1. Verification/constraints workshop, 2006 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

2. Combining Decision Diagrams f = A*E + B*F g = C*G +D*H B A F 01 E B E D C H 01 G D G B A C DD C B DD C DD 01 H GG FFF EEEEEE OBDD f * g =

3. Combining AND/OR BDDs f = A*E + B*F g = C*G +D*H B A F 01 E B E D C H 01 G D G AOBDD f * g = B A F 01 E B E D C H 01 G D G AND

4. OBDDs vs AOBDD f * g = B A F 01 E B E D C H 01 G D G AND B A C DD C B DD C DD 01 H GG FFF EEEEEE AOBDD OBDD

5. 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

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

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

8. A graphical model (X,D,C): X = {X 1,…X n } variables D = {D 1, … D n } domains C = {F 1,…,F t }functions (constraints, CPTS, cnfs) Graphical models A D BC E F All these tasks are NP-hard identify special cases approximate Examples: Constraint networks Belief networks Cost networks Markov random fields Influence diagrams

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

10. 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.

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

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

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

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

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

16. 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

17. Classic OR Search Space 01010101 0101010101010101 0101010101010101010101010101010101010101010101010101010101010101 01010101010101010101010101010101 0101 E C F D B A 01 Ordering: A B E C D F A D BC E F

18. AND/OR Search Space A OR 0 AND 1 B OR B 0 AND 1 0 1 E OR C EC EC EC DFDF DFDF DFDF DFDF AND 01010101 01010101 01010101 01010101 0101 0101 0101 0101 A D BC E F A D B CE F Primal graphDFS tree A D BC E F A D BC E F

19. AND/OR vs. OR A OR 0 AND 1 B OR B 0 AND 1 0 1 E OR C EC EC EC DFDF DFDF DFDF DFDF AND 01010101 01010101 01010101 01010101 0101 0101 0101 0101 E 01010101 0 C 101010101010101 F 0101010101010101010101010101010101010101010101010101010101010101 D 01010101010101010101010101010101 0 B 101 A 01 AND/OR OR A D BC E F A D B CE F 1 1 1 0 1 0 AND/OR size: exp(4), OR size exp(6)

20. AND/OR vs. OR with Constraints A OR 0 AND 1 B OR B 0 AND 1 0 1 E OR C EC EC EC DFDF DFDF DFDF DFDF AND 01010101 01010101 01010101 01010101 0101 0101 0101 0101 E 01010101 0 C 101010101010101 F 0101010101010101010101010101010101010101010101010101010101010101 D 01010101010101010101010101010101 0 B 101 A 01 AND/OR OR A D BC E F A D B CE F No-goods (A=1,B=1) (B=0,C=0)

21. AND/OR vs. OR with Constraints E 010101 C 11010111 F 01010101010101010101010101010101 D 0101010101010101 0 B 10 A 01 A OR 0 AND 1 B OR B 0 AND 1 0 E OR C EC EC DF DFDF DF AND 0101 01010101 0101 101 0101 101 E 010101 C 11010111 F 01010101010101010101010101010101 D 0101010101010101 0 B 10 A 01 E 010101 C 11010111 F 01010101010101010101010101010101 D 0101010101010101 0 B 10 A 01 AND/OR OR A D BC E F A D B CE F No-goods (A=1,B=1) (B=0,C=0

22. DFS algorithm (#CSP example) A 0 B 0 EC DFDF 01010101 OR AND OR AND OR AND 0101 A D BC E F AND node: Combination operator (product) OR node: Marginalization operator (summation) A D B CE F 1110110 1 0 21 2 2 0 0 2 1 2 4 Value of node = number of solutions below it solution

23. Pseudo-Trees ( Freuder 85, Bayardo 95, Bodlaender and Gilbert, 91 ) (a) Graph 461 3275 (b) DFS tree depth=3 (c) pseudo- tree depth=2 (d) Chain depth=6 46 1 3 27 5 27 1 4 3 5 6 4 6 1 3 2 7 5 m <= w* log n

24. 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)

25. From AND/OR Tree A D BC E F A D B CE F GH J K G H J K A OR 0 AND 1 B OR B 0 AND 1 0 1 E OR C EC EC EC DFDF DFDF DFDF DFDF AND 0101 0101 0101 0101 OR AND 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01

26. To an AND/OR Graph A D BC E F A D B CE F GH J K G H J K A OR 0 AND 1 B OR B 0 AND 1 0 1 E OR C EC EC EC DFDF DFDF DFDF DFDF AND 0101 0101 0101 0101 OR AND 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01

27. 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.

