Tableau Systems for Logics of Subinterval Structures over Dense Orderings

  • Authors:

  • Davide Bresolin;Valentin Goranko;Angelo Montanari;Pietro Sala

  • Affiliations:

  • Department of Computer Science, University of Verona, Verona, Italy;School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa;Department of Mathematics and Computer Science, University of Udine, Udine, Italy;Department of Mathematics and Computer Science, University of Udine, Udine, Italy

  • Venue:
  • TABLEAUX '07 Proceedings of the 16th international conference on Automated Reasoning with Analytic Tableaux and Related Methods
  • Year:
  • 2007

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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 strictsubinterval 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.