1 / 48

From AND/OR Search to AND/OR BDDs

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. A. B. B. A. C. C. C. C. B. B. D. D. D. D. D. D. D. D. E. E. G. G. E. E. E.

ahmed-salazar
Download Presentation

From AND/OR Search to AND/OR BDDs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. A B B A C C C C B B D D D D D D D D E E G G E E E E E E F H 0 1 0 1 F F F G G H 0 1 Combining Decision Diagrams g = C*G +D*H f = A*E + B*F OBDD f * g =

  3. A A C C B B B B D D D D AND E E E E G G G G F F H H 0 0 1 1 0 0 1 1 Combining AND/OR BDDs g = C*G +D*H f = A*E + B*F AOBDD f * g =

  4. A B B A C C C C B B D D D D D D D D AND E E G G E E E E E E F H 0 1 0 1 F F F G G H 0 1 OBDDs vs AOBDD f * g = 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. E A B red green red yellow green red green yellow yellow green yellow red A D B F G C Constraint Networks A Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (red, green, blue) Constraints: Constraint graph A E D F B G C Semantics: set of all solutions Primary task: find a solution

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

  8. A F B C E D Graphical models • 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 • All these tasks are NP-hard •  identify special cases •  approximate

  9. P(X) P(Y|X) P(Z|X) P(T|Y) P(R|Y) P(L|Z) P(M|Z) Tree-solving is easy CSP – consistency (projection-join) Belief updating (sum-prod) #CSP (sum-prod) MPE (max-prod) 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 ABC DGF G D A B BDEF F C EFH E M K H FHK L J Inference algorithm: Time: exp(tree-width) Space: exp(tree-width) HJ KLM treewidth = 4 - 1 = 3 treewidth = (maximum cluster size) - 1

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

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

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

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

  16. Outline • Background in Graphical models • AND/OR searchtrees and Graphs • Minimal AND/OR graphs • From AND/OR graphs to AOMDDs • Compilation of AOMDDs • AOMDDs and earlier BDDs

  17. A F B C E D A B E C D F Classic OR Search Space Ordering: A B E C D F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

  18. A A F F B B C C E E D D A B OR A E C AND 0 1 D F OR B B AND 0 0 1 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 AND/OR Search Space A F B C E D Primal graph DFS tree

  19. A F B C E D A B A 0 1 E C B 0 1 0 1 D F A E 0 1 0 1 0 1 0 1 0 1 B C 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 AND/OR vs. OR OR A AND 0 1 AND/OR OR B B AND 0 0 1 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 AND/OR size: exp(4), OR size exp(6) 0 OR 1 1 1 0 1

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

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

  22. A F B C solution E D OR A 11 AND 6 5 1 B OR B 5 6 E C AND 1 0 1 1 4 2 OR E C E C E C 1 1 4 2 1 1 D F AND 0 0 1 0 1 0 1 0 1 0 1 0 1 1 2 1 2 1 1 0 1 0 0 1 OR D F D F D F D F D F D F 1 1 1 0 2 1 2 1 1 1 1 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 DFS algorithm (#CSP example) A 0 B 4 0 OR node: Marginalization operator (summation) E C 2 2 AND node: Combination operator (product) 0 2 0 1 0 1 1 1 D F D F 2 1 2 0 0 1 0 1 0 1 0 1 1 1 1 0 1 1 0 0 Value of node = number of solutions below it

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

  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 spaceO(exp(w* log n)

  25. 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 J J J J G G J J G G J J J J G G J J G G G G J J G G J J G G G G 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 A J A From AND/OR Tree F B B C K E C E D D F G J G H OR A H K AND 0 1 OR B B AND 0 0 1 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND OR AND OR AND

  26. A J A To an AND/ORGraph F B B C K E C E D D F G J G H OR A H K AND 0 1 OR B B AND 0 0 1 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND 0 1 0 1 OR J J G G AND 0 1 0 1 0 1 0 1 OR H H H H K K K K AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

  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 theclosure 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. A J A F B B C K E C E D D F G J G H H K 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 context(B) = {A, B} context(C) = {A,B,C} context(D) = {D} context(F) = {F}

  29. 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 J J J J G G J J G G J J J J G G J J G G G G J J G G J J G G G G 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 A J A Caching F B B C K E C E D D F G J G H context(B) = {A, B} context(C) = {A,B,C} context(D) = {D} context(F) = {F} OR A H K AND 0 1 OR B B AND 0 0 1 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND OR AND OR AND

  30. A J A Caching F B B C K E C E D D F context(D)={D} context(F)={F} G J G H OR A H K AND 0 1 OR B B AND 0 0 1 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND 0 1 0 1 OR J J G G AND 0 1 0 1 0 1 0 1 OR H H H H K K K K AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

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ll Four Search Spaces Full OR search tree 126 nodes Context minimal OR search graph 28 nodes Context minimal AND/OR search graph 18 AND nodes Full AND/OR search tree 54 AND nodes

  32. Properties of minimal AND/OR graphs Complexity: • MinimalAND/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=0 i=w* j Space: O(exp w*) Time: O(exp w*) Space: O(n) Time: O(exp(w* log n)) Space: O(exp(j) ) Time: O(exp(m_j+j )

  34. C A F D B E A 5 7 0 1 B B 7 5 6 5 6 4 0 1 0 1 C C C C 4 3 4 E E E 4 E 2 0 5 2 0 1 0 1 D D D D 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 B Primal graph C E D F Cache Table (C) 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. A 0 1 B B 0 1 0 1 C C C C 0 1 0 1 0 1 0 1 Full AND/OR search tree A A 0 1 0 1 A 0 1 B B B 1 B B 1 0 1 1 0 1 C C C C 1 C 1 1 1 An OBDD 1 Backtrack free AND/OR search tree OR Search Graphs vs OBDDs A B C redundant Minimal AND/OR search graph

  37. A B A B A D C 0 1 0 0 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) B redundancy B 1 1 D C D D C 1 1 0 D 1 1 0

  38. A B A B A D C 0 1 0 0 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) B redundancy B 1 1 D C D D C 1 1 0 D 1 1 0

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

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

  41. B D D D C D B A A 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 A 0 1 B 0 1 C 0 1 0 1 0 1 0 1 Combining AOBDD (apply) A A A B B B C D D C = *

  42. A A D G B C B F C F A m7 E A H D E G H B primal graph pseudo tree B C F 0 1 m6 m6 m3 0 1 m3 D E G H A A B F 0 1 0 1 B C F C9 C5 0 1 0 1 0 1 m2 C m4 m1 m5 0 1 m1 m5 m4 G H m2 D E A A A C C B E C D B B C D D A D G C B B E E E F G G A A B F C C B C H A F A G H H F F G F H F F C C B H B G H F C D F B G A B A A D 0 1 0 1 0 1 0 1 0 1 0 1 F E H D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 C8 E G D C1 C4 C6 C7 C2 F C3 0 1 H G Example: (f+h) * (a+!h) * (a#b#g) * (f+g) * (b+f) * (a+e) * (c+e) * (c#d) * (b+c) m7

  43. A m7 B C F 0 1 m6 A 0 1 m3 D E G H 0 1 m7 B E G G F B H D C B F G G F B D F C H F A F A C C D C E C D H C A B F H F C B B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Example (continued)

  44. D G C B F A E A H B primal graph C F D E G H 0 1 C G F F C B G F H A B D C E H F D C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 AOBDD vs. OBDD A B B C C C C D D D D D D E E E F F F F G G G G G A H H B C 1 0 D AOBDD 18 nonterminals 47 arcs OBDD 27 nonterminals 54 arcs E F G H

  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.

More Related