From AND/OR Search to AND/OR BDDs Rina Dechter Information and Computer Science, UC-Irvine, and Radcliffe Institue of Advanced Study, Cambridge Joint work with Robert Mateescu and John Cobb Verification/constraints workshop, 2006

Combining Decision Diagrams f = A*E + B*F g = C*G +D*H B A F 1 E D C H 1 G B A C D 1 H G F E OBDD f * g =

Combining AND/OR BDDs f = A*E + B*F g = C*G +D*H AND AOBDD f * g = B A 1 E D C H 1 G B A F 1 E D C H G AND AOBDD f * g =

OBDDs vs AOBDD f * g = AND AOBDD OBDD B A F 1 E D C H G B A C D 1 H G 1 E D C H G AND AOBDD B A C D 1 H G F E OBDD

Outline Background in Graphical models AND/OR search trees and Graphs Minimal AND/OR graphs From AND/OR graphs to AOMDDs Compilation of AOMDDs AOMDDs and earlier BDDs

Constraint Networks A Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (red, green, blue) Constraints: A B red green red yellow green red green yellow yellow green yellow red Constraint graph C A B D E F G A E D F B G C Semantics: set of all solutions Primary task: find a solution

Propositional Satisfiability  = {(¬C), (A v B v C), (¬A v B v E), (¬B v C v D)}.

Graphical models A D B C E F A graphical model (X,D,C): X = {X1,…Xn} variables D = {D1, … Dn} domains C = {F1,…,Ft} functions (constraints, CPTS, cnfs) Examples: Constraint networks Belief networks Cost networks Markov random fields Influence diagrams Path width All these tasks are NP-hard  identify special cases  approximate

Tree-solving is easy Belief updating (sum-prod) MPE (max-prod) CSP – consistency (projection-join) Belief updating (sum-prod) P(X) P(Y|X) P(Z|X) P(T|Y) P(R|Y) P(L|Z) P(M|Z) MPE (max-prod) #CSP (sum-prod) Trees are processed in linear time and memory

Transforming into a Tree By Inference (thinking) Transform into a single, equivalent tree of sub-problems By Conditioning (guessing) Transform into many tree-like sub-problems.

Inference and Treewidth ABC DGF A D G B BDEF F C EFH E K M H L FHK J Inference algorithm: Time: exp(tree-width) Space: exp(tree-width) HJ KLM treewidth = 4 - 1 = 3 treewidth = (maximum cluster size) - 1

Conditioning and Cycle cutset P J A L B E D F M O H K G N C P J A L B E D F M O H K G N C P J A L B E D F M O H K G N C P J L B E D F M O H K G N C P J L B E D F M O H K G N A Cycle cutset = {A,B,C} B P J L E D F M O H K G N C P J L E D F M O H K G N C P J L E D F M O H K G N C

Search over the Cutset Inference may require too much memory Graph B K G L D F H M J E Graph Coloring problem Inference may require too much memory Condition (guessing) on some of the variables A=yellow A=green C B K G L D F H M J E C B K G L D F H M J E

Search over the Cutset (cont) B K G L D F H M J E Graph Coloring problem Inference may require too much memory Condition on some of the variables A=yellow A=green B=red B=blue B=green B=red B=yellow B=blue C K G L D F H M J E C K G L D F H M J E C K G L D F H M J E C K G L D F H M J E C K G L D F H M J E C K G L D F H M J E

Inference vs. Conditioning By Inference (thinking) Exponential in treewidth Time and memory E K F L H C B A M G J D ABC BDEF DGF EFH FHK HJ KLM Variable-elimination Directional resolution Join-tree clustering By Conditioning (guessing) Exponential in cycle-cutset Time-wise, linear memory A=yellow A=green B=blue B=red B=green C K G L D F H M J E A B Backtracking Branch and bound Depth-first search

Outline Background in Graphical models AND/OR search trees and Graphs Minimal AND/OR graphs From AND/OR graphs to AOMDDs Compilation of AOMDDs AOMDDs and earlier BDDs

Classic OR Search Space D B C E F Ordering: A B E C D F E C F D B A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

AND/OR Search Space Primal graph DFS tree A D B C E F A A D B C E F A AND 1 B OR AND 1 E OR C AND 1 OR D F AND 1

AND/OR vs. OR AND/OR AND/OR size: exp(4), OR size exp(6) OR A D B C E F A D B C E F OR A AND 1 AND/OR OR B B AND 1 1 OR E C E C E C E C AND 1 1 1 1 1 1 1 1 OR D F D F D F D F D F D F D F D F AND 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 AND/OR size: exp(4), OR size exp(6) E 1 C F D B A E 1 C F D B A OR 1 1 1 1

AND/OR vs. OR with Constraints No-goods (A=1,B=1) (B=0,C=0) A D B C E F A D B C E F A OR AND 1 AND/OR OR B B AND 1 1 OR E C E C E C E C AND 1 1 1 1 1 1 1 1 OR D F D F D F D F D F D F D F D F AND 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 E 1 C F D B A E 1 C F D B A E 1 C F D B A OR

