An In-Depth Investigation of Interval Temporal Logic Model Checking with Regular Expressions

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...

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“…the latter being a hybrid CTL * l p formula. Thus, by Points (3), (5), and (6), we obtain that ϕ(x) and ϕ LTL p (x) are congruent.…”

Section: From Hs Ct To Finitary Ctl *mentioning

confidence: 85%

Interval vs. Point Temporal Logic Model Checking

Bozzelli

Molinari

Montanari

et al. 2018

ACM Trans. Comput. Logic

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“…, w |W | }. 7 We consider its negation ¬ψ and build from it a tree-structure of blocks T K ,¬ψ . Each block of T K ,¬ψ has a type, forward or backward, and it is associated with a subformula of ¬ψ.…”

Section: The Reduction Of the MC Problem For Aa Formulas To Tb(sat) 1×mmentioning

confidence: 99%

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

Bozzelli¹,

Molinari²,

Montanari³

et al. 2018

Information and Computation

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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.

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“…As for the semantics of HS, in this paper we follow the approach of [3], where the intervals correspond to the traces of a finite Kripke structure K (state-based semantics) and each abstract interval proposition p u ∈ P u denotes a regular language of finite words over 2 AP . More specifically, every abstract interval proposition p u is a (propositional-based) regular expression over AP .…”

Section: The Interval Temporal Logic Hsmentioning

confidence: 99%

“…In that work, the authors prove the decidability of MC with regular expressions for some very restricted fragments of epistemic HS, giving some rough upper bounds to its computational complexity. A deeper insight into the problem of MC for HS with regular expressions can be found in [3] where, under the assumption of a state-based semantics, it is proved that MC with regular expressions for full HS is decidable, and that a large class of HS fragments can be checked in polynomial working space (see the third column of Table 1).…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we study the problems of MC for the two (syntactically) maximal (symmetric) fragments AABBE and AAEBE with regular expressions, which are not covered by [3], proving that the complexity of both problems is AEXP pol -complete. AEXP pol denotes the complexity class of problems decided by exponential-time bounded alternating Turing Machines with a polynomially bounded number [2,11] (e.g., the first-order theory of real addition with order [11]).…”

Section: Introductionmentioning

confidence: 99%

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

Bozzelli¹,

Molinari

Montanari

et al. 2017

Electron. Proc. Theor. Comput. Sci.

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In this paper, we investigate the model checking (MC) problem for Halpern and Shoham's interval temporal logic HS. In the last years, interval temporal logic MC has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained under the homogeneity assumption, that constrains a proposition letter to hold over an interval if and only if it holds over each component state. Recently, Lomuscio and Michaliszyn proposed a way to relax such an assumption by exploiting regular expressions to define the behaviour of proposition letters over intervals in terms of their component states. When homogeneity is assumed, the exact complexity of MC is a difficult open question for full HS and for its two syntactically maximal fragments AABBE and AAEBE. In this paper, we provide an asymptotically optimal bound to the complexity of these two fragments under the more expressive semantic variant based on regular expressions by showing that their MC problem is AEXP pol -complete, where AEXP pol denotes the complexity class of problems decided by exponential-time bounded alternating Turing Machines making a polynomially bounded number of alternations.

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An In-Depth Investigation of Interval Temporal Logic Model Checking with Regular Expressions

Cited by 5 publications

References 15 publications

Interval vs. Point Temporal Logic Model Checking

Interval vs. Point Temporal Logic Model Checking

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

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

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