Complete and Terminating Tableau for the Logic of Proper Subinterval Structures over Dense Orderings

We introduce special pseudo-models for the interval logic of proper subintervals over dense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminating tableau for that logic. The case of proper subintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work

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

Complete and Terminating Tableau for the Logic of Proper Subinterval Structures Over Dense Orderings

We introduce special pseudo-models for the interval logic of propersubintervalsoverdense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminatingtableau for that logic. The case of propersubintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work

Tableau Systems for Logics of Subinterval Structures over Dense Orderings

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