Advertisement

On the Complexity of the Equational Theory of Relational Action Algebras

  • Wojciech Buszkowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4136)

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andréka, H., Mikulaś, S.: Lambek calculus and its relational semantics: completeness and incompleteness. Journal of Logic, Language and Information 3, 1–37 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 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. 3.
    Buszkowski, W.: Some decision problems in the theory of syntactic categories. Zeitschrift f. math. Logik und Grundlagen der Mathematik 28, 539–548 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 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)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Buszkowski, W.: The finite model property for BCI and related systems. Studia Logica 57, 303–323 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Buszkowski, W.: Mathematical Linguistics and Proof Theory, 683-736, in [24]Google Scholar
  7. 7.
    Buszkowski, W.: Finite models of some substructural logics. Mathematical Logic Quarterly 48, 63–72 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 9.
    W. Buszkowski, On action logic: Equational theories of action algebras. Journal of Logic and Computation ( to appear )Google Scholar
  10. 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)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hardin, C., Kozen, D.: On the complexity of the Horn theory of REL, manuscript (2003)Google Scholar
  12. 12.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory. In: Languages and Computation, Addison-Wesley, Reading (1979)Google Scholar
  13. 13.
    Hoare, C., Jifeng, H.: The weakest prespecification. Fundamenta Informaticae 9, 51-84, 217–252 (1986)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jipsen, P.: From semirings to residuated Kleene algebras. Studia Logica 76, 291–303 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 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)CrossRefGoogle Scholar
  16. 16.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Kozen, D.: On the complexity of reasoning in Kleene algebras. Information and Computation 179, 152–162 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly 65, 154–170 (1958)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 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. 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)CrossRefGoogle Scholar
  21. 21.
    Palka, E.: An infinitary sequent system for the equational theory of *-continuous action lattices. Fundamenta Informaticae (to appear )Google Scholar
  22. 22.
    Pratt, V.: Action logic and pure induction. In: van Eijck, J. (ed.) JELIA 1990. LNCS, vol. 478, pp. 97–120. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  23. 23.
    Redko, V.N.: On defining relations for the algebra of regular events. Ukrain. Mat. Z. 16, 120–126 (1964) (In Russian)MathSciNetGoogle Scholar
  24. 24.
    van Benthem, J., ter Meulen, A. (eds.): Handbook of Logic and Language. Elsevier, Amsterdam, The MIT Press, Cambridge Mass. (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Wojciech Buszkowski
    • 1
  1. 1.Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, Faculty of Mathematics and Computer ScienceUniversity of Warmia and Mazury in Olsztyn 

Personalised recommendations