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Spanning the Spectrum from Safety to Liveness

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Book cover Automated Technology for Verification and Analysis (ATVA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9364))

Abstract

Of special interest in formal verification are safety specifications, which assert that the system stays within some allowed region, in which nothing “bad” happens. Equivalently, a computation violates a safety specification if it has a “bad prefix” – a prefix all whose extensions violate the specification. The theoretical properties of safety specifications as well as their practical advantages with respect to general specifications have been widely studied. Safety is binary: a specification is either safety or not safety. We introduce a quantitative measure for safety. Intuitively, the safety level of a language L measures the fraction of words not in L that have a bad prefix. In particular, a safety language has safety level 1 and a liveness language has safety level 0. Thus, our study spans the spectrum between traditional safety and liveness. The formal definition of safety level is based on probability and measures the probability of a random word not in L to have a bad prefix. We study the problem of finding the safety level of languages given by means of deterministic and nondeterministic automata as well as LTL formulas, and the problem of deciding their membership in specific classes along the spectrum (safety, almost-safety, fraction-safety, etc.). We also study properties of the different classes and the structure of deterministic automata for them.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 278410, and from The Israel Science Foundation (grant no 1229/10).

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Notes

  1. 1.

    The definition of safety we consider here is given in [1, 2], it coincides with the definition of limit closure defined in [7], and is different from the definition in [16], which also refers to the property being closed under stuttering.

  2. 2.

    Note that \(\varSigma =2^{AP}\) and our probability distribution is such that for each atomic proposition a and for each position in a computation, the probability that a holds in the position is \(\frac{1}{2}\).

  3. 3.

    An anomaly of this definition is the language \(L=\varSigma ^\omega \). While \(Pr(L)=1\), making its safety level 0, we also have that L is a safety language. Thus, L is the only language that is both safety and has safety level 0.

  4. 4.

    We did not find in the literature an NLOGSPACE-complete result for deciding liveness of a DPW. The proof, however, follows standard considerations, and we give it in the full version.

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Authors and Affiliations

  1. School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel

    Rachel Faran & Orna Kupferman

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  1. Rachel Faran
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  2. Orna Kupferman
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Correspondence to Rachel Faran .

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Editors and Affiliations

  1. Universität des Saarlandes, Saarbrücken, Germany

    Bernd Finkbeiner

  2. East China Normal University, Shanghai, China

    Geguang Pu

  3. Chinese Academy of Sciences, Beijing, China

    Lijun Zhang

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Faran, R., Kupferman, O. (2015). Spanning the Spectrum from Safety to Liveness. In: Finkbeiner, B., Pu, G., Zhang, L. (eds) Automated Technology for Verification and Analysis. ATVA 2015. Lecture Notes in Computer Science(), vol 9364. Springer, Cham. https://doi.org/10.1007/978-3-319-24953-7_13

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  • DOI: https://doi.org/10.1007/978-3-319-24953-7_13

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