The dark side of interval temporal logic: marking the undecidability border

et al. 2016

Automated Reasoning

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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.

Section: Introductionmentioning

confidence: 99%

Interval Temporal Logic Model Checking: The Border Between Good and Bad HS Fragments

Bozzelli

et al. 2016

Automated Reasoning

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“…In [8], it has been shown that the satisfiability problem for HS interpreted over all relevant (classes of) linear orders is highly undecidable. Since then, a lot of work has been done on satisfiability for HS fragments, which showed that undecidability rules over them [3], [10], [13]. However, meaningful exceptions exist, e.g., the interval logic of temporal neighbourhood AA and the logic of sub-intervals D [4]- [6], [17].…”

Section: Introductionmentioning

confidence: 99%

Complexity of ITL Model Checking: Some Well-Behaved Fragments of the Interval Logic HS

2015 22nd International Symposium on Temporal Representation and Reasoning (TIME)

Peron³

2015

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Model checking has been successfully used in many computer science fields, including artificial intelligence, theoretical computer science, and databases. Most of the proposed solutions make use of classical, point-based temporal logics, while little work has been done in the interval temporal logic setting. Recently, a non-elementary model checking algorithm for Halpern and Shoham's modal logic of time intervals HS over finite Kripke structures (under the homogeneity assumption) and an EXPSPACE model checking procedure for two meaningful fragments of it have been proposed. In this paper, we show that more efficient model checking procedures can be developed for some expressive enough fragments of HS

“…(Actually, the three Allen's modalities meets A, started-by B, and finished-by E, together with the corresponding inverse modalities A, B, and E, suffice for expressing the entire set of relations.) The satisfiability problem for HS is undecidable over all relevant classes of linear orders [13], and most of its fragments (with meaningful exceptions) are undecidable as well [8,19].…”

Section: Introductionmentioning

confidence: 99%

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

Bozzelli¹,

Electron. Proc. Theor. Comput. Sci.

et al. 2017

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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.