Compiling Constraint Networks into AND/OR Multi-valued Decision Diagrams (AOMDDs)

  • Robert Mateescu
  • Rina Dechter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4204)


Inspired by AND/OR search spaces for graphical models recently introduced, we propose to augment Ordered Decision Diagrams with AND nodes, in order to capture function decomposition structure. This yields AND/OR multi-valued decision diagram (AOMDD) which compiles a constraint network into a canonical form that supports polynomial time queries such as solution counting, solution enumeration or equivalence of constraint networks. We provide a compilation algorithm based on Variable Elimination for assembling an AOMDD for a constraint network starting from the AOMDDs for its constraints. The algorithm uses the apply operator which combines two AOMDDs by a given operation. This guarantees the complexity upper bound for the compilation time and the size of the AOMDD to be exponential in the treewidth of the constraint graph, rather than pathwidth as is known for ordered binary decision diagrams (OBDDs).


Search Tree Primal Graph Binary Decision Diagram Constraint Network Constraint Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Dechter, R., Mateescu, R.: Mixtures of deterministic-probabilistic networks and their and/or search space. In: UAI 2004, pp. 120–129 (2004)Google Scholar
  2. 2.
    Dechter, R., Mateescu, R.: The impact of and/or search spaces on constraint satisfaction and counting. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 731–736. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Mateescu, R., Dechter, R.: The relationship between and/or search and variable elimination. In: UAI 2005, pp. 380–387 (2005)Google Scholar
  4. 4.
    Dechter, R., Mateescu, R.: And/or search spaces for graphical models. Artificial Intelligence (forthcoming, 2006)Google Scholar
  5. 5.
    Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  6. 6.
    McMillan, K.L.: Symbolic Model Checking. Kluwer Academic, Dordrecht (1993)MATHGoogle Scholar
  7. 7.
    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transaction on Computers 35, 677–691 (1986)MATHCrossRefGoogle Scholar
  8. 8.
    Brayton, R., McMullen, C.: The decomposition and factorization of boolean expressions. In: ISCAS, Proceedings of the International Symposium on Circuits and Systems, pp. 49–54 (1982)Google Scholar
  9. 9.
    Bertacco, V., Damiani, M.: The disjunctive decomposition of logic functions. In: ICCAD, International Conference on Computer-Aided Design, pp. 78–82 (1997)Google Scholar
  10. 10.
    McMillan, K.L.: Hierarchical representation of discrete functions with application to model checking. In: Computer Aided Verification, pp. 41–54 (1994)Google Scholar
  11. 11.
    Fargier, H., Vilarem, M.: Compiling csps into tree-driven automata for interactive solving. Constraints 9(4), 263–287 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Freuder, E.C., Quinn, M.J.: Taking advantage of stable sets of variables in constraint satisfaction problems. In: IJCAI 1985, pp. 1076–1078 (1985)Google Scholar
  13. 13.
    Bertele, U., Brioschi, F.: Nonserial Dynamic Programming. Academic Press, London (1972)MATHGoogle Scholar
  14. 14.
    Dechter, R.: Bucket elimination: A unifying framework for reasoning. Artificial Intelligence 113, 41–85 (1999)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bayardo, R., Miranker, D.: A complexity analysis of space-bound learning algorithms for the constraint satisfaction problem. In: AAAI 1996, pp. 298–304 (1996)Google Scholar
  16. 16.
    Darwiche, A.: Recursive conditioning. Artificial Intelligence 125(1-2), 5–41 (2001)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Wilson, N.: Decision diagrams for the computation of semiring valuations. In: IJCAI 2005, pp. 331–336 (2005)Google Scholar
  18. 18.
    Darwiche, A., Marquis, P.: A knowledge compilation map. Journal of Artificial Intelligence Research (JAIR) 17, 229–264 (2002)MATHMathSciNetGoogle Scholar
  19. 19.
    Huang, J., Darwiche, A.: Dpll with a trace: From sat to knowledge compilation. In: IJCAI 2005, pp. 156–162 (2005)Google Scholar
  20. 20.
    McAllester, D., Collins, M., Pereira, F.: Case-factor diagrams for structured probabilistic modeling. In: UAI 2004, pp. 382–391 (2004)Google Scholar
  21. 21.
    Dechter, R., Pearl, J.: Tree clustering for constraint networks. Artificial Intelligence 38, 353–366 (1989)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Cobb, J.: Joining and/or networks. Report, Radcliffe fellowship, Cambridge, MA (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Robert Mateescu
    • 1
  • Rina Dechter
    • 1
  1. 1.Donald Bren School of Information and Computer ScienceUniversity of CaliforniaIrvine

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