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Abstract Local Completeness

A Local Form of Abstract Non-interference

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Verification, Model Checking, and Abstract Interpretation (VMCAI 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15530))

Abstract

Abstract interpretation offers sound and decidable approximations for undecidable queries related to program behavior. The effectiveness of an abstract interpretation process relies entirely on the abstract domain itself, and the worst-case scenario is when the abstract interpreter responds with “don’t know”, meaning that anything could happen during runtime. The concept of completeness relates to the answer precision degree when performing computations within the abstract domain. However, completeness for a whole language is an ideal domain property, usually holding only on trivial situations [20]; for this reason, a local notion of completeness, holding on a specific program input, has been deeply investigated [5]. In this paper, we characterize an intermediate notion holding for sets of input selected by abstraction. In other words, completeness holds for a set of concrete inputs determined by one abstract input. In this sense, it is a form of local abstract completeness required locally on one specific abstract value. We provide a simple proof system for proving this weakening of completeness and several examples. Notably, this proof system is both language and domain-agnostic and can be readily incorporated to support static program analysis.

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Notes

  1. 1.

    By \(\le _{\textsf{A}}\) we denote the partial order relation on \(\textsf{A}\).

  2. 2.

    Note that, the abstract program semantics so far defined is, in general, an over-approximation of the bca  , i.e., \(\forall \texttt {r}\in {{\,\mathrm{{\mathfrak {L}}}\,}},\forall \texttt{a}\in \textsf{A}.\,{\llbracket \texttt {r} \rrbracket }^{\textsf{A}}\texttt{a}\le _{\textsf{A}}{\llbracket \texttt {r} \rrbracket }^{\sharp }_{\textsf{A}}\texttt{a}\).

  3. 3.

    The choice of considering functions with potentially different input and output domains aims to underline the generality of the definition, independent from the field of application.

  4. 4.

    In this case, for the sake of readability, we avoid annotating the abstract domains on which we define the abstraction.

  5. 5.

    The name (abstract) non-interference is an inheritance from language-based security where the notion has been introduced for modeling when a (property of a) sensitive input was not interfering with (a property of) an observable output [15].

  6. 6.

    In the machine learning context, a similar notion recently introduced is abstract robustness [24].

  7. 7.

    Note that if the \(\textsf{b}\) is not expressible then we need \(\textsf{A}(\texttt{c})\subseteq (\!|\textsf{b}|\!)\) for implying local ani  , similarly for \(\lnot \textsf{b}\).

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Funding

This work was supported by the project SERICS (PE00000014) under the MUR National Recovery and Resilience Plan funded by the European Union - NextGenerationEU and by PRIN2022PNRR “RAP-ARA” (PE6) - codice MUR: P2022HXNSC.

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Authors and Affiliations

  1. Computer Science Department, University of Verona, Verona, Italy

    Isabella Mastroeni

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  1. Isabella Mastroeni
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Correspondence to Isabella Mastroeni .

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  1. Indian Institute of Technology Bombay, Mumbai, Maharashtra, India

    Krishna Shankaranarayanan

  2. University of Colorado Boulder, Boulder, CO, USA

    Sriram Sankaranarayanan

  3. University of Colorado Boulder, Boulder, CO, USA

    Ashutosh Trivedi

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Mastroeni, I. (2025). Abstract Local Completeness. In: Shankaranarayanan, K., Sankaranarayanan, S., Trivedi, A. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2025. Lecture Notes in Computer Science, vol 15530. Springer, Cham. https://doi.org/10.1007/978-3-031-82703-7_1

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