AND/OR vs. OR with Constraints No-goods (A=1,B=1) (B=0,C=0 A D B C E F A D B C E F OR A AND 1 AND/OR OR B B AND 1 OR E C E C E C AND 1 1 1 1 1 1 OR D F D F D F D F AND 1 1 1 1 1 1 1 1 A A A 1 1 1 OR B B B 1 1 1 E E E 1 1 1 1 1 1 1 1 1 C C C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 D D D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 F F F 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

DFS algorithm (#CSP example) B C E F A D B C E F solution OR AND B 1 E C D F 2 4 5 6 11 A B 4 OR node: Marginalization operator (summation) E 2 C 2 AND node: Combination operator (product) 1 2 1 1 1 D 2 F 1 D 2 F 1 1 1 1 1 1 1 1 1 Value of node = number of solutions below it

Pseudo-Trees (Freuder 85, Bayardo 95, Bodlaender and Gilbert, 91) 4 1 6 2 3 2 7 5 3 m <= w* log n (a) Graph 4 7 1 1 2 7 3 5 5 3 5 4 2 7 6 6 4 6 (b) DFS tree depth=3 (c) pseudo- tree depth=2 (d) Chain depth=6

Tasks and value of nodes V( n) is the value of the tree T(n) for the task: Consistency: v(n) is 0 if T(n) inconsistent, 1 othewise. Counting: v(n) is number of solutions in T(n) Optimization: v(n) is the optimal solution in T(n) Belief updating: v(n), probability of evidence in T(n). Partition function: v(n) is the total probability in T(n). Goal: compute the value of the root node recursively using dfs search of the AND/OR tree. AND/OR search tree and algorithms are ([Freuder & Quinn85], [Collin, Dechter & Katz91], [Bayardo & Miranker95],[Darwiche 2001], [Bacchus et. Al, 2003]) Space: O(n) Time: O(exp(m)), where m is the depth of the pseudo-tree Time: O(exp(w* log n)) BFS is time and space O(exp(w* log n)

From AND/OR Tree A J A F B B C K E C E D D F G G H J H K OR AND OR AND 1 OR B B AND 1 1 OR E C E C E C E C AND 1 1 1 1 1 1 1 1 OR D F D F D F D F D F D F D F D F G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K AND OR AND OR AND

To an AND/OR Graph A J A F B B C K E C E D D F G G H J H K 1 OR B B AND 1 1 OR E C E C E C E C AND 1 1 1 1 1 1 1 1 OR D F D F D F D F D F D F D F D F =============================================================== AND 1 1 OR G G J J AND 1 1 1 1 OR H H H H K K K K AND 1 1 1 1 1 1 1 1

AND/OR Search Graphs Any two nodes that root identical subtrees/subgraphs (are unifiable) can be merged Minimal AND/OR search graph: of R relative to tree T is the closure under merge of the AND/OR search tree of R relative to T, where inconsistent subtrees are pruned. Canonicity: The minimal AND/OR graph relative to T is unique for all constraints equivalent to R.

Context based caching Caching is possible when context is the same context = parent-separator set in induced pseudo-graph = current variable + ancestors connected to subtree below A D B C E F G H J K context(B) = {A, B} context(C) = {A,B,C} context(D) = {D} context(F) = {F}

Caching context(B) = {A, B} context(C) = {A,B,C} context(D) = {D} J A F B B C K E C E D D F G J context(B) = {A, B} context(C) = {A,B,C} context(D) = {D} context(F) = {F} G H OR A H K AND 1 OR B B AND 1 1 OR E C E C E C E C AND 1 1 1 1 1 1 1 1 OR D F D F D F D F D F D F D F D F G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K G H 1 1 G H J K 1 1 J K AND OR AND OR AND

Caching context(D)={D} context(F)={F} A J A F B B C K E C E D D F G G OR A H K AND 1 OR B B AND 1 1 OR E C E C E C E C AND 1 1 1 1 1 1 1 1 OR D F D F D F D F D F D F D F D F AND 1 1 OR G G J J AND 1 1 1 1 OR H H H H K K K K AND 1 1 1 1 1 1 1 1

All Four Search Spaces A A B B C C D D E E F F A E C B F D 1 C D F E B A 1 C D F E B A Full OR search tree 126 nodes Context minimal OR search graph 28 nodes A OR AND B E F 1 C D A OR AND B E F 1 C D Context minimal AND/OR search graph 18 AND nodes Full AND/OR search tree 54 AND nodes

Properties of minimal AND/OR graphs Complexity: Minimal AND/OR R relative to pseudo-tree T is O(exp(w*)) where w* is the tree-width of R along T. Minimal OR search graph is O(exp(pw*)) where pw* is path-width w* ≤ pw*, pw* ≤ w*log n Canonicity: The minimal AND/OR search graph is unique (canonical) for all equivalent formulas (Boolean or Constraints), consistent with its pseudo tree.

