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Refinement to Imperative HOL

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Abstract

Many algorithms can be implemented most efficiently with imperative data structures. This paper presents Sepref, a stepwise refinement based tool chain for the verification of imperative algorithms in Isabelle/HOL. As a back end we use imperative HOL, which allows to generate verified imperative code. On top of imperative HOL, we develop a separation logic framework with powerful proof tactics. We use this framework to verify basic imperative data structures and to define a refinement calculus between imperative and functional programs. We provide a tool to automatically synthesize a concrete imperative program and a refinement proof from an abstract functional program, selecting implementations of abstract data types according to a user-provided configuration. As a front end to describe the abstract programs, we use the Isabelle Refinement Framework, for which many algorithms have already been formalized. Our tool chain is complemented by a large selection of verified imperative data structures. We have used Sepref for several verification projects, resulting in efficient verified implementations that are competitive with unverified ones in Java or C\(++\).

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Notes

  1. For technical reasons, we formalize a partial heap as a full heap with an address range. Assertions must not depend on heap content outside this address range.

  2. Again, for technical reasons, we additionally check that the program does not modify addresses outside the heap’s address range, and that it does not deallocate memory.

  3. The Isabelle/HOL function package [15] allows for convenient definition of such functions.

  4. The control structures must be the same, and the abstract operations must match the corresponding concrete operations.

  5. We applied \(\alpha \)-conversion to give the newly created variables convenient names.

  6. Again, we applied \(\alpha \)-conversion, to make the generated variable names more readable.

  7. A workaround for this particular example is to simultaneously return the head and tail of the list. However, for more complex operations, e. g. returning the first element that satisfies a predicate P, this technique gets unwieldy.

  8. Paper under review at the time of writing, the formalization is available at https://www21.in.tum.de/~lammich/max_flow/.

  9. Paper under review at the time of writing, the tool and further information is available at https://www21.in.tum.de/~lammich/grat/.

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Acknowledgements

We thank Rene Meis for formalizing the basics of separation logic for imperative HOL. Moreover, we thank Thomas Tuerk for interesting discussions about automation of separation logic. Finally, we thank Max Haslbeck, Simon Wimmer, and the anonymous reviewers for useful comments on draft versions of this paper.

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  1. TU Munich, Boltzmannstr. 3, 85748, Garching, Germany

    Peter Lammich

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Lammich, P. Refinement to Imperative HOL. J Autom Reasoning 62, 481–503 (2019). https://doi.org/10.1007/s10817-017-9437-1

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