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The Density of Linear-Time Properties

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Automated Technology for Verification and Analysis (ATVA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10482))

Abstract

Finding models for linear-time properties is a central problem in verification and planning. We study the distribution of linear-time models by investigating the density of linear-time properties over the space of ultimately periodic words. The density of a property over a bound n is the ratio of the number of lasso-shaped words of length n, that satisfy the property, to the total number of lasso-shaped words of length n. We investigate the problem of computing the density for both linear-time properties in general and for the important special case of \(\omega \)-regular properties. For general linear-time properties, the density is not necessarily convergent and can oscillates indefinitely for certain properties. However, we show that the oscillation is bounded by the growth of the sets of bad- and good-prefix of the property. For \(\omega \)-regular properties, we show that the density is always convergent and provide a general algorithm for computing the density of \(\omega \)-regular properties as well as more specialized algorithms for certain sub-classes and their combinations.

This work was partly supported by the ERC Grant 683300 (OSARES) and by the Deutsche Telekom Foundation.

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Notes

  1. 1.

    For more on counting complexities and the counting problem for linear-time temporal logic we refer the reader to [9, 23, 24].

  2. 2.

    The formula is an example of a 0-convergent liveness formula.

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Authors and Affiliations

  1. Saarland University, Saarbrücken, Germany

    Bernd Finkbeiner & Hazem Torfah

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  1. Bernd Finkbeiner
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  2. Hazem Torfah
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Correspondence to Hazem Torfah .

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Editors and Affiliations

  1. Indian Institute of Science, Bangalore, India

    Deepak D'Souza

  2. Chennai Mathematical Institute, Kelambakkam, India

    K. Narayan Kumar

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Finkbeiner, B., Torfah, H. (2017). The Density of Linear-Time Properties. In: D'Souza, D., Narayan Kumar, K. (eds) Automated Technology for Verification and Analysis. ATVA 2017. Lecture Notes in Computer Science(), vol 10482. Springer, Cham. https://doi.org/10.1007/978-3-319-68167-2_10

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  • DOI: https://doi.org/10.1007/978-3-319-68167-2_10

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