Tableau Systems for Logics of Subinterval Structures over Dense Orderings

We construct a sound, complete, and terminating tableau system for the interval temporal logic Dsquare subset. 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 Dsquare subset-structures, and show that every formula satisfiable in Dsquare subset is satisfiable in such pseudo-models, thereby proving small-model property and decidability in PSPACE of Dsquare subset, a result established earlier by Shapirovsky and Shehtman by means of filtration. We also show how to extend our results to the interval logic Dsquare subset interpreted over dense interval structures with proper (irreflexive) subinterval relation, which differs substantially from Dsquare subset and is generally more difficult to analyze. Up to our knowledge, no complete deductive systems and decidability results for Dsquare subset have been proposed in the literature so far.