Overview. Overview. Example: Full Adder Diagnosis. Example: Full Adder Diagnosis. Valued CSP. Diagnosis using Bounded Search and Symbolic Inference

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1 Overview Diagnosis using Bounded Search and Smbolic Inference Martin Sachenbacher and Brian C. Williams MIT CSAIL Soft Constraints Framework Characteriing Structure in Soft Constraints Eploiting Structure in Soft Constraints Set-based Search Decomposition-based Search Hbrids Overview Soft Constraints Framework Characteriing Structure in Soft Constraints Eploiting Structure in Soft Constraints Set-based Search Decomposition-based Search Hbrids Eample: Full Adder Diagnosis Variables describe modes of gates Gates are either in good ( ) or broken ( ) mode Valued CSP A constraint network set of variables set of domains set of constraints valuation structure Eample: Full Adder Diagnosis AND-gates broken with % probabilit OR-, XOR-gates broken with 5% probabilit Probabilistic valuation structure A constraint is a function mapping assignments over to valuations in. The complete valuation of an assignment. is

2 Eample: Full Adder Diagnosis Model gates as soft constraints Inferring Solutions Constraint network Value of optimal solution obtained b combining the constraints: Gate is good Gate is broken Time: O(ep(n)) Space: O(ep(n)) Branch-and and-bound Search Overview Time: O(ep(n)) Space: O(n) Each search node is a soft constraint subproblem Lower Bound (lb): Optimistic estimate of best solution in subtree Upper Bound (ub): Best solution found so far Prune, if lb ub. Soft Constraints Framework Characteriing Structure in Soft Constraints Eploiting Structure in Soft Constraints Set-based Search Decomposition-based Search Hbrids Structure in Soft Constraints Structure can be characteried as independence properties of functions. Strong independence: Function independent of all assignments to some variables. Weak independence: Function independent of some assignments to some variables. Strong Independence Support Weak Strong Subset of variables that function depends upon. 2

3 Strong Independence Strong Independence Hpergraph Weak Independence Weak Independence E.g. for constraint : if, then value is regardless of and Sharing of common subassignments. Weak Independence Overview Soft Constraints Framework Characteriing Structure in Soft Constraints Eploiting Structure in Soft Constraints Decomposition-based Search Set-based Search Hbrids Algebraic Decision Diagram 3

4 Eploiting Strong Independence Principle: Strong independence allows to decompose the problem into subproblems with smaller sets of variables. Tree Decomposition Subproblems Tree Decomposition BnB with Tree Decomposition Variables χ Constraints λ Algorithm BTD (Terriou and Jégou CP-3) Record solutions for subproblems ( structural goods ) BnB with Tree Decomposition EO- Model: Constraint Graph Algorithm BTD (Terriou and Jégou CP-3) Record solutions for subproblems ( structural goods ) Time: O(ep(ma i χ i )) Space: O(ep(ma i,j χ i -χ j )) 4

5 EO- Model: Tree Decomposition Ssa267-4 Circuit: Graph Ssa267-3 Circuit: Tree Overview Soft Constraints Framework Characteriing Structure in Soft Constraints Eploiting Structure in Soft Constraints Decomposition-based Search Set-based Search Hbrids Eploiting Weak Independence Set-based Branch-and and-bound Principle: Weak independence allows to consider sets of assignments at once instead of individual assignments. Encode e.g. using ADD Each search node is a set of soft constraint subproblems Lower Bound Function (f lb ): Optimistic estimates of best solutions in subtree Upper Bound (ub): Best solution found so far Prune, if f lb ub. 5

6 Domain Splitting Generalie to search over sets: Partition domains into sets Choose subset for unassigned variable Eample: 4-Queens4 Variables: Rows Domains: Columns Constraints: Q Q Q Q Eample Eample Domain splitting with partitions Solution Solution Eample Domain splitting with partitions Results C++ Implementation of SBBTD on Pentium 4 with GB RAM, using ADD librar from CUDD package Weighted version of 6-Queens-Problem (6 variables, 36 constraints, domain sie 6) Using partition {{,...,5}}: out of memor (> GB) Using partition {{},{},, {5}}: out of time (> min) Using partition {{,...,7},{8,...,5}}: 4.8 sec Solution 6

7 Overview Soft Constraints Framework Characteriing Structure in Soft Constraints Eploiting Structure in Soft Constraints Set-based Search Decomposition-based Search Hbrids Hbrid Algorithm SBBTD Eploits both strong independence using tree decomposition, and weak independence using setbased search. Partition, all else Upper bound = Upper bound = Upper bound =.44 <u=, =>.47 <u=, =>.92 <u=, =>.47 <u=, =>.92 <v=, w=>.95 7

