1. 4/18/2005EE5621 EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS Lecture 7, 4/25/2005 University of Washington, Department of Electrical Engineering Spring 2005 Instructor: Professor Jeff A. Bilmes

2. 4/18/2005EE5622 Material Read Fahiem Bacchus “Constraints” Tutorial on the web (dated Oct 25 th, 2001). Homework: Due Monday, May 2nd. –Do: problems 5.1, 5.3, 5.5, 5.8, 5.9, 5.11 Reminder: Midterm next Monday, May 2 nd (will cover up to and including Chapter 5, and Bacchus tutorial)

3. 4/18/2005EE5623 Outline For Today Bacchus tutorial –note: some of this material slightly overlaps with chapter 5, but using more precise notation.

4. 4/18/2005EE5624 CSP –Set of variables: { V 1, …, V n } –Set of domains for each V i, Dom[V i ] –Set of constraints {C 1, …, C m } –Each constraint over a set of variables gives set of satisfying values of that set of vars. E.g.: VarsOf[C i ] = { V 2, V 4, V 7 } Constraint might be a table of possible values: V2V2 V4V4 V7V7 135 147 249 349

5. 4/18/2005EE5625 CSP Goal is to: –1) find one set of variable values that satisfy all constraints –2) find all variable values that satisfy all constraints. This can be NP-hard in general. –Ex: 3-SAT (or 3-CNF) (x 1 Ç : x 1 Ç : x 2 ) Æ (x 3 Ç x 2 Ç x 4 ) Æ ( : x 1 Ç : x 3 Ç : x 4 ) There is no polynomial-time algorithm to find a satisfying assignment to all variables (in a general 3-CNF formula) unless P=NP

6. 4/18/2005EE5626 CSP We’ve talked about consistency –node-consistency –arc-consistency AC-3, and AC-4 –Path consistency –(i,j) consistency A CSP is (i,j)-consistent if whenever we choose consistent values for any set of i variables, we can find values for any set of j variables such that the values of all i+j variables are consistent. –Strongly k-consistent if it is (j,1) consistent, for j=1 to k. –(1,j)-consistent, called i inverse consistency. obtained by pruning domain values from variables.

7. 4/18/2005EE5627 Backtracking Generic Backtracking: –do a tree search, as soon as a variable becomes inconsistent, then back out to previous variable going to its next value. –Quite suboptimal Constraint propagation –enforce consistency in sub-CSP below a given tree- node Intelligent Backtracking –keep track of why things fail, and jump back to those reasons (rather than blindly to previous variable) –Dynamic variable order heuristics can improve search considerably.

8. 4/18/2005EE5628 No-Goods A “no-good”: set of assignments guaranteed not to be contained in any solution. All of backtracking and other forms of search can be seen as “no-good processing”, essentially discovering no-goods, and unioning them together to get larger no-goods. –larger no-goods are better since they cover more –We re-use the no-goods at other sections of the search –We can’t store all no-goods since there are an exponential number of them. –All of these are heuristic algorithms (none solve the problem in polynomial time of course).

9. 4/18/2005EE5629 Propositional Encoding Propositional Logic. –a proposition “p” is stated, meaning it is asserted to be true. We can encode CSP using propositional logic. –Primitive constraints: C is a constriant, if a 2 C, then a is an assignment such as: We can represent all constraints in C as disjunction of conjunctions:

10. 4/18/2005EE56210 Propositional Encoding –Exclusivity: each variable assigned only a single value, and we can represent this also:

11. 4/18/2005EE56211 Propositional Encoding –Exhaustiveness: Each variable must be assigned a value. –All three combined, we get:

12. 4/18/2005EE56212 No-good reasoning Resolution: We can use resolution to union together no-goods. No Goods Equivalent Clauses

13. 4/18/2005EE56213 No-good reasoning Resolution: We can use resolution to union together no-goods. No Goods Equivalent Clauses Exhaustiveness New Unioned No-Good

14. 4/18/2005EE56214 No-good reasoning Other forms of no-goods can be combined together as well. Most forms of no-good processing is done more implicitly using various heuristics. Key is that there are too many no-goods to remember, even after unioning. We must strive for ways to remember what will be useful in future, and forget what we won’t ever need again, but never remember too much so that memory requirements becomes exponential.

