Skip to main content

On the Complexity of the Equational Theory of Relational Action Algebras

  • Conference paper
Relations and Kleene Algebra in Computer Science (RelMiCS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4136))

Included in the following conference series:

  • 588 Accesses

Abstract

Pratt [22] defines action algebras as Kleene algebras with residuals. In [9] it is shown that the equational theory of *-continuous action algebras (lattices) is Π\(^{0}_{1}\)–complete. Here we show that the equational theory of relational action algebras (lattices) is Π\(^{0}_{1}\) –hard, and some its fragments are Π\(^{0}_{1}\)–complete. We also show that the equational theory of action algebras (lattices) of regular languages is Π\(^{0}_{1}\)–complete.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Andréka, H., Mikulaś, S.: Lambek calculus and its relational semantics: completeness and incompleteness. Journal of Logic, Language and Information 3, 1–37 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bar-Hillel, Y., Gaifman, C., Shamir, E.: On categorial and phrase structure grammars, Bulletin Res. Council Israel F9, 155-166 (1960)

    Google Scholar 

  3. Buszkowski, W.: Some decision problems in the theory of syntactic categories. Zeitschrift f. math. Logik und Grundlagen der Mathematik 28, 539–548 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buszkowski, W.: The equivalence of unidirectional Lambek categorial grammars and context-free grammars. Zeitschrift f. math. Logik und Grundlagen der Mathematik 31, 369–384 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buszkowski, W.: The finite model property for BCI and related systems. Studia Logica 57, 303–323 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buszkowski, W.: Mathematical Linguistics and Proof Theory, 683-736, in [24]

    Google Scholar 

  7. Buszkowski, W.: Finite models of some substructural logics. Mathematical Logic Quarterly 48, 63–72 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buszkowski, W.: Relational models of Lambek logics. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) Theory and Applications of Relational Structures as Knowledge Instruments. LNCS, vol. 2929, pp. 196–213. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  9. W. Buszkowski, On action logic: Equational theories of action algebras. Journal of Logic and Computation ( to appear )

    Google Scholar 

  10. Buszkowski, W., Kołowska-Gawiejnowicz, M.: Representation of residuated semigroups in some algebras of relations (The method of canonical models.). Fundamenta Informaticae 31, 1–12 (1997)

    MathSciNet  MATH  Google Scholar 

  11. Hardin, C., Kozen, D.: On the complexity of the Horn theory of REL, manuscript (2003)

    Google Scholar 

  12. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory. In: Languages and Computation, Addison-Wesley, Reading (1979)

    Google Scholar 

  13. Hoare, C., Jifeng, H.: The weakest prespecification. Fundamenta Informaticae 9, 51-84, 217–252 (1986)

    MathSciNet  MATH  Google Scholar 

  14. Jipsen, P.: From semirings to residuated Kleene algebras. Studia Logica 76, 291–303 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kozen, D.: On Kleene algebras and closed semirings, in: Proc. MFCS 1990, Lecture Notes in Comp. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 26–47. Springer, Heidelberg (1990)

    Chapter  Google Scholar 

  16. Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kozen, D.: On the complexity of reasoning in Kleene algebras. Information and Computation 179, 152–162 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly 65, 154–170 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ono, H.: Semantics for Substructural Logics. In: Schroeder- Heister, P., Dosen, K. (eds.) Substructural Logics, pp. 259–291. Clarendon Press, Oxford (1993)

    Google Scholar 

  20. Orłowska, E., Radzikowska, A.M.: Double residuated lattices and their applications. In: de Swart, H. (ed.) RelMiCS 2001. LNCS, vol. 2561, pp. 171–189. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  21. Palka, E.: An infinitary sequent system for the equational theory of *-continuous action lattices. Fundamenta Informaticae (to appear )

    Google Scholar 

  22. Pratt, V.: Action logic and pure induction. In: van Eijck, J. (ed.) JELIA 1990. LNCS, vol. 478, pp. 97–120. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  23. Redko, V.N.: On defining relations for the algebra of regular events. Ukrain. Mat. Z. 16, 120–126 (1964) (In Russian)

    MathSciNet  Google Scholar 

  24. van Benthem, J., ter Meulen, A. (eds.): Handbook of Logic and Language. Elsevier, Amsterdam, The MIT Press, Cambridge Mass. (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Buszkowski, W. (2006). On the Complexity of the Equational Theory of Relational Action Algebras. In: Schmidt, R.A. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2006. Lecture Notes in Computer Science, vol 4136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11828563_7

Download citation

  • DOI: https://doi.org/10.1007/11828563_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37873-0

  • Online ISBN: 978-3-540-37874-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics