Interval-based Synthesis

We introduce the synthesis problem for Halpern and Shoham's modal logic of intervals extended with an equivalence relation over time points, abbreviated HSeq. In analogy to the case of monadic second-order logic of one successor, the considered synthesis problem receives as input an HSeq formula phi and a finite set Sigma of propositional variables and temporal requests, and it establishes whether or not, for all possible evaluations of elements in Sigma in every interval structure, there exists an evaluation of the remaining propositional variables and temporal requests such that the resulting structure is a model for phi. We focus our attention on decidability of the synthesis problem for some meaningful fragments of HSeq, whose modalities are drawn from the set A (meets), Abar (met by), B (begins), Bbar (begun by), interpreted over finite linear orders and natural numbers. We prove that the fragment ABBbareq is decidable (non-primitive recursive hard), while the fragment AAbarBBbar turns out to be undecidable. In addition, we show that even the synthesis problem for ABBbar becomes undecidable if we replace finite linear orders by natural numbers.Comment: In Proceedings GandALF 2014, arXiv:1408.556

A decidable weakening of Compass Logic based on cone-shaped cardinal directions

We introduce a modal logic, called Cone Logic, whose formulas describe properties of points in the plane and spatial relationships between them. Points are labelled by proposition letters and spatial relations are induced by the four cone-shaped cardinal directions. Cone Logic can be seen as a weakening of Venema's Compass Logic. We prove that, unlike Compass Logic and other projection-based spatial logics, its satisfiability problem is decidable (precisely, PSPACE-complete). We also show that it is expressive enough to capture meaningful interval temporal logics - in particular, the interval temporal logic of Allen's relations "Begins", "During", and "Later", and their transposes

Metric propositional neighborhood logic with an equivalence relation

The propositional interval logic of temporal neighborhood (PNL for short) features two modalities that make it possible to access intervals adjacent to the right (modality \u27e8 A\u27e9) and to the left (modality \u27e8 A\uaf \u27e9) of the current interval. PNL stands at a central position in the realm of interval temporal logics, as it is expressive enough to encode meaningful temporal conditions and decidable (undecidability rules over interval temporal logics, while PNL is NEXPTIME-complete). Moreover, it is expressively complete with respect to the two-variable fragment of first-order logic extended with a linear order FO 2[<]. Various extensions of PNL have been studied in the literature, including metric, hybrid, and first-order ones. Here, we study the effects of the addition of an equivalence relation 3c to Metric PNL (MPNL 3c). We first show that the finite satisfiability problem for PNL extended with 3c is still NEXPTIME-complete. Then, we prove that the same problem for MPNL 3c can be reduced to the decidable 0\u20130 reachability problem for vector addition systems and vice versa (EXPSPACE-hardness immediately follows)

Maximal decidable fragments of Halpern and Shoham's modal logic of intervals

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 (AAbarBBbar logic). AAbarBBbar properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BBbar of Allen's ``begun by/begins'' relations and propositional neighborhood logic AAbar, in its many variants (including metric ones). We prove that the satisfiability problem for AAbarBBbar, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AAbarBBbar turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AAbarBBbar 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, Q, and R

A decidable weakening of Compass Logic based on cone-shaped cardinal directions

We introduce a modal logic, called Cone Logic, whose formulas describeproperties of points in the plane and spatial relationships between them.Points are labelled by proposition letters and spatial relations are induced bythe four cone-shaped cardinal directions. Cone Logic can be seen as a weakeningof Venema's Compass Logic. We prove that, unlike Compass Logic and otherprojection-based spatial logics, its satisfiability problem is decidable(precisely, PSPACE-complete). We also show that it is expressive enough tocapture meaningful interval temporal logics - in particular, the intervaltemporal logic of Allen's relations "Begins", "During", and "Later", and theirtransposes

Model Checking Well-Behaved Fragments of HS: The (Almost) Final Picture

Model checking is one of the most powerful and widespread tools for system verification with applications in many areas of computer science and artificial intelligence. The large majority of model checkers deal with properties expressed in point-based temporal logics, such as LTL and CTL. However, there exist relevant properties of systems which are inherently interval-based. Model checking algorithms for interval temporal logics (ITLs) have recently been proposed to check interval properties of computations. As the model checking problem for full Halpern and Shoham\u2019s ITL (HS for short) turns out to be decidable, but computationally heavy, research has focused on its well-behaved fragments. In this paper, we provide an almost final picture of the computational complexity of model checking for HS fragments with modalities for (a subset of) Allen\u2019s relations meets, met by, starts, and end

Pspace-Completeness of the Temporal Logic of Sub-Intervals and Suffixes

In this paper, we establish Pspace-completeness of the finite satisfiability and model checking problems for the fragment of Halpern and Shoham interval logic with modality ?E?, for the "suffix" relation on pairs of intervals, and modality ?D?, for the "sub-interval" relation, under the homogeneity assumption. The result significantly improves the Expspace upper bound recently established for the same fragment, and proves the rather surprising fact that the complexity of the considered problems does not change when we add either the modality for suffixes (?E?) or, symmetrically, the modality for prefixes (?B?) to the logic of sub-intervals (featuring only ?D?)

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

Model Checking the Logic of Allen's Relations Meets and Started-by is P^NP-Complete

In the plethora of fragments of Halpern and Shoham's modal logic of time intervals (HS), the logic AB of Allen's relations Meets and Started-by is at a central position. Statements that may be true at certain intervals, but at no sub-interval of them, such as accomplishments, as well as metric constraints about the length of intervals, that force, for instance, an interval to be at least (resp., at most, exactly) k points long, can be expressed in AB. Moreover, over the linear order of the natural numbers N, it subsumes the (point-based) logic LTL, as it can easily encode the next and until modalities. Finally, it is expressive enough to capture the {\omega}-regular languages, that is, for each {\omega}-regular expression R there exists an AB formula {\phi} such that the language defined by R coincides with the set of models of {\phi} over N. It has been shown that the satisfiability problem for AB over N is EXPSPACE-complete. Here we prove that, under the homogeneity assumption, its model checking problem is {\Delta}^p_2 = P^NP-complete (for the sake of comparison, the model checking problem for full HS is EXPSPACE-hard, and the only known decision procedure is nonelementary). Moreover, we show that the modality for the Allen relation Met-by can be added to AB at no extra cost (AA'B is P^NP-complete as well)

Interval vs. Point Temporal Logic Model Checking: an Expressiveness Comparison

Model checking is a powerful method widely explored in formal verification to check the (state-transition) model of a system against desired properties of its behaviour. Classically, properties are expressed by formulas of a temporal logic, such as LTL, CTL, and CTL*. These logics are "point-wise" interpreted, as they describe how the system evolves state-by-state. On the contrary, Halpern and Shoham\u27s interval temporal logic (HS) is "interval-wise" interpreted, thus allowing one to naturally express properties of computation stretches, spanning a sequence of states, or properties involving temporal aggregations, which are inherently "interval-based". In this paper, we study the expressiveness of HS in model checking, in comparison with that of the standard logics LTL, CTL, and CTL*. To this end, we consider HS endowed with three semantic variants: the state-based semantics, introduced by Montanari et al., which allows branching in the past and in the future, the linear-past semantics, allowing branching only in the future, and the linear semantics, disallowing branching. These variants are compared, as for their expressiveness, among themselves and to standard temporal logics, getting a complete picture. In particular, HS with linear (resp., linear-past) semantics is proved to be equivalent to LTL (resp., finitary CTL*)