15. 4/18/2005EE56215 No-good reasoning No-goods are encountered during search –depth-first search (standard for CSP) –When values found inconsistent, they become nogood. –Logic can explain what we do: if at tree-node n, all values of a variable V fail, we can use resolution above to create a new no-good not containing V, but containing all other reasons variable has failed. –No-goods are stored as a set of levels i.e., a tree-node n, a set of variables have been assigned. To store the no-good, we store the current set of levels (implicitly storing the values of those variables). We also store the current variable’s assignmente. E.g., –(1,3,5,9,V  a ) is a no good that says the current assignments to variables at levels 1,3,5, and 9 + assigning V to a is a no good.

16. 4/18/2005EE56216 No-good reasoning What are no-goods good for? –For Intelligent Backtracking logic can justify intelligent backtracking (e.g., to previous level or to much earlier in search tree) –For domain pruning if we find a no-good {1,2,4,V  a }, where V is a future variable, we can prune a from the domain of V at the current time (but we need to restore it when we return from level 4)

17. 4/18/2005EE56217 Generic Tree-Search Template

18. 4/18/2005EE56218 Generic Tree-Search Template FINDALL : if we should find all solutions. Note that if we find a solution, it is not another solution so it becomes a no-good for the rest of the search. pickNextVariable : determines next variable to split on –this is important, can have large impact on search. “Fail First” principle later –In any case, picks variables with empty domains first to find deadends, and next picks variables with only one value in the domains checkConsistent : checks that newly assigned value is consistent propagateConstraints : propagate any constraints now available by the newly assigned variable (e.g., forward checking or arc consistency to future vars) TreeSearch : might want to backtrack below current level and if so, we backtrack back to that level right there, in inner loop computeBacktrackLevel : if all values of V failed, we might want to backtrack to level earlier than previous level setNoGoodOfAssigmentatBTLevel : if any no-goods can be computed, they are done so here.

19. 4/18/2005EE56219 Generic Backtracking

20. 4/18/2005EE56220 Generic Backjumping

21. 4/18/2005EE56221 Conflict-Directed Backjumping

22. 4/18/2005EE56222 Value Specific CBJ

23. 4/18/2005EE56223 More advanced options Domain pruning –keep a current domain of values for each variable, prune out values we know are impossible, restore them when they become possible again. Constraint propagation –Use forward checking or AC-3 to for each sub-CSP prune from the domains of variables values that we know are not possible. –Other more advanced algorithms are: MAC: Maintain Arc Consistency: It does (1,1)-consistency in each sub-CSP. Can be combined with CBJ. GAC: generalized arc consistency (rather than waiting for constraint that has all but one value instantiated (FC) or two values instantiated (MAC), we can do more).

24. 4/18/2005EE56224 Dynamic Variable Orderings The pickNextVar routine need not always choose the next variable in all circumstances. In the most general case, we choose both the next variable and a value of the next variable at the same time, and each time we choose this, it can be different and might depend on anything that has previously happened during the search. This is shown on algorithm in next slide:

25. 4/18/2005EE56225 General Dynamic Variable order

26. 4/18/2005EE56226 Dynamic Variable order Current Practice –In graph coloring, a heuristic that works well is the “Brelaz heuristic” first color vertex of maximum degree. Next, select uncolored vertex of maximum saturation (maximum number of *different* colored vertices connected to it). –break ties by the degree of the vertex intersected with the uncolored nodes. –Maximizing number of different colored vertices puts most constraints on new variable choosen.

27. 4/18/2005EE56227 Dynamic Variable order Brelaz heuristic is essentially same as: –choose next variable with minimum remaining consistent values –break ties by number of constraints variable has with set of currently unassigned variables. This is example of the “fail first” principle –the best search order is one that minimizes the expected length (depth) of each branch (i.e., prefer short trees since search is exponential in the depth).

28. 4/18/2005EE56228 Other good sources “Constraint Processing,” Rina Dechter, Morgan Kaufmann, 2003. “Principles of Constraint Programming”, Krzysztof R. Apt, Cambridge, 2003 “Increasing Tree-Search Efficiency for Constraint Satisfaction Problems”, R. Haralick and G. L. Elliot, Artificial Intelligence 14, 1980, pgs. 263—313.