Pietro Sala

Tableaux for Logics of Subinterval Structures over Dense Orderings

In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec.

An optimal Tableau-based decision algorithm for propositional neighborhood logic

In this paper we focus our attention on the decision problem for Propositional Neighborhood Logic (PNL for short). PNL is the proper subset of Halpern and Shohams modal logic of intervals whose modalities correspond to Allens relations meets and met by. We show that the satisfiability problem for PNL over the integers is NEXPTIME-complete. Then, we develop a sound and complete tableau-based decision procedure and we prove its optimality.

What's Decidable about Halpern and Shoham's Interval Logic? The Maximal Fragment ABBL

The introduction of Halpern and Shohams modal logic of intervals (later on called HS) dates back to 1986. Despite its natural semantics, this logic is undecidable over all interesting classes of temporal structures. This discouraged research in this area until recently, when a number of non-trivial decidable fragments have been found. This paper is a contribution towards the complete classification of HS fragments. Different combinations of Allens interval relations begins (B), meets (A), and later (L), and their inverses Abar, Bbar, and Lbar, have been considered in the literature. We know from previous work that the combination ABBbarAbar is decidable over finite linear orders and undecidable everywhere else. We extend these results by showing that ABBbarLbar is decidable over the class of all (resp., dense, discrete) linear orders, and that it is maximal w.r.t decidability over these classes: adding any other interval modality immediately leads to undecidability.

A decidable spatial logic with cone-shaped cardinal directions

We introduce a spatial modal logic based on cone-shaped cardinal directions over the rational plane and we prove that, unlike projection-based ones, such as, for instance, Compass Logic, its satisfiability problem is decidable (PSPACE-complete). We also show that it is expressive enough to subsume meaningful interval temporal logics, thus generalizing previous results in the literature, e.g., its decidability implies that of the subinterval/superinterval temporal logic interpreted over the rational line.

A general tableau method for propositional interval temporal logics: Theory and implementation ✩

In this paper, we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema’s CDT logic interpreted over partial orders (BCDT + for short). It combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented.

Optimal Tableaux for Right Propositional Neighborhood Logic over Linear Orders

The study of interval temporal logics on linear orders is a meaningful research area in computer science and artificial intelligence. Unfortunately, even when restricted to propositional languages, most interval logics turn out to be undecidable. Decidability has been usually recovered by imposing severe syntactic and/or semantic restrictions. In the last years, tableau-based decision procedures have been obtained for logics of the temporal neighborhood and logics of the subinterval relation over specificclasses of temporal structures. In this paper, we develop an optimal NEXPTIME tableau-based decision procedure for the future fragment of Propositional Neighborhood Logic over the wholeclass of linearly ordered domains.

Interval temporal logics over finite linear orders: the complete picture

Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. In this paper, we identify all fragments of Halpern and Shohams interval temporal logic HS whose finite satisfiability problem is decidable. We classify them in terms of both relative expressive power and complexity. We show that there are exactly 62 expressively-different decidable fragments, whose complexity ranges from NP-complete to non-primitive recursive (all other HS fragments have been already shown to be undecidable).

Optimal tableau systems for propositional neighborhood logic over all, dense, and discrete linear orders

In this paper, we focus our attention on tableau systems for the propositional interval logic of temporal neighborhood (Propositional Neighborhood Logic, PNL for short). PNL is the proper subset of Halpern and Shohams modal logic of intervals whose modalities correspond to Allens relations meets and met by. We first prove by a model-theoretic argument that the satisfiability problem for PNL over the class of all (resp., dense, discrete) linear orders is decidable (and NEXPTIME-complete). Then, we develop sound and complete tableau-based decision procedures for all the considered classes of orders, and we prove their optimality. (As a matter of fact, decidability with respect to the class of all linear orders had been already proved via a reduction to the decidable satisfiability problem for the two-variable fragment of first-order logic of binary relational structures over ordered domains).

Tableau Systems for Logics of Subinterval Structures over Dense Orderings

We construct a sound, complete, and terminating tableau system for the interval temporal logic

Interval Temporal Logics over Strongly Discrete Linear Orders: the Complete Picture

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