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Algorithms for monitoring real-time properties

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Abstract

Real-time logics are popular specification languages for reasoning about systems intended to meet timing constraints. Numerous formalisms have been proposed with different underlying time models that can be characterized along two dimensions: dense versus discrete time and point-based versus interval-based. We present monitoring algorithms for the past-only fragment of metric temporal logics that differ along these two dimensions, analyze their complexity, and compare them on a class of formulas for which the point-based and the interval-based settings coincide. Our comparison reveals similarities and differences between the monitoring algorithms and highlights key concepts underlying our and prior monitoring algorithms. For example, point-based algorithms are conceptually simpler and more efficient than interval-based ones as they are invoked only at time points occurring in the monitored trace and their reasoning is limited to just those time points.

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Notes

  1. We do not use \({\mathbb {R}}_{\ge 0}\) as dense time domain because of representation issues. Namely, each element in \({\mathbb {Q}}_{\ge 0}\) can be finitely represented, which is not the case for \({\mathbb {R}}_{\ge 0}\). Choosing \({\mathbb {Q}}_{\ge 0}\) instead of \({\mathbb {R}}_{\ge 0}\) is without loss of generality for the satisfiability of properties specified in real-time logics like metric interval temporal logic [1].

  2. Note that \(\frac{p}{q} - \frac{p^{\prime }}{q^{\prime }}=\frac{p\cdot q^{\prime }- p^{\prime }\cdot q}{q\cdot q^{\prime }}\) and that \(\mathcal {O}(m^2)\) is an upper bound on the multiplication of two m bit integers. There are more sophisticated algorithms for multiplication that run in \(\mathcal {O}(m \log m \log \log m)\) time [32] and \(\mathcal {O}(m\log m 2^{\log ^* m})\) time [13], where \(\log ^* m\) denotes the iterated logarithm of m. For simplicity, we use the quadratic upper bound.

  3. In case \(r(I)=\infty \), we actually store fewer intervals \(K_2\) in \(\varDelta ^{\prime }_2\) then those appearing in the equality (5.4). Details are provided when explaining the contents of \(\varDelta _\phi \).

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Acknowledgements

We thank Kevin Baldor and Jianwei Niu for answering our questions about their work. The Nokia Research Center, Switzerland and the Zurich Information Security and Privacy Center (ZISC) partly supported this work.

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Authors and Affiliations

  1. Department of Computer Science, Institute of Information Security, ETH Zurich, Zurich, Switzerland

    David Basin

  2. NEC Europe Ltd., Heidelberg, Germany

    Felix Klaedtke

  3. Institut für Informatik, Technische Universität München, Munich, Germany

    Eugen Zălinescu

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  1. David Basin
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  2. Felix Klaedtke
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  3. Eugen Zălinescu
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Correspondence to Eugen Zălinescu.

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A preliminary version of this article appeared in the proceedings of the 2nd International Conference on Runtime Verification [7].

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Basin, D., Klaedtke, F. & Zălinescu, E. Algorithms for monitoring real-time properties. Acta Informatica 55, 309–338 (2018). https://doi.org/10.1007/s00236-017-0295-4

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