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Deciding Program Properties via Complete Abstractions on Bounded Domains

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Static Analysis (SAS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13790))

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Abstract

Abstract interpretation provides an over-approximation of program behaviours that is used to prove the absence of bugs. When the computed approximation in the chosen abstract domain is as precise as possible, we say the analysis is complete and false alarms cannot arise. Unfortunately for any non trivial abstract domain there is some program whose analysis is incomplete. In this paper we want to characterize the classes of complete programs on some non-trivial abstract domains for studying their expressiveness. To this aim we introduce the notion of bounded domains for posets with ascending chains of bounded length only. We show that any complete program on bounded domains can be rewritten in an equivalent canonical form without nontrivial loops. This result proves that program termination on the class of complete programs on bounded domain is decidable. Moreover, semantic equivalence between programs in the above class can be reduced to determining the equivalence of a set of guarded statements. We show how our approach can be applied to a quite large class of programs. Indeed, abstract domains defined on Boolean abstractions that are complete for the same functions can be composed by preserving boundedness and completeness also w.r.t. any expressible guard. This suggests that new complete bounded abstract domains can be tailored on the guards and functions appearing in the program. Their existence is sufficient to prove decidability of termination and program equivalence for such programs.

Research supported by MIUR PRIN Project 201784YSZ5 ASPRA–Analysis of Program Analyses. Partially funded by RYC-2016-20281, by ESF Investing in your future and by Madrid regional government as part of the program S2018/TCS-4339 co-funded by EIE Funds of the European Union.

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Notes

  1. 1.

    Namely, the identical abstraction, making abstract and concrete semantics the same, and the top abstraction, making all programs equivalent by abstract semantics.

  2. 2.

    Note that this is different from establishing termination for all input in S, which should be addressed separately.

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, Pisa, Italy

    Roberto Bruni, Roberta Gori & Nicolas Manini

  2. IMDEA Software Institute, Madrid, Spain

    Nicolas Manini

  3. Universidad Politécnica de Madrid, Madrid, Spain

    Nicolas Manini

Authors
  1. Roberto Bruni
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  2. Roberta Gori
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  3. Nicolas Manini
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Correspondence to Nicolas Manini .

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Editors and Affiliations

  1. VMware Research and University of Illinois Urbana-Champaign, Urbana, IL, USA

    Gagandeep Singh

  2. Inria and ENS/PSL, Paris, France

    Caterina Urban

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Bruni, R., Gori, R., Manini, N. (2022). Deciding Program Properties via Complete Abstractions on Bounded Domains. In: Singh, G., Urban, C. (eds) Static Analysis. SAS 2022. Lecture Notes in Computer Science, vol 13790. Springer, Cham. https://doi.org/10.1007/978-3-031-22308-2_9

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  • DOI: https://doi.org/10.1007/978-3-031-22308-2_9

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