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Pré-Publication, Document De Travail Année : 2017

A Regular Metric Temporal Logic

Khushraj Madnani
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Paritosh Pandya
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Résumé

We study an extension of MTL in pointwise time with regular expression guarded modality Reg I (re) where re is a regular expression over subformulae. We study the decidability and ex-pressiveness of this extension, called RegMTL, as well as its fragment SfrMTL where only star-free extended regular expressions are allowed. Using the technique of temporal projections, we show that RegMTL has decidable satisfiability by giving an equisatisfiable reduction to MTL. Moreover, we identify a subset MITL[UReg] for which our (polynomial time computable) equi-satisfiable reduction gives rise to formulae of MITL. Thus, MITL[UReg] has elementary decidability. As our second main result, we show that SfrMTL is equivalent to partially ordered (or very weak) 1-clock alternating timed automata. We also identify natural fragments of logic TPTL which correspond to our logics. Digital Object Identifier 10.4230/LIPIcs... 1 Introduction Temporal logics provide constructs to specify qualitative ordering between events in time. Real time logics are quantitative extensions of temporal logics with the ability to specify real time constraints amongst events. The main modality in Metric Temporal Logic (MTL) is the until modality a U I b which, when asserted at a point specifies that there is a future within a time distance in I where b holds, and a holds continuously till then. Two notions of MTL have been studied in the literature : continuous and pointwise. It is known [1] that satisfiability checking of MTL is undecidable in the continuous semantics even for finite words, while for the pointwise case, this is decidable [10]. The complexity of the satisfiability problem for MTL over finite timed words is known to be non-primitive recursive (NPR) in the pointwise semantics, while if the intervals I allowed in the until modalities are non-punctual, then the complexity drops to EXPSPACE in both the pointwise and continuous semantics. The fragment of MTL with only non-punctual intervals is denoted MITL, and was introduced in [1]. A non-punctual interval has the form x, y where x < y, x ∈ N, y ∈ N ∪ {∞}. There are various natural extensions of temporal logics have been studied both in classical and timed logic domain. Wolper extended LTL with certain grammar operators to achieve MSO completeness. Baziramwabo, McKenzie and Thérien extended LTL with modular and group modalities, and showed that the latter is as expressive as regular languages [2]. Counting LTL is an extension of LTL with threshold counting. It has been shown that this extension does not increase the expressive power of LTL [6]. As another extension, LTL with just modulo counting modalities has been studied by [7]. In timed logics, Raskin's Ph.D thesis studied various extensions of MITL with the ability to count over the entire model. Rabinovich et. al. extended continuous MITL with counting (called the C modality) and Pnueli modalities [13] and showed that these extensions are more expressive than MITL. The counting modalities C n (φ), specify that the number of points that satisfy φ within licensed under Creative Commons License CC-BY
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Dates et versions

hal-01483315 , version 1 (08-03-2017)

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  • HAL Id : hal-01483315 , version 1

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Shankara Narayanan Krishna, Khushraj Madnani, Paritosh Pandya. A Regular Metric Temporal Logic. 2017. ⟨hal-01483315⟩
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