Abstract. In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations "meets", "met by", "begun by", and "begins" of Allen's interval algebra (AĀBB logic). AĀBB properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BB of Allen's "begun by/begins" relations and propositional neighborhood logic AĀ, in its many variants (including metric ones). We prove that the satisfiability problem for AĀBB, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AĀBB turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AĀBB is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, and R.

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In the last years, model checking with interval temporal logics is emerging as a viable alternative to model checking with standard point-based temporal logics, such as LTL, CTL, CTL * , and the like. The behavior of the system is modelled by means of (finite) Kripke structures, as usual. However, while temporal logics which are interpreted "point-wise" describe how the system evolves state-by-state, and predicate properties of system states, those which are interpreted "interval-wise" express properties of computation stretches, spanning a sequence of states. A proposition letter is assumed to hold over a computation stretch (interval) if and only if it holds over each component state (homogeneity assumption). A natural question arises: is there any advantage in replacing points by intervals as the primary temporal entities, or is it just a matter of taste?In this paper, we study the expressiveness of Halpern and Shoham's interval temporal logic (HS) in model checking, in comparison with those of LTL, CTL, and CTL * . To this end, we consider three semantic variants of HS: the state-based one, introduced by Montanari et al. in [34,30], that allows time to branch both in the past and in the future, the computationtree-based one, that allows time to branch in the future only, and the trace-based variant, that disallows time to branch. These variants are compared among themselves and to the aforementioned standard logics, getting a complete picture. In particular, we show that HS with trace-based semantics is equivalent to LTL (but at least exponentially more succinct), HS with computation-tree-based semantics is equivalent to finitary CTL * , and HS with state-based semantics is incomparable with all of them (LTL, CTL, and CTL * ). The work has been supported by the GNCS project Formal Methods for Verification and Synthesis of Discrete and Hybrid Systems. The work by A. Molinari and A. Montanari has also been supported by the project (PRID) ENCASE -Efforts in the uNderstanding of Complex interActing SystEms. * This work is an extended and revised version of [8].Structure of the paper. In Section 2, we introduce basic notation and preliminary notions. In Subsection 2.1 we define Kripke structures and interval structures, in Subsection 2.2 we recall the well-known PTLs LTL, CTL, and CTL * , and in Subsection 2.3 we present the interval temporal logic HS. Then, in Subsection 2.4 we define the three semantic variants of HS (HS st , HS ct , and HS lin ). Finally, in Subsection 2.5 we provide a detailed example which gives an intuitive account of the three semantic variants and highlights their differences. In the next three sections, we analyze and compare their expressiveness. In Section 3 we show the expressive equivalence of LTL and HS lin . Then, in Section 4 we prove the expressive equivalence of HS ct and finitary CTL * . Finally, in Section 5 we compare the expressiveness of HS st , HS ct , and HS lin . Conclusions summarize the work done and outline some directions for future research.0 ), where the set of node...

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. Their computational behavior mainly depends on two parameters: the set of modalities they feature and the linear orders over which they are interpreted. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders as well as over its relevant subclasses (the class of finite linear orders, Z, N, and Z−). We classify them in terms of both their relative expressive power and their complexity, which ranges from NP-completeness to non-primitive recursiveness

Interval temporal logics provide a general 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 Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders. We classify them in terms of both their relative expressive power and their complexity. We show that there are exactly 44 expressively different decidable fragments, whose complexity ranges from NP to EXPSPACE. In addition, we identify some new undecidable fragments (all the remaining HS fragments were already known to be undecidable over strongly discrete linear orders). We conclude the paper by an analysis of the specific case of natural numbers, whose behavior slightly differs from that of the whole class of strongly discrete linear orders. The number of decidable fragments over N raises up to 47: three undecidable fragments become decidable with a non-primitive recursive complexity.

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Abstract. The model checking problem has thoroughly been explored in the context of standard point-based temporal logics, such as LTL, CTL, and CTL ⇤ , whereas model checking for interval temporal logics has been brought to the attention only very recently. In this paper, we prove that the model checking problem for the logic of Allen's relations started-by and finished-by is highly intractable, as it can be proved to be EXPSPACE-hard. Such a lower bound immediately propagates to the full Halpern and Shoham's modal logic of time intervals (HS). In contrast, we show that other noteworthy HS fragments, namely, Propositional Neighbourhood Logic extended with modalities for the Allen relation starts (resp., finishes) and its inverse started-by (resp., finished-by), turn out to have-maybe unexpectedly-the same complexity as LTL (i.e., they are PSPACE-complete), thus joining the group of other already studied, well-behaved albeit less expressive, HS fragments.

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