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Refinement to Imperative/HOL

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Interactive Theorem Proving (ITP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9236))

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Abstract

Many algorithms can be implemented most efficiently with imperative data structures that support destructive update. In this paper we present an approach to automatically generate verified imperative implementations from abstract specifications in Isabelle/HOL. It is based on the Isabelle Refinement Framework, for which a lot of abstract algorithms are already formalized.

Based on Imperative/HOL, which allows to generate verified imperative code, we develop a separation logic framework with automation to make it conveniently usable. On top of this, we develop an imperative collection framework, which provides standard implementations for sets and maps like hash tables and array lists. Finally, we define a refinement calculus to refine abstract (functional) algorithms to imperative ones.

Moreover, we have implemented a tool to automate the refinement process, replacing abstract data types by efficient imperative implementations from our collection framework. As a case study, we apply our tool to automatically generate verified imperative implementations of nested depth-first search and Dijkstra’s shortest paths algorithm, which are considerably faster than the corresponding functional implementations. The nested DFS implementation is almost as fast as a C++ implementation of the same algorithm.

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Notes

  1. 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. 2.

    The distinction between values and arrays is dictated by Imperative/HOL.

  3. 3.

    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.

  4. 4.

    Specified in Isabelle’s long goal format, which is more readable for large propositions.

  5. 5.

    Inserting x twice is redundant, but gives a nice example for our transformations.

  6. 6.

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

  7. 7.

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

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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.

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Authors and Affiliations

  1. Technische Universität München, Munich, Germany

    Peter Lammich

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  1. Peter Lammich
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Correspondence to Peter Lammich .

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Editors and Affiliations

  1. Department of Informatics, King's College London, London, United Kingdom

    Christian Urban

  2. PLA University of Science and Technology, Nanjing, China

    Xingyuan Zhang

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Lammich, P. (2015). Refinement to Imperative/HOL. In: Urban, C., Zhang, X. (eds) Interactive Theorem Proving. ITP 2015. Lecture Notes in Computer Science(), vol 9236. Springer, Cham. https://doi.org/10.1007/978-3-319-22102-1_17

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  • DOI: https://doi.org/10.1007/978-3-319-22102-1_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22101-4

  • Online ISBN: 978-3-319-22102-1

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