Interval vs. Point Temporal Logic Model Checking

Abstract

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...

On the Complexity of Model Checking for Syntactically Maximal Fragments of the Interval Temporal Logic HS with Regular Expressions

“…In particular, while the modalities for B and E are linear-time (they allow us to select prefixes and suffixes of the current trace), the modalities for A and B (resp., A and E) are branching-time in the future (resp., in the past) since they allow us to nondeterministically extend a trace in the future (resp., in the past). As shown in [6], for the considered semantics, the logics HS and CTL * are expressively incomparable already under the homogeneity assumption. However, under the homogeneity assumption, the use of the past branching-time modalities A and E is necessary for capturing requirements which cannot be expressed in CTL * .…”

“…However, under the homogeneity assumption, the use of the past branching-time modalities A and E is necessary for capturing requirements which cannot be expressed in CTL * . For instance, the requirement "each state reachable from the initial one where p holds has a predecessor where p holds as well" cannot be expressed in CTL * , but can be easily expressed in the fragment AE [6]. In the more expressive setting based on regular expressions, the future branching-time modalities A and B are already sufficient for capturing requirements which cannot be expressed in CTL * , such as the following branching-time bounded response property: "for each state reachable from the initial one where a request req occurs, there is a computation from this state such that the request is followed by a response res within an even number of steps".…”

“…Only few, hard issues are left open in this picture, mostly regarding the precise complexity of the full logic and its maximal fragments. A comparison of different semantic solutions (i.e., state-based semantics, trace-based semantics and computation-tree-based semantics), together with an expressiveness comparison with standard point-based temporal logics LTL, CTL, and CTL * can be found in [6].…”

An Auxiliary Logic on Trees: on the Tower-Hardness of Logics Featuring Reachability and Submodel Reasoning

“…An example of this is given by the framework of temporal logics, formalisms that describe the evolution of reactive systems [24]. Among the various temporal logics, from the classical linear temporal logic (LTL) [39] and computation tree logic (CTL) [13], as well as their fragments [2,33], to the more recently developed interval temporal logics [7,8], the main common feature of this framework is perhaps the ability to check whether the system can evolve to a certain configuration, i.e. a reachability query.…”

Attack Graph Implementation and Visualization for Cyber Physical Systems

“…The expressiveness of Halpern and Shoham's interval temporal logic (HS) is studied by [32] in the context of model checking (MC) in comparison with those of the standard point-based temporal logics (PTLs), linear temporal logic (LTL), computation tree logic (CTL), and CTL * (a superset of CTL). The results show that HS with trace-based semantics is equivalent to LTL, HS with computation-tree-based semantics is equivalent to finitary CTL * , and HS with state-based semantics is incomparable with LTL, CTL, and CTL * .…”