f12-csp-sat - Constraint Satisfaction Problems Search when states are factored Until now we assumed states are blackboxes We will now assume that states

# f12-csp-sat - Constraint Satisfaction Problems Search when...

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Constraint Satisfaction Problems
Search when states are factored Until now, we assumed states are black- boxes. We will now assume that states are made up of “state-variables” and their “values” Two interesting problem classes CSP & SAT (Constraint Satisfaction Problems) Planning
December 2, 1998 Sqalli, Tutorial on Constraint Satisfaction Problems 4 Constraint Satisfaction Problems (a brief animated overview) Values Constraints Problem Statement Variables CSP Algorithm Solution X Z Y Coloring Problem CSP Representation Search backtracking, variable/value heuristics Inference Consistency enforcement, forward checking X: red Y: blue Z: green Red green blue Red green blue Red green blue Z X Y Constraint Graph
Red green blue Red green blue Red green blue Z X Y Constraint Graph As a piece of code that takes a partial assignment and returns Legal or not
Example: N-queen problem N=4 Variables: Queen per column Values: N rows that queen can be in Constraints: no pair in same row, column or diagonal
Constraint Graphs will be hyper-graphs for non-binary CSPs
“Real world” CSP problems.. Most assignment problems including Time-tabling Variables: Courses; Values: Rooms, times Jobshop Scheduling Variables: jobs; values: machines Sudoku; KenKen Cross-word puzzle Boolean satisfiability Many other AI problems can be compiled to CSP problems We will see how to compile planning into a CSP problem
Complexity of CSP.. Boolean Satisfiability is a special case of discrete variable CSP problem So, CSP is NP-hard Specific types of CSP may be tractable. E.g. if all the variables are boolean and all the constraints are binary, you have 2-SAT which is tractable. The topology of the “constraint graph” also affects the complexity of the CSP problem E.g. If the constraint graph is a chain graph or a multi-tree, we can solve it polynomially Do you remember the similar result for Bayes Nets?
CSP vs. SAT CSP allows multi-valued variables; while SAT focuses on boolean variables Can convert CSP into SAT Make a variable V with values 1, 2, 3 into propositions V-1, V-2, V-3 Also need to add a constraint saying that no more than one of them can be true » Usually, as a set of binary mutex constraints ( not V-i or not V-j ) CSP algorithms are written assuming constraints as procedures; SAT algorithms are written assuming clauses as declaratively represented
CSP/SAT vs. IP/LP Linear programming Maximize a linear objective function 3x+4y+5z over real valued variables Subject to a set of linear constraints x+y <= 2; y+3z <=4 etc Can be understood in two phases Feasibility of the polytope Finding the optimal corner of the polytope Integer programming Linear programming, but (some of) the variables have to take integer values
General Search vs. CSP Blackbox State External Child-generator State-space can be infinite External goal test Goals can occur at any depth

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• Spring '06
• Staff
• Economics, Constraint satisfaction, Constraint satisfaction problem, constraint graph

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