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Exact Reasoning: conditioning search and hybrids

Exact Reasoning: conditioning search and hybrids. COMPSCI 276 Fall 2007. Probabilistic Inference Tasks. Belief updating: Finding most probable explanation (MPE) Finding maximum a-posteriory hypothesis Finding maximum-expected-utility (MEU) decision. Belief Updating. Smoking.

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Exact Reasoning: conditioning search and hybrids

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  1. Exact Reasoning:conditioning search and hybrids COMPSCI 276 Fall 2007

  2. Probabilistic Inference Tasks • Belief updating: • Finding most probable explanation (MPE) • Finding maximum a-posteriory hypothesis • Finding maximum-expected-utility (MEU) decision CS 276

  3. Belief Updating Smoking lung Cancer Bronchitis X-ray Dyspnoea P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ? CS 276

  4. Conditioning generates the probability tree Complexity of conditioning: exponential time, linear space CS 276

  5. Idea: conditioning until of a (sub)problem gets small Conditioning+Elimination CS 276

  6. A=0 A=0 A=0 A B C B C B B F F Loop-cutset decomposition • You condition until you get a polytree A=1 A=0 A=1 A=1 A=1 B C B F P(B|F=0) = P(B, A=0|F=0)+P(B,A=1|F=0) Loop-cutset method is time exp in loop-cutset size And linear space CS 276

  7. A 0 1 B 0 1 0 1 E 0 1 0 1 0 1 0 1 C 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 OR search space A F B C Ordering: A B E C D F E D CS 276

  8. 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 CS 276

  9. 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 OR vs AND/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 0 OR 1 1 1 0 CS 276 1

  10. 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 AND/OR vs. OR 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) OR CS 276

  11. 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. OR 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 CS 276

  12. A F B C E D A B E C D F (A=1,B=1) (B=0,C=0) AND/OR vs. OR 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 CS 276 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

  13. OR space vs. AND/OR space CS 276

  14. A F B C E D A B E C D F OR A 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 1 1 0 0 0 1 0 1 OR D D F D F F D F D F D F D F D F AND 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 1 0 1 0 1 AND/OR search tree for graphical models • The AND/OR search tree of R relative to a spanning-tree, T, has: • Alternating levels of: OR nodes (variables) and AND nodes (values) • Successor function: • The successors of OR nodes X are all its consistent values along its path • The successors of AND <X,v> are all X child variables in T • A solution is a consistent subtree • Task: compute the value of the root node CS 276

  15. From DFS trees to pseudo-trees (Freuder 85, Bayardo 95) 1 4 1 6 2 3 2 7 5 3 (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 CS 276

  16. OR 1 AND a b c OR 3 5 AND a b c a b c OR 4 4 4 7 7 7 2 2 2 6 6 6 AND a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c From DFS trees to Pseudo-trees OR 1 AND a b c OR 2 7 AND a b c a b c DFS tree depth = 3 OR 3 3 3 5 5 5 AND a b c a b c a b c a b c a b c a b c OR 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 AND a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c pseudo- tree depth = 2 CS 276

  17. Finding min-depth backbone trees • Finding min depth DFS, or pseudo tree is NP-complete, but: • Given a tree-decomposition whose tree-width is w*, there exists a pseudo tree T of G whose depth, satisfies (Bayardo and Mirankar, 1996, bodlaender and Gilbert, 91): m <= w* log n, CS 276

  18. A F C bucket-A A E B D AB bucket-B bucket-C ABC ABE bucket-E bucket-D BDE AEF bucket-F A A OR AND 0 1 B OR B B AND 0 0 1 1 C E OR C E C E C E C E AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D F 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 Generating pseudo-trees from Bucket trees A A (A) B B (AB) C C (AC) (BC) E E (AE) (BE) (AE) D D (BD) (DE) F F (AF) (EF) d: A B C E D F Bucket-tree based on d Induced graph Bucket-tree Bucket-tree used as pseudo-tree AND/OR search tree CS 276

