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Weighted Model Counting Without Parameter Variables

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Theory and Applications of Satisfiability Testing – SAT 2021 (SAT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12831))

Abstract

Weighted model counting (WMC) is a powerful computational technique for a variety of problems, especially commonly used for probabilistic inference. However, the standard definition of WMC that puts weights on literals often necessitates WMC encodings to include additional variables and clauses just so each weight can be attached to a literal. This paper complements previous work by considering WMC instances in their full generality and using recent state-of-the-art WMC techniques based on pseudo-Boolean function manipulation, competitive with the more traditional WMC algorithms based on knowledge compilation and backtracking search. We present an algorithm that transforms WMC instances into a format based on pseudo-Boolean functions while eliminating around 43 % of variables on average across various Bayesian network encodings. Moreover, we identify sufficient conditions for such a variable removal to be possible. Our experiments show significant improvement in WMC-based Bayesian network inference, outperforming the current state of the art.

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Notes

  1. 1.

    Ace [11] implements most of the Bayesian network encodings and can also be used for compilation (and thus inference). It is available at http://reasoning.cs.ucla.edu/ace/.

  2. 2.

    Example 2 demonstrates what we mean by implication clauses.

  3. 3.

    http://www.cril.univ-artois.fr/kc/bn2cnf.html.

  4. 4.

    Note that since cd05 and cd06 are minimum-cardinality WMC encodings, they are not supported by most WMC algorithms.

  5. 5.

    Adding scaling factor \(\omega \) to the definition allows us to remove clauses that consist entirely of a single parameter variable. The idea of extracting some of the structure of the WMC instance into an external multiplicative factor was loosely inspired by the bklm16 encoding, where it is used to subsume the most commonly occurring probability of each CPT [3].

  6. 6.

    For convenience and without loss of generality we assume that \(w(p) \ne 0\) for all \(p \in X_P\).

  7. 7.

    Recall that cd05 and cd06 are incompatible with DPMC.

  8. 8.

    https://github.com/vardigroup/DPMC.

  9. 9.

    https://cs.rochester.edu/u/kautz/Cachet/.

  10. 10.

    http://reasoning.cs.ucla.edu/c2d/.

  11. 11.

    http://www.cril.univ-artois.fr/kc/d-DNNF-reasoner.html.

  12. 12.

    Each instance was run on the same processor across all algorithms and encodings.

  13. 13.

    The data on this (along with the implementation of Algorithm 1) is available at https://github.com/dilkas/wmc-without-parameters.

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Acknowledgments

We thank the anonymous reviewers for their helpful comments. The first author was supported by the EPSRC Centre for Doctoral Training in Robotics and Autonomous Systems, funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016834/1). The second author was supported by a Royal Society University Research Fellowship. This work has made use of the resources provided by the Edinburgh Compute and Data Facility (ECDF) (http://www.ecdf.ed.ac.uk/).

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  1. University of Edinburgh, Edinburgh, UK

    Paulius Dilkas & Vaishak Belle

  2. Alan Turing Institute, London, UK

    Vaishak Belle

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  1. Laboratoire MIS, University of Picardie Jules Verne, Amiens, France

    Chu-Min Li

  2. IIIA-CSIC, Spanish National Research Council (CSIC), Bellaterra, Barcelona, Spain

    Felip Manyà

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Dilkas, P., Belle, V. (2021). Weighted Model Counting Without Parameter Variables. In: Li, CM., Manyà, F. (eds) Theory and Applications of Satisfiability Testing – SAT 2021. SAT 2021. Lecture Notes in Computer Science(), vol 12831. Springer, Cham. https://doi.org/10.1007/978-3-030-80223-3_10

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