Searching AND/OR Graphs AO(j): searches depth-first, cache i-context j = the max size of a cache table (i.e. number of variables in a context) i=0 j i=w* Space: O(n) Time: O(exp(w* log n)) Space: O(exp w*) Time: O(exp w*) Space: O(exp(j) ) Time: O(exp(m_j+j )

Context-Based Caching context(A) = {A} context(B) = {B,A} context(C) = {C,B} context(D) = {D} context(E) = {E,A} context(F) = {F} A E C B F D A B Primal graph C E D F A B E C D 1 5 2 3 6 4 7 Cache Table (C) B C Value 5 1 2 Space: O(exp(2))

Outline Background in Graphical models AND/OR search trees and Graphs Minimal AND/OR graphs From AND/OR search graphs to AOMDDs Compilation of AOMDDs AOMDDs and earlier BDDs

OR Search Graphs vs OBDDs 1 B C Full AND/OR search tree A B C f(ABC) 1 A B C A 1 B C Backtrack free AND/OR search tree A 1 B C A 1 B C An OBDD redundant Minimal AND/OR search graph

AND/OR Search Graphs; AOBDDs F(A,B,C,D)= (0,1,1,1), (1,0,1,0), (1,1,1,0) AOBDD(F) A 1 A 1 B B redundancy B 1 1 1 C D D C D D 1 1 1 1

AND/OR Search Graphs; AOBDDs F(A,B,C,D)= (0,1,1,1), (1,0,1,0), (1,1,1,0) AOBDD(F) A 1 A 1 B B redundancy B 1 1 1 C D D C D D 1 1 1 1

AOBDD Conventions Point dead-ends to terminal 0 1 B C D A 1 B C D Point dead-ends to terminal 0 Point goods to terminal 1

AOBDD Conventions A B D C Introduce Meta-nodes A 1 B C D A 1 B C D

Combining AOBDD (apply) f(ABC) 1 A A B D g(ABC) 1 A A B B B C D C D A 1 B C A 1 B D A 1 B C D = *

Example: (f+h) * (a+!h) * (a#b#g) * (f+g) * (b+f) * (a+e) * (c+e) * (c#d) * (b+c) A B C F D E G H pseudo tree H G E D C F B A m7 m7 D C B F A E G H primal graph A 1 B C D E m6 A 1 B F G H m3 A B C F D E G H m3 m6 A 1 B G F m2 A B F G H A 1 E C m4 A 1 F H m1 A B C D E B 1 C C9 C 1 D m5 B 1 F C5 m5 m4 m2 m1 C 1 D C8 A 1 E C6 C 1 E C7 A 1 B G C3 C E A A B F G F 1 H C1 A 1 H C2 A F H F 1 G C4 C D

Example (continued) m7 A m7 A B C D F G H E B m6 m3 A B C D E A B F G 1 B C D F G H E m7 B A 1 B C D E m6 A 1 B F G H m3 A B C F D E G H

AOBDD vs. OBDD AOBDD 18 nonterminals 47 arcs OBDD 27 nonterminals F A E G H primal graph B B C C C C A 1 B C D F G H E D D D D D D E E E F F F F G G G G G A H H B C 1 D A B C F D E G H AOBDD 18 nonterminals 47 arcs OBDD 27 nonterminals 54 arcs E F G H

Complexity Complexity of apply: Complexity of apply is bounded quadratically by the product of input AOBDDs restricted to each branch in the output pseudo-tree. Complexity of VE-AOBDD: is exponential in the tree-width along the pseudo-tree.

AOMDDs and tree-BDDs Tree-BDDs (McMillan1994) are : AND/OR BDDS are

Related work Related work in Search Related to compilation schemes: Backjumping + learning (Freuder 1988,Dechter 1990,Bayardo and Mirankar 1996) Recursive-conditioning (Darwiche 2001) Value elimination (Bacchus et. Al. 2003) Search over tree-decompositions (Trrioux et. Al. 2002) Related to compilation schemes: Minimal AND/OR – related to tree-OBDDs (McMillan 94), d-DNNF (Darwiche et. Al. 2002) Case-factor diagrams (Mcallester, Collins, Pereira, 2005) Tutorial references: Dechter, Constraint processing, Morgan Kauffman, 2003 Dechter, Tractable Structures of Constraint Satisfaction problems, In Handbook of constraint solving, forthcoming.

Conclusion AND/OR search should always be used. AND/OR BDDs are superior to OBDDs. A search algorithm with good and no-good learning generates an OBDD or an AND/OR Bdds. Dynamic variable ordering can be incorporated With caching, AND/OR search is similar to inference (variable-elimination) The real tradeoff should be rephrased: Time vs space rather than search vs inference, or search vs model-checking. Search methods are more sensitive to this tradeoff.