28. 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 BC E F A D B CE F GH J K G H J K context(B) = {A, B} context(C) = {A,B,C} context(D) = {D} context(F) = {F}

29. Caching A D BC E F A D B CE F GH J K G H J K A OR 0 AND 1 B OR B 0 AND 1 0 1 E OR C EC EC EC DFDF DFDF DFDF DFDF AND 0101 0101 0101 0101 OR AND 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 context(B) = {A, B} context(C) = {A,B,C} context(D) = {D} context(F) = {F}

30. Caching A D BC E F A D B CE F GH J K G H J K A OR 0 AND 1 B OR B 0 AND 1 0 1 E OR C EC EC EC DFDF DFDF DFDF DFDF AND 0101 0101 0101 0101 OR AND 0 G HH 0101 01 1 G HH 0101 01 0 J KK 0101 01 1 J KK 0101 01 context(D)={D} context(F)={F}

31. All Four Search Spaces Full OR search tree 126 nodes Full AND/OR search tree 54 AND nodes Context minimal OR search graph 28 nodes Context minimal AND/OR search graph 18 AND nodes 01010101 01010101 01 0101 0101 C D F E B A 01 A E C B F D

32. 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.

33. 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=0i=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 ) j

34. Context-Based Caching A D B EC F context(A) = {A} context(B) = {B,A} context(C) = {C,B} context(D) = {D} context(E) = {E,A} context(F) = {F} Cache Table (C) BCValue 005 012 102 110 A 0 B 0 E C DD 01 1 E C DD 01 1 B 0 E C 1 E C 5 5 52 3 6 20 4 5 4 4 46 7 7 A E C B F D Primal graph Space: O(exp(2))

35. 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

36. OR Search Graphs vs OBDDs A B C ABCf(ABC) 0000 0010 0100 0111 1000 1011 1100 1111 A 01 B 01 C 01 C 01 B 01 C 01 C 01 Full AND/OR search tree A 01 B 1 C 1 B 01 C 1 C 1 Backtrack free AND/OR search tree A 01 B 1 B 01 C 1 Minimal AND/OR search graph redundant A 01 B 1 C 1 An OBDD

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

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

39. AOBDD Conventions A B D C A 01 B 1 C 1 D 1 D 0 Point dead-ends to terminal 0 Point goods to terminal 1 A 01 B 01 0 C 01 1 D 01 D 01

40. AOBDD Conventions A B D C A 01 B 1 C 1 D 1 D 0 A 01 B 01 0 C 01 1 D 01 D 01 Introduce Meta-nodes

41. Combining AOBDD (apply) A B C ABCf(ABC) 0000 0010 0100 0111 1000 1011 1100 1111 0 A 01 B 01 1 C 01 * A B D ABDg(ABC) 0000 0010 0100 0111 1001 1010 1101 1110 A B D C 0 A 01 B 01 1 D 01 D 01 = A 01 B 01 0 C 01 1 D 01 D 01

42. H G E D C F B A F 01 H 01 01 C1C1 A 01 H 01 01 C2C2 A 01 B 01 G 01 G 01 B 01 01 C3C3 F 01 G 01 01 C4C4 A 01 E 01 01 C6C6 C 01 E 01 01 C7C7 C 01 D 01 D 01 01 C8C8 B 01 C 01 01 C9C9 B 01 F 01 01 C5C5 m1m1 m3m3 m7m7 m6m6 m4m4 m2m2 m5m5 Example: (f+h) * (a+!h) * (a#b#g) * (f+g) * (b+f) * (a+e) * (c+e) * (c#d) * (b+c) A 01 F 01 H 01 F 01 H 01 01 m1m1 A 01 B 01 G 01 F 01 B 01 G 01 01 m2m2 A 01 E 01 C 01 01 m4m4 C 01 D 01 D 01 01 m5m5 A 01 B 01 F 01 G 01 H 01 F 01 G 01 B 01 F 01 F 01 H 01 01 m3m3 A 01 B 01 C 01 C 01 D 01 E 01 D 01 B 01 C 01 C 01 01 m6m6 m7m7 A F H D CBF AE G H primal graph A B CF DEGH pseudo tree A B F GH C D A B C DE A B CF DEGH A B F G C E A

43. B A Example (continued) m7m7 A 01 B 01 F 01 G 01 H 01 F 01 G 01 B 01 F 01 F 01 H 01 01 m3m3 A 01 B 01 C 01 C 01 D 01 E 01 D 01 B 01 C 01 C 01 01 m6m6 A B CF DEGH m7m7 A 01 B 01 C 01 0 D 01 1 F 01 G 01 H 01 C 01 D 01 E 01 F 01 G 01 B 01 C 01 F 01 C 01 F 01 H 01

44. AOBDD vs. OBDD D C B F A E G H 10 B CCC DDDDD EE FFF GGGG H AOBDD 18 nonterminals 47 arcs OBDD 27 nonterminals 54 arcs A 01 B 01 C 01 0 D 01 1 F 01 G 01 H 01 C 01 D 01 E 01 F 01 G 01 B 01 C 01 F 01 C 01 F 01 H 01 A B CF DEGH A B C F D E G H D CBF AE G H primal graph

45. 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.

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

47. Related work Related work in Search 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.

48. 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.