Model checking for fragments of the interval temporal logic HS at the low levels of the polynomial time hierarchy

Abstract

Some temporal properties of reactive systems, such as actions with duration, accomplishments, and temporal aggregations, which are inherently interval-based, can not be properly dealt with by the standard, point-based temporal logics LTL, CTL and CTL*, as they give a state-by-state account of system evolution. Conversely, interval temporal logics-which feature intervals, instead of points, as their primitive entities-are highly expressive formalisms for temporal representation and reasoning that naturally allow one to deal with them. In this paper, we study the model checking (MC) problem for Halpern and Shoham's modal logic of time intervals (HS), interpreted on Kripke structures, under the homogeneity assumption, according to which a proposition letter holds over a finite computation path (interval) if and only if it holds at all of its states. HS is the best known interval-based temporal logic, which has one modality for each of the 13 possible ordering relations between pairs of intervals (the so-called Allen's relations), apart from equality. We focus on the MC problem for some HS fragments featuring modalities for (a subset of) Allen's relations meet, met-by, started-by, and finished-by, showing that it is in P NP , a class to which other pointbased logics (e.g., CTL+ and FCTL) are known to belong. Additionally, we provide some complexity lower bounds to the problem. All the algorithms we propose can be efficiently implemented by means of a polynomial-time procedure which iteratively invokes a SATsolver, enabling us to directly exploit the great speed of SAT-solvers.

Complexity Assessments for Decidable Fragments of Set Theory. I: A Taxonomy for the Boolean Case*

“…Quite recently, regarding model-checking in the framework of Halpern and Shoham's modal logic of time intervals, we have [27,28].…”

Parametric Interval Temporal Logic over Infinite Words

“…Model checking of (finite) Kripke structures against HS has been investigated only recently [25,26,27,29,30,6,7,4,9]. The idea is to interpret each finite path of a Kripke structure as an interval, whose labelling is defined on the basis of the labelling of the component states, that is, a proposition letter holds over an interval if and only if it holds over each component state (homogeneity assumption [36]).…”

Complexity analysis of a unifying algorithm for model checking interval temporal logic

“…Model checking of (finite) Kripke structures against HS has been investigated only very recently [13,14,16,2,5,4,3,18,19]. The idea is to interpret each finite path of a Kripke structure as an interval, whose labelling is defined on the basis of the labelling of the component states: a proposition letter holds over an interval if and only if it holds over each component state (homogeneity assumption [24]).…”

“…Paths having the same h-level BE-descriptor (i) are indistinguishable with respect to the fulfillment of HS formulas having nesting depth of modalities for prefixes (B) and suffixes (E) at most h, and (ii) admit a bounded minimal representative (h-level BE-certificate) whose length is at most a tower of exponentials of height h. The model-checking procedure for full HS based on BE-descriptors is only sketched in [16] and, in particular, the succinctness of BE-descriptors has not been investigated so far. In subsequent papers [3,19,4,17], the focus has been on some syntactical fragments of HS: the fragment featuring only the modalities for the contains relation (D), and fragments featuring modalities for a subset of the Allen relations meets (A), started-by (B), finished-by (E) and their transposed relations A, B, and E, respectively (see Table 1 for a graphical intuition of relations). The complete picture of known results is reported in Figure 1.…”