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On Limitations of Structured (Deterministic) DNNFs

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Abstract

The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of them, by definition they are restricted structured deterministic DNNFs and more general than ordered binary decision diagrams (OBDDs). In this paper limitations of structured (deterministic) DNNFs and in particular SDDs are considered. The main result in the paper is a quasipolynomial simulation of structured (deterministic) DNNFs by equivalent (unambiguous) nondeterministic OBDDs. The result is tight in the sense that there cannot be a polynomial simulation due to a known separation result. One may ask whether there are simulations in smaller size for restricted structured (deterministic) DNNFs. Answering a question posed by Bova and Szeider (2017) in the negative it is shown that not every SDD with disjunctions of constant fan-in can be transformed into equivalent OBDDs of polynomial size.

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Acknowledgements

The authors would like to thank the referees for several valuable comments and suggestions which helped to improve the readability of the paper. In particular the construction in Section 4 is due to a refree’s advice.

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Authors and Affiliations

  1. LS2 Informatik, TU Dortmund, Dortmund, Germany

    Beate Bollig

  2. Department of Theoretical Computer Science, University of Bonn, Bonn, Germany

    Matthias Buttkus

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  1. Beate Bollig
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  2. Matthias Buttkus
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Correspondence to Beate Bollig.

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Bollig, B., Buttkus, M. On Limitations of Structured (Deterministic) DNNFs. Theory Comput Syst 64, 799–825 (2020). https://doi.org/10.1007/s00224-019-09960-w

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