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On Preprocessing Techniques and Their Impact on Propositional Model Counting

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Abstract

This paper is concerned with preprocessing techniques for propositional model counting. We have considered several elementary preprocessing techniques: backbone identification, occurrence reduction, vivification, as well as equivalence, AND and XOR gate identification and replacement. All those techniques have been implemented in a preprocessor pmc, freely available on the Web. In order to assess the benefits which can be gained by taking advantage of pmc, we performed many experiments, based on benchmarks coming from several data sets. More precisely, we made a differential evaluation of each elementary preprocessing technique in order to evaluate its impact on the number of variables of the instance, its size, as well as the treewidth of its primal graph. We also considered two combinations of preprocessings: \( eq \), based on equivalence-preserving techniques only, and \({\#eq}\), which additionally exploits techniques preserving only the number of models. Several approaches to model counting have also been considered downstream in our experiments: “direct” model counters, including the exact ones Cachet, sharpSAT, and an approximate one SampleCount, as well as the compilation-based model counters C2D, Dsharp, SDD and cnf2obdd have been used. The experimental results we have obtained show that each elementary preprocessing technique is useful, and that some synergetic effects can be achieved by combining them.

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Notes

  1. In weighted model counting (WMC), each literal is associated with a real number, the weight of an interpretation is the product of the weights of the literals it sets to true, and the weight of a formula is the sum of the weights of its models. Accordingly, WMC amounts to model counting when each literal has weight 1.

  2. d-DNNF is the language of deterministic, decomposable negation normal form formulae; this language supports the model counting query in polynomial time [15].

  3. Literals are degenerate AND gates and degenerate XOR gates; however \(\mathtt {equivSimpl}\) may detect equivalences that would not be detected by \(\mathtt {ANDgateSimpl}\) or by \(\mathtt {XORgateSimpl}\); this explains why \(\mathtt {equivSimpl}\) is used.

  4. “Solved” means that the number of models has been found for all methods but SampleCount, and that an approximation has been computed when SampleCount is considered.

  5. We made similar measurements with the other model counters considered in the paper, and got similar observations. The corresponding results are available from the authors on demand.

  6. Similarly, it is easy to check that the Equivalent Literal Substitution (ELS) technique which consists in computing first the strongly connected components of the binary implication graph of \(\Sigma \), then in replacing in \(\Sigma \) every literal by a representative of its equivalence class is less effective than \(\mathtt {equivSimpl}\).

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Acknowledgments

This work has been partially supported by the project BR4CP ANR-11-BS02-008 of the French National Agency for Research.

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  1. CRIL-CNRS and Université d’Artois, Lens, France

    Jean-Marie Lagniez & Pierre Marquis

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  1. Jean-Marie Lagniez
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  2. Pierre Marquis
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Correspondence to Pierre Marquis.

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This is a revised and extended version of the conference paper “Preprocessing for Propositional Model Counting”, which appeared in the proceedings of AAAI’14, pp. 2688–2694, 2014.

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Lagniez, JM., Marquis, P. On Preprocessing Techniques and Their Impact on Propositional Model Counting. J Autom Reasoning 58, 413–481 (2017). https://doi.org/10.1007/s10817-016-9370-8

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