Axiomatizing tropical semirings

Luca Aceto, Z. Ésik, Anna Ingólfsdóttir

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

Original languageEnglish
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer Verlag
Pages42-56
Number of pages15
Volume2030
ISBN (Print)3540418644
Publication statusPublished - 2001
Event4th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2001 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2001 - Genova, Italy
Duration: Apr 2 2001Apr 6 2001

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2030
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other4th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2001 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2001
CountryItaly
CityGenova
Period4/2/014/6/01

Fingerprint

Semiring
Algebra
Idempotent
Carrier set
Equational Theory
Binary operation
Free Algebras
Axiomatization
Subset

Keywords

  • Commutative idempotent weak semirings
  • Complete axiomatizations
  • Convexity
  • Equational logic
  • Exponential time complexity
  • Relative axiomatizations
  • Tropical semirings
  • Varieties

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Aceto, L., Ésik, Z., & Ingólfsdóttir, A. (2001). Axiomatizing tropical semirings. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 2030, pp. 42-56). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2030). Springer Verlag.

Axiomatizing tropical semirings. / Aceto, Luca; Ésik, Z.; Ingólfsdóttir, Anna.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 2030 Springer Verlag, 2001. p. 42-56 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2030).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Aceto, L, Ésik, Z & Ingólfsdóttir, A 2001, Axiomatizing tropical semirings. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 2030, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2030, Springer Verlag, pp. 42-56, 4th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2001 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2001, Genova, Italy, 4/2/01.
Aceto L, Ésik Z, Ingólfsdóttir A. Axiomatizing tropical semirings. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 2030. Springer Verlag. 2001. p. 42-56. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Aceto, Luca ; Ésik, Z. ; Ingólfsdóttir, Anna. / Axiomatizing tropical semirings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 2030 Springer Verlag, 2001. pp. 42-56 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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