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Automated Verification of Functional Correctness of Race-Free GPU Programs

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Verified Software. Theories, Tools, and Experiments (VSTTE 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9971))

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Abstract

We study an automated verification method for functional correctness of parallel programs running on GPUs. Our method is based on Kojima and Igarashi’s Hoare logic for GPU programs. Our algorithm generates verification conditions (VCs) from a program annotated by specifications and loop invariants and pass them to off-the-shelf SMT solvers. It is often impossible, however, to solve naively generated VCs in reasonable time. A main difficulty stems from quantifiers over threads due to the parallel nature of GPU programs. To overcome this difficulty, we additionally apply several transformations to simplify VCs before calling SMT solvers.

Our implementation successfully verifies correctness of several GPU programs, including matrix multiplication optimized by using shared memory. In contrast to many existing tools, our verifier succeeds in verifying fully parameterized programs: parameters such as the number of threads and the sizes of matrices are all symbolic. We empirically confirm that our simplification heuristics is highly effective for improving efficiency of the verification procedure.

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Notes

  1. 1.

    We choose these initial values to explain what happens when the control branches. These initial values do not satisfy the precondition on the first line, so the asserted invariant is not preserved during execution.

  2. 2.

    Some of the terms appearing in this expression are not well-typed. We could write \( assign (b_2, (\lambda t.i_2(t) < \textit{len}_0), b_1, (\lambda t.i_2(t)), (\lambda t.a_0(i_2(t))))\), but for brevity we abbreviate it as above.

  3. 3.

    In this case \(t+1, t+2, \dots \) are also \(\forall \)-bounds, but we do not take them into account. Practically, considering only t seems sufficient in many cases.

  4. 4.

    Currently we use Why3 only for manipulating formulas and calling SMT solvers, although it provides a programming language WhyML.

  5. 5.

    alt-ergo.lri.fr, www.cs.nyu.edu/acsys/cvc3, cvc4.cs.nyu.edu, www.eprover.org, z3.codeplex.com.

  6. 6.

    Several examples are found at https://fmt.ewi.utwente.nl/redmine/projects/vercors- verifier/wiki/Examples.

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Authors and Affiliations

  1. Kyoto University, Kyoto, Japan

    Kensuke Kojima, Akifumi Imanishi & Atsushi Igarashi

  2. JST CREST, Tokyo, Japan

    Kensuke Kojima & Atsushi Igarashi

Authors
  1. Kensuke Kojima
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  2. Akifumi Imanishi
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  3. Atsushi Igarashi
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Corresponding author

Correspondence to Kensuke Kojima .

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Editors and Affiliations

  1. IRISA, University of Rennes 1, Rennes, France

    Sandrine Blazy

  2. Department of Computer Science, University of Toronto, Toronto, Ontario, Canada

    Marsha Chechik

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Kojima, K., Imanishi, A., Igarashi, A. (2016). Automated Verification of Functional Correctness of Race-Free GPU Programs. In: Blazy, S., Chechik, M. (eds) Verified Software. Theories, Tools, and Experiments. VSTTE 2016. Lecture Notes in Computer Science(), vol 9971. Springer, Cham. https://doi.org/10.1007/978-3-319-48869-1_7

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  • DOI: https://doi.org/10.1007/978-3-319-48869-1_7

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