Connecting Knowledge Compilation Classes Width Parameters

Abstract

The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width bounds, e.g., they have bounded treewidth or pathwidth. In this work, we give a systematic picture of many circuit classes considered in knowledge compilation and show how they can be systematically connected to width measures, through upper and lower bounds. Our upper bounds show that bounded-treewidth circuits can be constructively converted to d-SDNNFs, in time linear in the circuit size and singly exponential in the treewidth; and that bounded-pathwidth circuits can similarly be converted to uOBDDs. We show matching lower bounds on the compilation of monotone DNF or CNF formulas to structured targets, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable): any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth, and the same holds for uOBDDs (resp., n-OBDDs) when considering pathwidth. Unlike most previous work, our bounds apply to any formula of this class, not just a well-chosen family. Hence, we show that pathwidth and treewidth respectively characterize the efficiency of compiling monotone DNFs to uOBDDs and d-SDNNFs with compilation being singly exponential in the corresponding width parameter. We also show that our lower bounds on CNFs extend to unstructured compilation targets, with an exponential lower bound in the treewidth (resp., pathwidth) when compiling monotone CNFs of constant arity and degree to DNNFs (resp., nFBDDs).

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  • 25 January 2020

    The article title in the original publication contains an error. The correct title is presented in this Erratum. The online version of the original article can be found at: <ExternalRef><RefSource>https://doi.org/10.1007/s00224-019-09930-2</RefSource><RefTarget Address="10.1007/s00224-019-09930-2" TargetType="DOI"/></ExternalRef>

Notes

  1. 1.

    This observation is due to Stefan Mengel and is adapted from the recent article [22].

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Acknowledgments

We acknowledge Chandra Chekuri for his helpful comments at https://cstheory.stackexchange.com/a/38943/, as well as Stefan Mengel for pointing us to a notion of width for d-SDNNFs and suggesting a strengthening of our complexity upper bound in Theorem 4.2.

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Correspondence to Mikaël Monet.

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Amarilli, A., Capelli, F., Monet, M. et al. Connecting Knowledge Compilation Classes Width Parameters. Theory Comput Syst 64, 861–914 (2020). https://doi.org/10.1007/s00224-019-09930-2

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Keywords

  • Knowledge compilation
  • Treewidth
  • Pathwidth
  • Circuit
  • Boolean function