  19. AND/OR Search-tree properties(k= domain size, m = pseudo-tree depth. n = number of variables) • Theorem: Any AND/OR search tree based on a pseudo-tree is sound and complete (expresses all and only solutions) • Theorem: Size of AND/OR search tree is O(n km) Size of OR search tree is O(kn) • Theorem: Size of AND/OR search tree can be bounded by O(exp(w* log n)) • Related to:(Freuder 85; Dechter 90, Bayardo et. al. 96, Darwiche 1999, Bacchus 2003) • When the pseudo-tree is a chain we get an OR space CS 276

  20. Tasks and value of nodes • V( n) is the value of the tree T(n) for the task: • Counting: v(n) is number of solutions in T(n) • Consistency: v(n) is 0 if T(n) inconsistent, 1 othewise. • 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. • Theorem: Complexity of AO dfs search is • Space: O(n) • Time: O(n km) • Time: O(exp(w* log n)) CS 276

  21. A A B B C E D E D C A 0 1 B B 0 1 0 1 E D E D E D E D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C C C C C C C C 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Belief-updating on example CS 276

  22. A A B B C E D E D C A PA(0) PA(1) 0 1 B B PB|A(0|0) PB|A(1|0) PB|A(0|1) PB|A(1|1) 0 1 0 1 E D E D E D E D PE|AB(1|0,0) PE|AB(0|0,1) PE|AB(0|1,0) PE|AB(1|1,0) PE|AB(0|1,1) PE|AB(1|1,1) PE|AB(0|0,0) PE|AB(1|0,1) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C C C C C C C C PD|BC(0|0,0)× PC|A(0|0) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 CS 276

  23. A P(A=0) 0 B P(B=0|A=0) P(B=1|A=0) 0 1 E D E D P(E=0|A=0,B=0) P(E=1|A=0,B=0) P(E=0|A=0,B=1) P(E=1|A=0,B=1) 1 0 1 0 0 1 0 1 C C C C A A P(D=0|B=0,C=0)× P(C=0|A=0) P(D=0|B=0,C=1)× P(C=1|A=0) P(D=1|B=0,C=0)× P(C=0|A=0) P(D=1|B=0,C=1)× P(C=1|A=0) P(D=0|B=1,C=0)× P(C=0|A=0) P(D=0|B=1,C=1)× P(C=1|A=0) P(D=1|B=1,C=0)× P(C=0|A=0) P(D=1|B=1,C=1)× P(C=1|A=0) B 0 1 0 1 0 1 0 1 B C E D E D C CS 276

  24. A P(A=0) 0 B P(B=0|A=0) P(B=1|A=0) 0 1 E C E C P(E=0|A=0,B=0) P(E=1|A=0,B=0) P(E=0|A=0,B=1) P(E=1|A=0,B=1) P(C=0|A=0) P(C=1|A=0) P(C=0|A=0) P(C=1|A=0) 0 1 0 1 0 1 0 1 D D D D P(D=0|B=0,C=0) P(D=0|B=0,C=1) P(D=1|B=0,C=0) P(D=1|B=0,C=1) P(D=0|B=1,C=0) P(D=0|B=1,C=1) P(D=1|B=1,C=0) P(D=1|B=1,C=1) 0 1 0 1 0 1 0 1 (d) CS 276

  25. A A B B C E C D E D AND/OR Tree DFS Algorithm (Belief Updating) Result: P(D=1,E=0) Evidence: E=0 .24408 OR A .6 .4 .3028 AND .1559 0 1 .3028 .1559 OR B B .1 .9 .4 .6 .352 .623 AND .27 .104 0 0 1 1 .5 .7 OR .4 .88 .54 .89 .2 .52 E C E C E C E C .4 .5 .7 .2 .2 .8 .2 .8 .1 .9 .1 .9 .8 .9 .7 .5 .8 .9 .7 .5 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR .8 .9 .7 .5 .8 .9 .7 .5 D D D D D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR node: Marginalization operator (summation) AND node: Combination operator (product) CS 276 Value of node = updated belief for sub-problem below Evidence: D=1

  26. Complexity of AND/OR Tree Search • k = domain size • m = depth of pseudo-tree • n = number of variables • w*= treewidth CS 276

