - 71
Main Author Ésik, Zoltán
Format Book Chapter
LanguageEnglish
Published Germany Springer Berlin / Heidelberg 2011
Springer Berlin Heidelberg
SeriesLecture Notes in Computer Science
Subjects
Artificial intelligence
Computer programming / software development
Distinguished Ideal
Mathematical theory of computation
Matricial Theory
Matrix Theory
Rational Element
Regular Language
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ISBN3642248969
9783642248962
ISSN0302-9743
1611-3349
DOI10.1007/978-3-642-24897-9_3

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Summary:A Conway semiring is a semiring S equipped with a unary operation *:S → S, called “star”, satisfying the sum star and product star identities. A Conway semiring-semimodule pair consists of a Conway semiring S and a left S-semimodule V together with a function ω: S → V, called “omega power”, subject to the sum omega and product omega identities. A Kleene type theorem holds in all Conway semiring-semimodule pairs that can be instantiated to give the equivalence of Büchi automata and regular languages over ω-words. However, sometimes the star and omega power operations cannot be defined in an appropriate manner on the whole semiring S. To handle this situation, we introduce partial Conway semiring-semimodule pairs and develop their basic theory in connection with automata. We prove a Kleene theorem, applicable to all partial Conway semiring-semimodule pairs.
ISBN:3642248969
9783642248962
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-642-24897-9_3