Tableau Systems for Logics of Subinterval Structures over Dense Orderings

  • Davide Bresolin
  • Valentin Goranko
  • Angelo Montanari
  • Pietro Sala
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4548)

Abstract

We construct a sound, complete, and terminating tableau system for the interval temporal logic \({{\rm D}_\sqsubset}\) interpreted in interval structures over dense linear orderings endowed with strict subinterval relation (where both endpoints of the sub-interval are strictly inside the interval). In order to prove the soundness and completeness of our tableau construction, we introduce a kind of finite pseudo-models for our logic, called \({{\rm D}_\sqsubset}\)-structures, and show that every formula satisfiable in \({{\rm D}_\sqsubset}\) is satisfiable in such pseudo-models, thereby proving small-model property and decidability in PSPACE of \({{\rm D}_\sqsubset}\), a result established earlier by Shapirovsky and Shehtman by means of filtration. We also show how to extend our results to the interval logic \({{\rm D}_\sqsubset}\) interpreted over dense interval structures with proper (irreflexive) subinterval relation, which differs substantially from \({{\rm D}_\sqsubset}\) and is generally more difficult to analyze. Up to our knowledge, no complete deductive systems and decidability results for \({{\rm D}_\sqsubset}\) have been proposed in the literature so far.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    van Benthem, J.: The Logic of Time: A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse, 2nd edn. Kluwer, Norwell (1991)MATHGoogle Scholar
  2. 2.
    Blackburn, P., de Rijke, M., Venema, V.: Modal Logic. CUP (2001)Google Scholar
  3. 3.
    Bowman, H., Thompson, S.: A decision procedure and complete axiomatization of finite interval temporal logic with projection. Journal of Logic and Computation 13(2), 195–239 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: On Decidability and Expressiveness of Propositional Interval Neighborhood Logics. In: LFCS 2007. Proc. of the International Symposium on Logical Foundations of Computer Science. LNCS, vol. 4514, pp. 84–99. Springer, Heidelberg (2007)Google Scholar
  5. 5.
    Bresolin, D., Montanari, A.: A tableau-based decision procedure for a branching-time interval temporal logic. In: Schlingloff, H. (ed.) Proc. of the 4th Int. Workshop on Methods for Modalities, pp. 38–53 (2005)Google Scholar
  6. 6.
    Bresolin, D., Montanari, A.: A tableau-based decision procedure for Right Propositional Neighborhood Logic. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Bresolin, D., Montanari, A., Sala, P.: An optimal tableau-based decision algorithm for Propositional Neighborhood Logic. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Bresolin, D., Montanari, A., Sciavicco, G.: An optimal decision procedure for Right Propositional Neighborhood Logic. Journal of Automated Reasoning 38(1-3), 173–199 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goranko, V., Montanari, A., Sciavicco, G., Sala, P.: A general tableau method for propositional interval temporal logics: Theory and implementation. Journal of Applied Logic 4(3), 305–330 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Halpem, J.Y., Shoham, Y.: A propositional modal logic of time intervals. Journal of the ACM 38(4), 935–962 (1991)CrossRefGoogle Scholar
  11. 11.
    Shapirovsky, I.: On PSPACE-decidability in Transitive Modal Logic. In: Schmidt, R., Pratt-Hartmann, I., Reynolds, M., Wansing, H. (eds.) Advances in Modal Logic, vol. 5, pp. 269–287. King’s College Publications, London (2005)Google Scholar
  12. 12.
    Shapirovsky, I., Shehtman, V.: Chronological future modality in Minkowski spacetime. In: Balbiani, P., Suzuki, N.Y., Wolter, F., Zakharyaschev, M. (eds.) Advances in Modal Logic, vol. 4, pp. 437–459. King’s College Publications, London (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Davide Bresolin
    • 1
  • Valentin Goranko
    • 2
  • Angelo Montanari
    • 3
  • Pietro Sala
    • 3
  1. 1.Department of Computer Science, University of Verona, VeronaItaly
  2. 2.School of Mathematics, University of the Witwatersrand, JohannesburgSouth Africa
  3. 3.Department of Mathematics and Computer Science, University of Udine, UdineItaly

Personalised recommendations