Abstract
Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains at several inequivalent definitions of Kleene algebras and related algebraic structures [2,13,14,5,6,1,9,7].
In this paper we establish some new relationships among these structures. Our main results are:
-
•There is a Kleene algebra in the sense of [6] that is not *-continuous.
-
•The categories of *-continuous Kleene algebras [5,6], closed semirings [1,9] and S-algebras [2] are strongly related by adjunctions.
-
•The axioms of Kleene algebra in the sense of [6] are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [2, p. 103].
-
•Right-handed Kleene algebras are not necessarily left-handed Kleene algebras. This verifies a weaker version of a conjecture of Pratt [14].
Supported by NSF grant CCR-8901061.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1975.
John Horton Conway. Regular Algebra and Finite Machines. Chapman and Hall, London, 1971.
John Horton Conway, personal communication, May 1990.
Harel, D., First-Order Dynamic Logic. Lecture Notes in Computer Science 68, Springer-Verlag, 1979.
Dexter Kozen, “On induction vs. *-continuity,” Proc. Workshop on Logics of Programs 1981, Spring-Verlag Lect. Notes in Comput. Sci. 131, ed. Kozen, 1981, 167–176.
Dexter Kozen, “A completeness theorem for Kleene algebras and the algebra of regular events,” Cornell TR90-1123, May 1990.
Werner Kuich, “The Kleene and Parikh Theorem in Complete Semirings,” in: Proc. 14th Colloq. Automata, Languages, and Programming, ed. Ottmann, Springer-Verlag Lecture Notes in Computer Science 267, 1987, 212–225.
Saunders Mac Lane. Categories for the Working Mathematician. Springer-Verlag, 1971.
Kurt Mehlhorn. Data Structures and Algorithms 2: Graph Algorithms and NP-Completeness. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1984.
K. C. Ng and A. Tarski, “Relation algebras with transitive closure,” Abstract 742-02-09, Notices Amer. Math. Soc. 24 (1977), A29–A30.
K. C. Ng. Relation Algebras with Transitive Closure. PhD Thesis, University of California, Berkeley, 1984.
Vaughan Pratt, “Semantical Considerations on Floyd-Hoare Logic,” Proc. 17th IEEE Symp. Found. Comput. Sci. 1976, 109–121.
Vaughan Pratt, “Dynamic algebras and the nature of induction,” Proc. 12th ACM Symp. on Theory of Computing, 1980, 22–28.
Vaughan Pratt, “Dynamic algebras as a well-behaved fragment of relation algebras,” in: D. Pigozzi, ed., Proc. Conf. on Algebra and Computer Science, Ames, Iowa, June 2–4, 1988; Spring-Verlag Lecture Notes in Computer Science, to appear.
Arto Salomaa, “Two complete axiom systems for the algebra of regular events,” J. Assoc. Comput. Mach. 13:1 (January, 1966), 158–169.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kozen, D. (1990). On kleene algebras and closed semirings. In: Rovan, B. (eds) Mathematical Foundations of Computer Science 1990. MFCS 1990. Lecture Notes in Computer Science, vol 452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0029594
Download citation
DOI: https://doi.org/10.1007/BFb0029594
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52953-8
Online ISBN: 978-3-540-47185-1
eBook Packages: Springer Book Archive