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Semipositivity in Separation Logic with Two Variables

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Dependable Software Engineering: Theories, Tools, and Applications (SETTA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9984))

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Abstract

In a recent work by Demri and Deters (CSL-LICS 2014), first-order separation logic restricted to two variables and separating implication was shown undecidable, where it was shown that even with only two variables, if the use of negations is unrestricted, then they can be nested with separating implication in a complex way to get the undecidability result. In this paper, we revisit the decidability and complexity issues of first-order separation logic with two variables, and proposes semi-positive separation logic with two-variables (SPSL2), where the use of negations is restricted in the sense that negations can only occur in front of atomic formulae. We prove that satisfiability of the fragment of SPSL2 where neither separating conjunction nor septraction (the dual operator of separating implication) occurs in the scope of universal quantifiers, is \(\textsc {nexptime}\)-complete. As a byproduct of the proof, we show that the finite satisfiability problem of first-order logic with two variables and a bounded number of function symbols is \(\textsc {nexptime}\)-complete (the lower bound holds even with only one function symbol and without unary predicates), which may be of independent interest beyond separation logic community.

Partially supported by the NSFC grants (Nos. 61100062, 61272135, 61472474, and 61572478).

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Acknowledgements

This work was partially done when I was a visiting researcher at LIAFA, Université Paris Diderot, from June 2014 to June 2015, supported by China Scholarship Council. My great thanks go to Stéphane Demri for the numerous discussions with him when I did this work. At last, I would like to thank the reviewers for their valuable comments.

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Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, People’s Republic of China

    Zhilin Wu

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  1. Zhilin Wu
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Correspondence to Zhilin Wu .

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  1. Carl von Ossietzky Universität, Oldenburg, Germany

    Martin Fränzle

  2. University of New Mexico, Albuquerque, New Mexico, USA

    Deepak Kapur

  3. Chinese Academy of Sciences, Beijing, China

    Naijun Zhan

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Wu, Z. (2016). Semipositivity in Separation Logic with Two Variables. In: Fränzle, M., Kapur, D., Zhan, N. (eds) Dependable Software Engineering: Theories, Tools, and Applications. SETTA 2016. Lecture Notes in Computer Science(), vol 9984. Springer, Cham. https://doi.org/10.1007/978-3-319-47677-3_12

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  • DOI: https://doi.org/10.1007/978-3-319-47677-3_12

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