  27. From Search Trees to Search Graphs • Any two nodes that root identical subtrees (subgraphs) can be merged CS 276

  28. From Search Trees to Search Graphs • Any two nodes that root identical subtrees (subgraphs) can be merged CS 276

  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 F B AND/OR Tree 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 CS 276

  30. A J A F B An AND/ORgraph 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 CS 276

  31. A C C A A F F B D D B B E E C E D F pseudo tree Merging Based on Context One way of recognizing nodes that can be merged: context (X) = ancestors of X in pseudo tree that are connected to X, or to descendants of X [ ] [A] [AB] [AB] [AE] [BC] CS 276

  32. A C A F B D B E C E D F AND/OR Search Graph Constraint Satisfaction – Counting Solutions Context [ ] [A] [AB] [AB] [AE] [BC] pseudo tree OR A AND 0 1 OR B B AND 1 1 0 0 C C OR C C E E E E AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D D D D F F F F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 CS 276 context minimal graph

  33. Context [ ] A A [A] B [AB] [AB] E C B C [BC] D E D AND/OR Tree DFS Algorithm (Belief Updating) Result: P(D=1,E=0) Evidence: E=0 .24408 A .6 .4 .3028 .1559 0 1 .3028 .1559 B B .1 .9 .4 .6 .352 .623 .27 .104 0 0 1 1 .5 .7 .4 .88 .54 .89 .2 .52 E C E C E C E C .4 .5 .7 .2 .2 .8 .2 .8 .1 .9 .1 .9 .8 .9 .7 .5 .8 .9 .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 .8 .9 .7 .5 .8 .9 .7 .5 D D D D D D D D .8 .9 .5 .8 .9 .7 .5 .7 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR node: Marginalization operator (summation) AND node: Combination operator (product) CS 276 Value of node = updated belief for sub-problem below Evidence: D=1

  34. Context [ ] A A [A] B [AB] [AB] E C B C [BC] D E D Cache table for D AND/OR Graph DFS Algorithm (Belief Updating) Result: P(D=1,E=0) Evidence: E=0 .24408 A .6 .4 .3028 .1559 0 1 .3028 .1559 B B .1 .9 .4 .6 .352 .623 .27 .104 0 0 1 1 .5 .7 .4 .88 .54 .89 .2 .52 E C E C E C E C .4 .5 .7 .2 .2 .8 .2 .8 .1 .9 .1 .9 .8 .9 .7 .5 .8 .9 .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 .8 .9 .7 .5 D D D D .8 .9 .5 .7 0 1 0 1 0 1 0 1 CS 276 Evidence: D=1

  35. F G B A J H E C C [ ] [ ] C D K K H H [C] [C] [C] [C] L L L A A [CK] [CK] [CH] [CH] K M N N B B [CKL] [CKL] [CHA] [CHA] N O O E E P [CKLN] [CKLN] [CHAB] [CHAB] F F [AB] [AB] O P P J J [CKO] [CKO] [CHAE] [CHAE] G G [AF] [AF] D D [CEJ] [CEJ] M M [CD] [CD] (C K H A B E J L N O D P M F G) (C K H A B E J L N O D P M F G) AND/OR context minimal graph C 0 1 H H K K 0 1 0 1 0 1 0 1 A A A A L L L L 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 B B B B B B B B N N N N N N N N 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 O O O O O O O O O O O O O O O O 0 1 0 1 0 1 0 1 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 F F F E E E E E E E E E E E E E E E E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 P P P P P P P P 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 G G G G J J J J J J J J J J J J J J J J 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 D D D D D D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 M M M M 0 1 0 1 0 1 0 1 CS 276

  36. F G B A J H E max context size = treewidth C [ ] C D K H [C] [C] L L A [CK] [CH] K M N B [CKL] [CHA] N O E P [CKLN] [CHAB] F [AB] O P J [CKO] [CHAE] G [AF] D [CEJ] M [CD] (C K H A B E J L N O D P M F G) How Big Is the Context? Theorem: The maximum context size for a pseudo tree is equal to the treewidth of the graph along the pseudo tree. CS 276

