An Optimal Decision Procedure for Right Propositional Neighborhood Logic

Abstract

Propositional interval temporal logics are quite expressive temporal logics that allow one to naturally express statements that refer to time intervals. Unfortunately, most such logics turn out to be (highly) undecidable. In order to get decidability, severe syntactic or semantic restrictions have been imposed to interval-based temporal logics to reduce them to point-based ones. The problem of identifying expressive enough, yet decidable, new interval logics or fragments of existing ones that are genuinely interval-based is still largely unexplored. In this paper, we focus our attention on interval logics of temporal neighborhood. We address the decision problem for the future fragment of Neighborhood Logic (Right Propositional Neighborhood Logic, RPNL for short), and we positively solve it by showing that the satisfiability problem for RPNL over natural numbers is NEXPTIME-complete. Then, we develop a sound and complete tableau-based decision procedure, and we prove its optimality.Key words interval temporal logic · tableaux-based decision procedures · right propositional neighborhood logic

Reasoning with Time Intervals: A Logical and Computational Perspective

“…The technical level required to understand such results is not trivial and involves the knowledge of model theory, complexity theory, and terminating and nonterminating tableaux, among others. For the interested reader, we give here three representative examples: the first one is the maximal incomplete and minimal complete sets of first-order interval relations in the class of all linearly ordered sets (Table 2), known from [35], the second one is the undecidability of full HS (a result known since [37], but presented here is a novel and simpler form), and the third one is the decidability of A (known since [44]). In the latter two cases, we focus our attention on the finite case, for the sake of simplicity.…”

“…NEXPTIMEmembership of AA has been proved in [27]. NEXPTIMEhardness of A, shown in [44], holds also for the class…”

“…NEXPTIME-membership of AA (also known as Propositional Neighborhood Logic, or PNL) has been shown in [27]. NEXPTIME-hardness of A, given in [44], holds also for finite satisfiability, and it can be easily adapted to the case of A. NEXPTIME-hardness of any fragment containing A or A immediately follows.…”

A new modal logic for reasoning about space: spatial propositional neighborhood logic

“…By using a technique similar to the one which can be found in [BMG06], it is possible to prove that τ (f ) is satisfiable over upright local SpPNL-models if and only if f is satisfiable, by providing a suitable translation between Compass Logic models and upright local SpPNL-models, and vice versa (notice that the universal operator is definable in SpPNL, as we will see in the Section 5).…”

Right Propositional Neighborhood Logic over Natural Numbers with Integer Constraints for Interval Lengths

“…For a long time, such sweeping undecidability results have discouraged attempts for practical applications and further research on interval logics. A renewed interest in the area has been recently stimulated by the discovery of some interesting decidable fragments of HS [7,8,9,12,13]. As an effect, the identification of expressive enough, yet decidable, fragments of HS has become one of the major topics of the current research agenda in interval temporal logics.…”