8 Upper bound =.44 Upper bound =.44 <u=, =>.47 <u=, =>.92 <u=, =>.47 <u=, =>.92 <v=, w=>.95 Eploiting goods recorded at v2 ( forward jump ) <v=, w=>.95 Upper bound =.44 Upper bound =.44 <u=, =>.47 <u=, =>.92 <u=, =>.47 <u=, =>.92 <v=, w=>.95 <v=, w=>.5 Cut b bound <v=, w=>.95 <v=, w=>.5 Cut b bound Cut b bound 8

9 Cut b bound Cut b bound Results Best solution =.44 # Nodes = 45 C++ Implementation of SBBTD on Pentium 4 with GB RAM, using ADD librar from CUDD package Weighted version of ssa432-3 circuit (435 variables, 27 constraints, domain sie 2) Time to compute tree decomposition: 5 sec. Using partition {{},{}}: out of time (> min) Using partition {{,}}: 3 sec. Cut b bound Finished. Future Work Material Determine optimal granularit of domain partitions? Combination with local filtering techniques? Material 9

10 Main Idea Diagnosis as soft constraint solving (DX-24) Efficient techniques for solving soft constraints? Branch-and-Bound Search: memor efficient, but time eponential (backtracking) Inference: no backtracking, but memor eponential Techniques to eploit structure: Decision diagrams, Tree Decompositions Still memor eponential. Idea: hbrid of branch-and-bound search, smbolic encoding and tree decomposition. Soft CSPs Unified framework for constraints and preferences. For each constraint/tuple, a valuation that reflects preference (e.g. cost, weight, priorit, probabilit, ). The valuation of an assignment is the combination of the valuations for each constraint, using a binar operator (with special aioms). Assignments are compared using a total order on valuations. The problem is to produce an assignment of minimum valuation. Formall: Valuation Structure Decision Diagrams Eample = set of valuations, used to assess assignments = minimum element of, corresponds to totall consistent assignments = maimum element of, corresponds to totall inconsistent assignments = total order on E, used to compare two valuations = operator used to combine two valuations (commutative, associative, monotonic, neutral element, annihilator ) Binar Decision Trees Recursive Shannon epansion Variable Ordering Impose arbitrar total ordering on variables Variables must obe ordering along all paths Propert: No conflicting assignments along path

11 Ordered Binar Decision Tree Ordered Binar Decision Tree Order Rule : Collapse Leaf Nodes Rule 2: Remove Redundant Tests No longer a tree. Rule 2: Remove Redundant Tests Rule 3: Isomorphic Subgraphs

12 Rule 3: Isomorphic Subgraphs Rule ma become applicable again Final Representation ROBDDs Reduced, Ordered Binar Decision Diagram (ROBDD) Contains onl variables from support Time and space compleit of operations depends on ROBDD sie rather than number of assignments Can be etended to functions with non-binar values and non-binar variables Algebraic Decision Diagrams (ADDs) [Bahar et al. 93] Multi-valued Decision Diagram (MDDs) [Kam 9] Significant compaction in man practical cases. Eample Full Adder Tree Decomposition Ever variable of the original problem must appear in at least one subproblem. Ever constraint of the original problem must appear in a subproblem, along with all variables in its scope (covering condition). If a variable occurs in two subproblems, it must appear in ever subproblem along the path that connects the two (connectedness condition). 2

13 Eample: Full Adder Variables Constraints Tree Decomposition Principle: Solve each subproblem, then combine solutions using dnamic programming. Time O(ep(tw)), where tree width tw is maimum number of variables in a subproblem minus one. Space O(ep(sw)), where separator width sw is maimum number of variables shared between subproblems. Finding decompositions with minimal width is NPhard, but good heuristics eist. The width is often small in practice. BTD (Terriou( and Jégou CP-3) Assign variables in subproblem, beginning with root of the tree decomposition. Inside a subproblem, use classical branch-andbound, considering onl the constraints of the subproblem. Once all variables in the subproblem have been assigned, consider its children (if an). Given a child, check if the current assignment, restricted to variables shared with the child, has been previousl recorded as a good. BTD (Terriou( and Jégou CP-3) If it is not previousl recorded, compute solution for the subproblem given the current assignment and upper bound, and record it as new good. Add recorded value to value of current assignment. If resulting value is below the upper bound, proceed with the net child, else backtrack. Projection Combination, 3

14 Combination Soft CSPs For each constraint/tuple: a valuation that reflects preference (e.g. cost, weight, probabilit, ). Valuations are combined using a binar operator (with special aioms). Assignments are compared using a total order on valuations. The problem is to produce an assignment of minimum valuation. Sinking Operation is a new constraint where all values of tuples have been replaced b Generalies the test to functions Constraint Constraint sink(f e2,.5) Set-based Branch-and and-bound Function ( : assignments, : value): value if then if then return let be an unassigned variable for each do return return Time: O(ep(n)) Space: O(ep(n)) Branch-and and-bound Search Function ( : assignment, : value): value if then if then return let be an unassigned variable for each do return return Time: O(ep(n)) Space: O(n) 4