Studia Logica

, Volume 89, Issue 1, pp 1–18 | Cite as

Infinitary Action Logic: Complexity, Models and Grammars

Article

Abstract

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the \({\Pi_{1}^{0}}\) –completeness of the equational theories of action lattices of subsets of a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics.

Keywords

Kleene algebra action algebra relation algebra categorial grammar 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bar-Hillel Y., Gaifman C. and Shamir E. (1960). ‘On categorial and phrase structure grammars’. Bull. Res. Council Israel F9: 155–166 Google Scholar
  2. 2.
    Buszkowski W. (1997) ‘Mathematical Linguistics and Proof Theory’. In: van Benthem J., ter Meulen A. (eds). Handbook of Logic and Language. Elsevier, The MIT Press, Amsterdam, Cambridge MA., pp. 683–736Google Scholar
  3. 3.
    Buszkowski, W., ‘On the complexity of the equational theory of relational action algebras’, in R. Schmidt (ed.), Relations and Kleene Algebra in Computer Science, LNCS 4136, 2006, pp. 106–119.Google Scholar
  4. 4.
    Buszkowski W. (2007). ‘On action logic: Equational theories of action algebras’. Journal of Logic and Computation 17.1: 199–217 Google Scholar
  5. 5.
    Dunn J.M. (1973). ‘A “Gentzen system” for positive relevant implication’ (abstract). Journal of Symbolic Logic 38: 356–357 Google Scholar
  6. 6.
    Dunn, J.M., ‘Relevance Logic and Entailment’, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. III, D. Reidel, Dordrecht, 1986, pp. 117–224.Google Scholar
  7. 7.
    Hardin, C., and D. Kozen, ‘On the complexity of the Horn theory of REL’, manuscript, Cornell University, 2003.Google Scholar
  8. 8.
    Hoare C., Jifeng H. ‘The weakest prespecification’, Fundamenta Informaticae 9 (1986), 51–84, 217–252Google Scholar
  9. 9.
    Jipsen P. (2004). ‘From semirings to residuated Kleene algebras’. Studia Logica 76: 291–303 CrossRefGoogle Scholar
  10. 10.
    Kozen, D., ‘On Kleene algebras and closed semirings’, Proc. of MFCS 1990, LNCS 452, 1990, pp. 26–47.Google Scholar
  11. 11.
    Kozen D. (1994). ‘A completeness theorem for Kleene algebras and the algebra of regular events’. Information and Computation 110(2): 366–390 CrossRefGoogle Scholar
  12. 12.
    Kozen D. (2002). ‘On the complexity of reasoning in Kleene algebra’. Information and Computation 179: 152–162 CrossRefGoogle Scholar
  13. 13.
    Kozen, D., and F. Smith, ‘Kleene algebra with tests: Completeness and decidability’, in D. van Dalen and M. Bezem (eds.), Proc. 10th Int. Workshop Computer Science Logic (CSL’96), LNCS 1258, Springer, 1996, pp. 244–259.Google Scholar
  14. 14.
    Lambek J. (1958). ‘The mathematics of sentence structure’. American Mathematical Monthly 65: 154–170 CrossRefGoogle Scholar
  15. 15.
    Mints G. (1976). ‘Cut Elimination Theorem in Relevant Logics’. Journal of Mathematical Sciences 6: 422–428 Google Scholar
  16. 16.
    Moortgat, M., ‘Categorial Type Logics’, in J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language, Elsevier, The MIT Press, Amsterdam, Cambridge MA, 1997, pp. 93–177.Google Scholar
  17. 17.
    Ono H. (1993) ‘Semantics of substructural logics’. In: Schroeder-Heister P., Dosen K. (eds). Substructural Logics. Clarendon Press, Oxford, pp. 259–291Google Scholar
  18. 18.
    Palka E. (2007). ‘An infinitary sequent system for the equational theory of *-continuous action lattices’. Fundamenta Informaticae 78.2: 295–309 Google Scholar
  19. 19.
    Palka, E., Logical systems for Kleene algebras and action algebras, PhD Thesis, 2007. In Polish, unpublished.Google Scholar
  20. 20.
    Pentus M. (2006). ‘Lambek calculus is NP-complete’. Theoretical Computer Science 357: 186–201 CrossRefGoogle Scholar
  21. 21.
    Pratt, V., ‘Action logic and pure induction’, in J. van Eijck (ed.), Logics in AI, LNAI 478, 1991, pp. 97–120.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceAdam Mickiewicz University in Poznań Research Group on Mathematical Linguistics Rovira i Virgili University in TarragonaPolandSpain
  2. 2.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland

Personalised recommendations