Tableau Systems for Logics of Subinterval Structures over Dense Orderings

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


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.


Modal Logic Simple Path Interval Model Expansion Rule Dense Ordering 
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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

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