Axiomatizing rational power series over natural numbers

S. L. Bloom, Z. Ésik

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

Iteration semi-rings are Conway semi-rings satisfying Conway's group identities. We show that the semi-rings Nrat {left double angle bracket} Σ* {right double angle bracket} of rational power series with coefficients in the semi-ring N of natural numbers are the free partial iteration semi-rings. Moreover, we characterize the semi-rings N∞rat {left double angle bracket} Σ* {right double angle bracket} as the free semi-rings in the variety of iteration semi-rings defined by three additional simple identities, where N is the completion of N obtained by adding a point of infinity. We also show that this latter variety coincides with the variety generated by the complete, or continuous semirings. As a consequence of these results, we obtain that the semi-rings N rat {left double angle bracket} Σ* {right double angle bracket}, equipped with the sum order, are free in the class of symmetric inductive *-semi-rings. This characterization corresponds to Kozen's axiomatization of regular languages.

Original languageEnglish
Pages (from-to)793-811
Number of pages19
JournalInformation and Computation
Volume207
Issue number7
DOIs
Publication statusPublished - Jul 2009

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Formal languages
Semiring
Natural number
Power series
Brackets
Angle
Iteration
Regular Languages
Axiomatization
Completion
Infinity

ASJC Scopus subject areas

  • Information Systems
  • Computational Theory and Mathematics
  • Theoretical Computer Science
  • Computer Science Applications

Cite this

Axiomatizing rational power series over natural numbers. / Bloom, S. L.; Ésik, Z.

In: Information and Computation, Vol. 207, No. 7, 07.2009, p. 793-811.

Research output: Contribution to journalArticle

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