Lecture 7 Constraint Satisfaction Problems

1. Lecture 7Constraint Satisfaction Problems CSE 573 Artificial Intelligence I Henry Kautz Fall 2001

2. But first… • Why is driving to Spokane like playing Backgammon?

3. Roadmap Backtracking Constraint Satisfaction Algorithms • Tree Search (guessing) • BFS • DFS • Iterative Deepening • Bidirectional • Best-first search • A* • Game tree • Davis-Putnam (logic) • Cutset conditioning (probability) • Inference (simplifying) • Forward Checking • Path Consistency (Waltz labeling, temporal algebra) • Resolution • “Bucket Algorithm” • Iterative improvement (wandering) • Hillclimbing (gradient descent) • Walksat & Simulated annealing • Genetic algorithms • Monte-Carlo Methods

4. Map Coloring

5. CSP • V is a set of variables v1, v2, …, vn • D is a set of finite domains D1, D2, …, Dn • C is a set of constraints C1, C2, …, Cm Each constraint specifies a restriction over joint values of a subset of the varibles • E.g.: v1 is Spain, v2 is France, v3 is Germany, … Di = { Red, Blue, Green} for all i For each adjacent vi, vj there is a constraint Ck (vi,vj) in { (R,G), (R,B), (G,R), (G,B), (B,R), (B,G) } • Important special case: Discrete Binary CSP

6. Variations • Find a solution that satisfies all constraints • Find all solutions • Find a “tightest form” for each constraint (v1,v2) in { (R,G), (R,B), (G,R), (G,B), (B,R), (B,G) }  (v1,v2) in { (R,G), (R,B), (B,G) } • Find a solution that minimizes some additional objective function • Note: in general path length to solution is not important!

7. Exploiting CSP Structure • Interleaveinference and search • At each internal node: • Select unassigned variable • Select a value in domain • Backtracking: try another value • Branching factor? • How to best select a variable? • At each node: • Propagate Constraints

8. Running Example: 4 Queens • Variables: • Constraints:

9. Running Example: 4 Queens • Variables: Q1 in {1,2,3,4} Q2 in {1,2,3,4} Q3 in {1,2,3,4} Q3 in {1,2,3,4} • Constraints:

10. Constraint Checking

11. Forward Checking

12. Arc Consistency Q3=3 inconsistent with Q4 in {2,3,4} Q2=1 and Q2=2 inconsistent with Q3 in {1} Only one guess!

13. Huffman-ClowesLabeling + - + + + + +

14. Waltz’s Filtering: Arc-Consistency • Lines: variables • Conjunctions: constraints • Initially Di = {+,-, ,  ) • Repeat until no changes: • Choose edge (variable) • Delete labels on edge not consistent with both endpoints

15. No labeling!

16. Path Consistency • Path consistency (3-consistency): • Check every triple of variables • More expensive! • k-consistency: • n-consistency: backtrack-free search

17. Variable and Value Selection • Select variable with smallest domain • Which values to try first? • Why different? • Tie breaking?

18. Variable and Value Selection • Select variable with smallest domain • Minimize branching factor • Most likely to propagate: most constrained variable heuristic • Which values to try first? • Most likely value for solution • Least propagation! Least constrained variable • Why different? • Every constraint must be eventually satisfied • Not every value must be assigned to a variable! • Tie breaking? • In general randomized tie breaking best – less likely to get stuck on same bad pattern of choices

19. N-queens Demo • Board size 15 • Delay 6 • Deterministic vs. Randomized tie breaking

20. Inference in CSP’s: So Far… • Constraint checking against partial assignments • Forward checking: each time a variable is instantiated, remove other inconsistent values • Keep track of current domains of vars • Arc consistency: • Iterate forward checking until no more changes • For each pair (Vi,Vj): for every value d in the current domain of Vi there is some value y in the domain of Vj such that Vi=x and Vj=y is permitted by the binary constraint between Vi and Vj • Variable / value selection

21. Coming Up • Leveraging randomized tie-breaking • Satisfiability • Boolean (0/1) variables • Non-binary constraints • Iterative Repair and Problem Hardness