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Quantified data automata for linear data structures: a register automaton model with applications to learning invariants of programs manipulating arrays and lists

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Abstract

We propose a new automaton model, called quantified data automata (QDA) over words, that can model quantified invariants over linear data structures, and study their theory, including closure properties, canonical minimality, and decidability of emptiness. We build poly-time active learning algorithms for them, where the learner is allowed to query the teacher with membership and equivalence queries. In order to express invariants in decidable logics, we invent a decidable subclass of QDAs, called elastic QDAs, and show translations to decidable theories of arrays and lists. We also prove that every QDA has a unique minimally-over-approximating elastic QDA, showing a robust technique for abstracting QDA-expressible properties to the decidable fragments expressed by elastic QDAs. We then give an application of these theoretically sound and efficient active learning algorithms to program verification by building a passive learning framework that efficiently learns adequate quantified linear data structure invariants from samples obtained from dynamic executions for a class of programs.

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Notes

  1. Index variables occur in the case of arrays and index into arrays.

  2. We make sure that the type of an interpretation (i.e., whether it is for formulas in the Array Property Fragment or in the decidable syntactic fragment of Strand) is always clear from the context.

  3. The benchmark suite and the source code of our implementation is available at http://www.cs.uiuc.edu/~madhu/cav13/.

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Acknowledgments

We would like to thank Xiaokang Qiu, who was involved in early discussions on finding an automaton model for the decidable fragment of Strand. This work was partially supported by NSF CAREER award #0747041 and NSF Expeditions in Computing ExCAPE Award #1138994.

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, 201 N. Goodwin Ave., Urbana, IL, 61801, USA

    Pranav Garg & P. Madhusudan

  2. Lehrstuhl für Informatik 7, RWTH Aachen University, 52056, Aachen, Germany

    Christof Löding & Daniel Neider

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  1. Pranav Garg
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  3. P. Madhusudan
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Correspondence to Pranav Garg.

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Garg, P., Löding, C., Madhusudan, P. et al. Quantified data automata for linear data structures: a register automaton model with applications to learning invariants of programs manipulating arrays and lists. Form Methods Syst Des 47, 120–157 (2015). https://doi.org/10.1007/s10703-015-0231-6

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