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A New Abstraction Framework for Affine Transformers

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Static Analysis (SAS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10422))

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Abstract

This paper addresses the problem of abstracting a set of affine transformers \(\overrightarrow{v}' = \overrightarrow{v} \cdot C + \overrightarrow{d}\), where \(\overrightarrow{v}\) and \(\overrightarrow{v}'\) represent the pre-state and post-state, respectively. We introduce a framework to harness any base abstract domain \(\mathcal {B}\) in an abstract domain of affine transformations. Abstract domains are usually used to define constraints on the variables of a program. In this paper, however, abstract domain \(\mathcal {B}\) is re-purposed to constrain the elements of C and \(\overrightarrow{d}\)—thereby defining a set of affine transformers on program states. This framework facilitates intra- and interprocedural analyses to obtain function and loop summaries, as well as to prove program assertions.

Supported, in part, by a gift from Rajiv and Ritu Batra; DARPA MUSE award FA8750-14-2-0270 and DARPA STAC award FA8750-15-C-0082; and by the UW-Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors, and do not necessarily reflect the views of the sponsoring agencies.

T. Reps has an ownership interest in GrammaTech, Inc., which has licensed elements of the technology discussed in this publication.

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Notes

  1. 1.

    k of the coefficients are always 0, and one coefficient is always 1 (i.e., the first column is always \((1\vert \ 0\ 0\ ...\ 0)^t\)). For this reason, we really need only \(k+k^2\) elements, but we will sometimes refer to \((k+1)^2\) elements for brevity.

  2. 2.

    We will continue to refer to the \(\text {MOS}\) domain directly, rather than “the instantiation of Observation 2 with an \(\text {AG}\) element containing \((k+1)^2\) columns” (à la Example 5).

  3. 3.

    The abstract domain \(\mathcal {I}_{\mathbb {Z}_{2^w}}^{(k+1)^2}\) is the product domain of \((k+1)^2\) interval domains, that is, \(\mathcal {I}_{\mathbb {Z}_{2^w}}^{(k+1)^2}= \mathcal {I}_{\mathbb {Z}_{2^w}}\times \mathcal {I}_{\mathbb {Z}_{2^w}}\times \ldots \times \mathcal {I}_{\mathbb {Z}_{2^w}}\). \(\mathcal {I}_{\mathbb {Z}_{2^w}}^{(k+1)^2}\) uses smash product to maintain a canonical representation for \(\bot _{\text {ATA}[\mathcal {I}_{\mathbb {Z}_{2^w}}^{(k+1)^2}]}\). Thus, if any of the coefficients in an abstract-domain element \(b\in \text {ATA}[\mathcal {I}_{\mathbb {Z}_{2^w}}^{(k+1)^2}]\) is \(\bot _{\mathcal {I}_{\mathbb {Z}_{2^w}}}\), then b is smashed to \(\bot _{\text {ATA}[\mathcal {I}_{\mathbb {Z}_{2^w}}^{(k+1)^2}]}\).

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Authors and Affiliations

  1. University of Wisconsin, Madison, WI, USA

    Tushar Sharma & Thomas Reps

  2. GrammaTech, Inc., Ithaca, NY, USA

    Thomas Reps

Authors
  1. Tushar Sharma
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  2. Thomas Reps
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Correspondence to Tushar Sharma .

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Editors and Affiliations

  1. Dipartimento di Matematica, University of Padova, Padua, Italy

    Francesco Ranzato

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Sharma, T., Reps, T. (2017). A New Abstraction Framework for Affine Transformers. In: Ranzato, F. (eds) Static Analysis. SAS 2017. Lecture Notes in Computer Science(), vol 10422. Springer, Cham. https://doi.org/10.1007/978-3-319-66706-5_17

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  • DOI: https://doi.org/10.1007/978-3-319-66706-5_17

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