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Parameterized linear temporal logics meet costs: still not costlier than LTL

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Abstract

We continue the investigation of parameterized extensions of linear temporal logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for parametric LDL (PLDL), which extends PLTL to be able to express all \(\omega \)-regular properties. Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.

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Notes

  1. Note that the bound in the parity condition with costs may depend on the trace while one typically uses global bounds for \(\text {cPLTL}\) (see, e.g., Sects. 4 and 5). However, for games in finite arenas (and thus also for model checking) these two variants coincide [22].

  2. Note that our definition is more involved than the one of Kupferman et al., since we require a cycle with non-zero cost instead of any circle.

  3. Here, we use our assumption on \(\kappa \) indicating the sign of the costs.

  4. The same disclaimer as for the parity condition with costs applies here. See Footnote 1.

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  1. Reactive Systems Group, Saarland University, 66123, Saarbrücken, Germany

    Martin Zimmermann

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  1. Martin Zimmermann
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Correspondence to Martin Zimmermann.

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Supported by the project “TriCS” (ZI 1516/1-1) of the German Research Foundation (DFG).

A preliminary version of this work appeared in the proceedings of GandALF 2015 [39].

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Zimmermann, M. Parameterized linear temporal logics meet costs: still not costlier than LTL. Acta Informatica 55, 129–152 (2018). https://doi.org/10.1007/s00236-016-0279-9

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