  37. G D A B BDEFG F C E EFH M K H FHKJ L J KLM Treewidth vs. Pathwidth TREE G ABC BDFG BDEF D A B treewidth = 3 = (max cluster size) - 1 F EFH C E M K H FHK L HJ KLM J CHAIN ABC pathwidth = 4 = (max cluster size) - 1 CS 276

  38. Complexity of AND/OR Graph Search • k = domain size • n = number of variables • w*= treewidth • pw*= pathwidth w* ≤ pw* ≤ w* log n CS 276

  39. AND/OR search algorithms • AO(i) • i = the max size of a cache table (i.e. number of variables in a context) i = 0 i = w* treewidth Space: Time: O( n ) O( exp (w* log n) ) O(n exp w* ) O(n exp w* ) CS 276

  40. Searching AND/OR Graphs • AO(i): searches depth-first, cache i-context • i = the max size of a cache table (i.e. number of variables in a context) i=0 i=w* i Space: O(exp w*) Time: O(exp w*) Space: O(n) Time: O(exp(w* log n)) Space: O(exp(i) ) Time: O(exp(m_i+i ) CS 276

  41. 0 1 0 1 0 1 0 1 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 F B Caching B C K E C E D D F G J G H OR A H K context(D)={D} context(F)={F} 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 CS 276

  42. A J A F B Caching 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 CS 276

  43. A F 0 1 A B C 0 1 0 1 B 0 1 0 1 0 1 0 1 E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C 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 A 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 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 B A 0 1 B E C 0 1 0 1 E 0 1 0 1 0 1 0 1 C 0 1 0 1 0 1 0 1 D F D OR AND A 0 1 0 1 0 1 0 1 0 1 AND F 0 1 OR 0 1 B B OR B B AND 0 0 1 1 AND 0 0 1 1 OR E C E C E C E C OR E C E C E C E C AND 0 1 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D F D F D F D F D F D F D F D F OR OR D F D F D F D F D F D F D F D F AND 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 0 1 0 1 All four search spaces Full OR search tree Context minimal OR search graph CS 276 Full AND/OR search tree Context minimal AND/OR search graph

  44. A F 0 1 A B C 0 1 0 1 B 0 1 0 1 0 1 0 1 E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C 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 A 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 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 B A 0 1 B E C 0 1 0 1 E 0 1 0 1 0 1 0 1 C 0 1 0 1 0 1 0 1 D F D OR AND A 0 1 0 1 0 1 0 1 0 1 AND F 0 1 OR 0 1 B B OR B B AND 0 0 1 1 AND 0 0 1 1 OR E C E C E C E C OR E C E C E C E C AND 0 1 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D F D F D F D F D F D F D F D F OR OR D F D F D F D F D F D F D F D F AND 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 0 1 0 1 All four search spaces Full OR search tree Context minimal OR search graph Time-space CS 276 Full AND/OR search tree Context minimal AND/OR search graph

  45. Available code • http://graphmod.ics.uci.edu/group/Software CS 276

  46. E E E E D D D D M M M M C C C A L L L L B B K K K H H H H F F F F G G G J J J J A A A A B K B K B K B K C G L C G L C G L C G L D F M D F M D F M D F M E H E H E H E H J J J J 1-cutset 3-cutset 2-cutset AND/OR w-cutset CS 276

  47. E D M C A L B K H F G J E E D D M M C C A L A L B B K K H H F F G J G J AND/OR w-cutset grahpical model pseudo tree 1-cutset tree CS 276

  48. Searching AND/OR Graphs • AO(i): searches depth-first, cache i-context • i = the max size of a cache table (i.e. number of variables in a context) i=0 i=w* i Space: O(exp w*) Time: O(exp w*) Space: O(n) Time: O(exp(w* log n)) Space: O(exp(i) ) Time: O(exp(m_i+i ) CS 276

  49. w-cutset Trees Over AND/OR space • Definition: • T_w is a w-cutset tree relative to backbone tree T, iff T_w is roots T and when removed, yields tree-width w. • Theorem: • AO(i) time complexity for backbone T is time O(exp(i+m_i)) and space O(i), m_i is the depth of the T_i tree. • Better than w-cutset: O(exp(i+c_i)) when c_i is the number of nodes in T_i CS 276

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