  Inspired by the recently introduced framework of AND/OR search spaces for
graphical models, we propose to augment Multi-Valued Decision Diagrams (MDD)
with AND nodes, in order to capture function decomposition structure and to
extend these compiled data structures to general weighted graphical models
(e.g., probabilistic models). We present the AND/OR Multi-Valued Decision
Diagram (AOMDD) which compiles a graphical model into a canonical form that
supports polynomial (e.g., solution counting, belief updating) or constant time
(e.g. equivalence of graphical models) queries. We provide two algorithms for
compiling the AOMDD of a graphical model. The first is search-based, and works
by applying reduction rules to the trace of the memory intensive AND/OR search
algorithm. The second is inference-based and uses a Bucket Elimination schedule
to combine the AOMDDs of the input functions via the the APPLY operator. For
both algorithms, the compilation time and the size of the AOMDD are, in the
worst case, exponential in the treewidth of the graphical model, rather than
pathwidth as is known for ordered binary decision diagrams (OBDDs). We
introduce the concept of semantic treewidth, which helps explain why the size
of a decision diagram is often much smaller than the worst case bound. We
provide an experimental evaluation that demonstrates the potential of AOMDDs.
