Tableau Systems for Logics of Subinterval Structures over Dense Orderings

Published: 01 Jan 2007, Last Modified: 14 Oct 2024TABLEAUX 2007EveryoneRevisionsBibTeXCC BY-SA 4.0
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.
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