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

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).

[1]  Stephan Merz,et al.  Model Checking , 2000 .

[2]  Nic Wilson,et al.  Decision Diagrams for the Computation of Semiring Valuations , 2005, IJCAI.

[3]  Rina Dechter,et al.  The Impact of AND/OR Search Spaces on Constraint Satisfaction and Counting , 2004, CP.

[4]  Umberto Bertelè,et al.  Nonserial Dynamic Programming , 1972 .

[5]  Fernando Pereira,et al.  Case-factor diagrams for structured probabilistic modeling , 2004, J. Comput. Syst. Sci..

[6]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[7]  Rina Dechter,et al.  Mixtures of Deterministic-Probabilistic Networks and their AND/OR Search Space , 2004, UAI.

[8]  Hélène Fargier,et al.  Compiling CSPs into Tree-Driven Automata for Interactive Solving , 2003, Constraints.

[9]  Adnan Darwiche,et al.  DPLL with a Trace: From SAT to Knowledge Compilation , 2005, IJCAI.

[10]  Michael J. Quinn,et al.  Taking Advantage of Stable Sets of Variables in Constraint Satisfaction Problems , 1985, IJCAI.

[11]  R. Bryant.  Graph-Based Algorithms for Boolean Function Manipulation12 , 1986 .

[12]  Rina Dechter,et al.  Bucket Elimination: A Unifying Framework for Reasoning , 1999, Artif. Intell..

[13]  Rina Dechter,et al.  Tree Clustering for Constraint Networks , 1989, Artif. Intell..

[14]  Valeria Bertacco,et al.  The disjunctive decomposition of logic functions , 1997, ICCAD 1997.

[15]  Kenneth L. McMillan,et al.  Symbolic model checking , 1992 .

[16]  Pierre Marquis,et al.  A Knowledge Compilation Map , 2002, J. Artif. Intell. Res..

[17]  Rina Dechter,et al.  The Relationship Between AND/OR Search and Variable Elimination , 2005, UAI.

[18]  Adnan Darwiche,et al.  Recursive conditioning , 2001, Artif. Intell..

[19]  Rina Dechter,et al.  AND/OR search spaces for graphical models , 2007, Artif. Intell..

[20]  Kenneth L. McMillan.  Hierarchical Representations of Discrete Functions, with Application to Model Checking , 1994, CAV.

[21]  Daniel P. Miranker,et al.  A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem , 1996, AAAI/IAAI, Vol. 1.