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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
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)
Blackburn, P., de Rijke, M., Venema, V.: Modal Logic. CUP (2001)
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)
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)
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)
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)
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)
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)
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)
Halpem, J.Y., Shoham, Y.: A propositional modal logic of time intervals. Journal of the ACM 38(4), 935–962 (1991)
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)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bresolin, D., Goranko, V., Montanari, A., Sala, P. (2007). Tableau Systems for Logics of Subinterval Structures over Dense Orderings. In: Olivetti, N. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2007. Lecture Notes in Computer Science(), vol 4548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73099-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-73099-6_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73098-9
Online ISBN: 978-3-540-73099-6
eBook Packages: Computer ScienceComputer Science (R0)