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On Finding Minimum Satisfying Assignments

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Principles and Practice of Constraint Programming (CP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9892))

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Abstract

Given a Satisfiability Modulo Theories (SMT) formula, a minimum satisfying assignment (MSA) is a partial assignment of minimum size that ensures the formula is satisfied. Minimum satisfying assignments find a number of practical applications that include software and hardware verification, among others. Recent work proposes the use of branch-and-bound search for computing MSAs. This paper proposes a novel counterexample-guided implicit hitting set approach for computing one MSA. Experimental results show significant performance gains over existing approaches.

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Notes

  1. 1.

    Proofs of Propositions 1 and 2 are omitted here due to lack of space. Note that they can be constructed following the ideas of [12, 22] where similar proofs are presented.

  2. 2.

    The Grow procedure is implemented as a sequence of SMT oracle calls, each increasing set X.

  3. 3.

    Indeed, one can observe that Algorithm 1 requires the minimum hitting set solver to report new hitting sets on demand, i.e. when a new counterexample is detected. This can be done in an incremental fashion [10], e.g. by adding new clauses when necessary and computing new solutions on demand while keeping all the information found during the previous calls.

  4. 4.

    https://github.com/CVC4/CVC4.

  5. 5.

    http://smtcomp.sourceforge.net/2015.

  6. 6.

    Note that the original distribution of MISTRAL does not have a command-line interface. But one can easily create one since the source code of the tool is available online at https://www.cs.utexas.edu/~tdillig/mistral.

  7. 7.

    We also tested the proposed approach on the standard SMTLIB benchmarks. However, minimum satisfying assignments for the majority of benchmarks in the QF_LIA category of the SMTLIB benchmarks have trivial minimum satisfying assignments, which contain all variables of the original formula. Therefore, considering these instances makes no sense.

  8. 8.

    For each variable x of the original CNF, two integer variables \(x_+\) and \(x_-\) are introduced s.t. \(0\le x_+ \le 1\) and \(0\le x_- \le 1\). Variables \(x_+\) and \(x_-\) cannot take value 0 or 1 at the same, which is forced by adding constraints of the form \(x_++x_-=1\). Each clause \(l_1\vee \ldots \vee l_m\) is translated into constraint \(x_{1*}+\ldots +x_{m*}\ge 1\), where each \(x_{i*}\) represents either \(x_{i+}\) or \(x_{i-}\) depending on the polarity of literal \(l_i\).

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Authors and Affiliations

  1. LaSIGE, Faculty of Science, University of Lisbon, Lisbon, Portugal

    Alexey Ignatiev, Alessandro Previti & Joao Marques-Silva

  2. ISDCT SB RAS, Irkutsk, Russia

    Alexey Ignatiev

Authors
  1. Alexey Ignatiev
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  2. Alessandro Previti
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  3. Joao Marques-Silva
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Correspondence to Alexey Ignatiev .

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Editors and Affiliations

  1. University of Nice Sophia Antipolis , Sophia Antipolis, France

    Michel Rueher

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Ignatiev, A., Previti, A., Marques-Silva, J. (2016). On Finding Minimum Satisfying Assignments. In: Rueher, M. (eds) Principles and Practice of Constraint Programming. CP 2016. Lecture Notes in Computer Science(), vol 9892. Springer, Cham. https://doi.org/10.1007/978-3-319-44953-1_19

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  • DOI: https://doi.org/10.1007/978-3-319-44953-1_19

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