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Efficient generation of small interpolants in CNF

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Abstract

Interpolation-based model checking (ITP) McMillan (in CAV, 2003) is an efficient and complete model checking procedure. However, for large problems, interpolants generated by ITP might become extremely large, rendering the procedure slow or even intractable. In this work we present a novel technique for interpolant generation in the context of model checking. The main novelty of our work is that we generate small interpolants in conjunctive normal form (CNF) using a twofold procedure: first we propose an algorithm that exploits resolution refutation properties to compute an interpolant approximation. Then we introduce an algorithm that takes advantage of inductive reasoning to turn the interpolant approximation into an interpolant. Unlike ITP, our approach maintains only the relevant subset of the resolution refutation. In addition, the second part of the procedure exploits the properties of the model checking problem at hand, in contrast to the general-purpose algorithm used in ITP. We developed a new interpolation-based model checking algorithm, called CNF-ITP. Our algorithm takes advantage of the smaller interpolants and exploits the fact that the interpolants are given in CNF. We integrated our method into a SAT-based model checker and experimented with a representative subset of the HWMCC’12 benchmark set. Our experiments show that, overall, the interpolants generated by our method are 117 times smaller than those generated by ITP. Our CNF-ITP algorithm outperforms ITP, and at times solves problems that ITP cannot solve. We also compared CNF-ITP to the successful IC3 Bradely (in VMCAI, volume 6538 of lecture notes in computer science, 2011) algorithm. We found that CNF-ITP outperforms IC3 Bradely (in VMCAI, volume 6538 of lecture notes in computer science, 2011) in a large number of cases.

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Notes

  1. Variable elimination [7] is an operation that eliminates all occurrences of a variable \(v\) from a CNF formula by replacing clauses containing \(v\) with the result of pairwise resolutions between all clauses containing the literal \(v\) and those containing the literal \(\lnot v\).

  2. Some works choose different ways of increasing \(k\). For example, \(k\) can be increased by the number of iterations executed in the inner loop: \(k=k+n\). In our experiments \(k=k+1\) yielded better results.

  3. https://bitbucket.org/alanmi/abc.

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Authors and Affiliations

  1. Computer Science Department, The Technion, Haifa, Israel

    Yakir Vizel

  2. Electrical Engineering Department, Princeton University, Princeton, USA

    Yakir Vizel

  3. Design Technology Solutions Group, Intel Corporation, Haifa, Israel

    Alexander Nadel & Vadim Ryvchin

Authors
  1. Yakir Vizel
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  2. Alexander Nadel
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  3. Vadim Ryvchin
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Correspondence to Yakir Vizel.

Additional information

A preliminary version of this work appeared in [24]. The authors would like to thank Håkan Hjort and Paul Inbar for their valuable comments.

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Vizel, Y., Nadel, A. & Ryvchin, V. Efficient generation of small interpolants in CNF. Form Methods Syst Des 47, 51–74 (2015). https://doi.org/10.1007/s10703-015-0224-5

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