Boolean affine approximation with binary decision diagrams

  • Authors:

  • Kevin Henshall;Peter Schachte;Harald Søndergaard;Leigh Whiting

  • Affiliations:

  • The University of Melbourne, Vic., Australia;The University of Melbourne, Vic., Australia;The University of Melbourne, Vic., Australia;The University of Melbourne, Vic., Australia

  • Venue:
  • CATS '09 Proceedings of the Fifteenth Australasian Symposium on Computing: The Australasian Theory - Volume 94
  • Year:
  • 2009

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Abstract

Selman and Kautz's work on knowledge compilation has established how approximation (strengthening and/or weakening) of a propositional knowledgebase can be used to speed up query processing, at the expense of completeness. In the classical approach, the knowledge-base is assumed to be presented as a propositional formula in conjunctive normal form (CNF), and Horn functions are used to over-and under-approximate it (in the hope that many queries can be answered efficiently using the approximations only). However, other representations are possible, and functions other than Horn can be used for approximations, as long as they have deduction-computational properties similar to those of the Horn functions. Zanuttini has suggested that the class of affine Boolean functions would be especially useful in knowledge compilation and has presented various affine approximation algorithms. Since CNF is awkward for presenting affine functions, Zanuttini considers both a sets-of-models representation and the use of modulo 2 congruence equations. Here we consider the use of reduced ordered binary decision diagrams (ROBDDs), a representation which is more compact than the sets of models and which (unlike modulo 2 congruences) can express any source knowledge-base. We present an ROBDD algorithm to find strongest affine upper-approximations of a Boolean function and we argue its correctness.