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Theory of Computing Systems

, Volume 63, Issue 3, pp 615–633 | Cite as

A Unifying Approach to Algebraic Systems Over Semirings

  • Peter KostolányiEmail author
Article
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Abstract

A fairly general definition of canonical solutions to algebraic systems over semirings is proposed. This is based on the notion of summation semirings, traditionally known as \({\Sigma }\)-semirings, and on assigning unambiguous context-free languages to variables of each system. The presented definition applies to all algebraic systems over continuous or complete semirings and to all proper algebraic systems over power series semirings, for which it coincides with the usual definitions of their canonical solutions. As such, it unifies the approaches to algebraic systems over semirings studied in literature. An equally general approach is adopted to study pushdown automata, for which equivalence with algebraic systems is proved. Finally, the Chomsky-Schützenberger theorem is generalised to the setting of summation semirings.

Keywords

Algebraic system Semiring Summation semiring Template 

Notes

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer Science, Faculty of Mathematics, Physics and InformaticsComenius University in BratislavaBratislavaSlovakia

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