Lecture 18. Questions? Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 1

Transcription

1 Lecture 18 Questions? Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 1

2 Outline Chapter 6 - Constraint Satisfaction Problems Path Consistency & Global Constraints Sudoku Example Backtracking Search for CSPs Local Search for CSPs Structure of Problems Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 2

3 Constraint Propagation Recall from last time: Node consistency - eliminate any values from the domain that is not consistent with unary constraints Arc consistency - remove any values in one domain that are not satisfied by a value in the other domain AC-3 algorithm - repeatedly make domains arc consistent until there are no more changes. If any domain becomes empty, there is no solution. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 3

4 Path Consistency Path consistency looks as triples of variables. A two-variable set { X i, X j } is path-consistent with respect to a third variable X m if for every assignment { X i =a, X j =b } consistent with constraints on { X i, X j }, there exists an assignment to X m that satisfies constraints on { X i, X m } and { X m, X j } Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 4

5 K-consistency Generalize notion of consistency. A CSP is k- consistent if, for any set of k-1 variables and for any consistent assignment to those variables, a consistent value can always be assigned to any k th variable. Node consistency is 1-consistency Arc consistency is 2-consistency Path consistency is 3-consistency, for binary constraints Unfortunately, exponential in time and space. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 5

6 Global Constraints Some global constraints are common enough to be handled by special-purpose algorithms. E.g. For AllDiff, if m variables are involved in the constraint, and if they have n possible distinct values altogether, and m > n, then the constraint cannot be satisfied. Algorithm: Remove any variable in constraint with a singleton domain and delete that variable's value from the domains of the remaining variables. Repeat as needed. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 6

7 Global Constraints Consider assignment { WA = red, NSW = red }. Note SA, NT, Q are effectively connected with AllDiff After running AC-3, domain of each variable reduced to { green, blue }, but need three colors for SA, NT, Q, so assignment is inconsistent. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 7

8 Global Constraints Other common global constraints include Resource constraint: Atmost (N, X 1, X k ), inconsistency if sum of the minimum domain values is greater than N. Bounds constraint: D i = [ lb i, ub i ], D j = [ lb j, ub j ], handled by bounds propagation. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 8

9 Sudoku Example 81 variables: A1,, A9,, I1,, I9, one for each square All empty square domains: { 1, 2, 3, 4, 5, 6, 7, 8, 9 } Pre-filled square domains are singleton value Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 9

10 Sudoku Example 27 different Alldiff constraints, one for each row, column, and box Alldiff (A1, A2, A3, A4, A5, A6, A7, A8, A9) Alldiff (B1, B2, B3, B4, B5, B6, B7, B8, B9) Alldiff (A1, B1, C1, D1, E1, F1, G1, H1, I1) Alldiff (A2, B2, C2, D2, E2, F2, G2, H2, I2) Alldiff (A1, A2, A3, B1, B2, B3, C1, C2, C3) Alldiff (A4, A5, A6, B4, B5, B6, C4, C5, C6) Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 10

11 Sudoku Example Assume Alldiffs have been expanded into binary constraints to apply AC-3. Consider E6. Column constraints remove 2, 3, 5, 6, 8, 9 from its domain. Box constraints additionally remove 1, 7. E6 domain is { 4 }. Now consider I6. Column constraints remove 2, 3, 4 (from E6), 5, 6, 8. Row constraint removes 1, leaving { 7 }. With 8 values in column 6, arc consistency can infer that A6 must be { 1 }. Monday, February 20 CS 430 Artificial Intelligence - Lecture

12 Sudoku Example Turns out that this particular puzzle can be solved using only arc consistency inference, thus it is an easy puzzle. Slightly harder puzzles can be solved using path consistency, but there are >250,000 different path constraints. Solving hardest puzzles requires something more than inference over constraints. However the hardest puzzles can be solved using a general CSP solver in less than 0.1 sec. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 12

13 KenKen Puzzles Row and column constraints like Sudoku Cage constraint: numbers in the cage must create an expression using the cage operator that results in the cage number. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 13

14 Backtracking Search for CSPs For non-trivial CSPs, inference alone is not enough to solve the problem. Eventually, need to search for a solution. Could apply standard DFS. States are partial assignments. Action would be to add var = value to the assignment. Unfortunately, for CSP with n variables of domain size d, branching factor at top level is nd, since any of d values can be assigned to any of n variables. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 14

15 Backtracking Search for CSPs Naive formulation ignores commutativity property of CSPs. The order in which the variables are assigned does not matter. Only need to consider single variable assignment at each node in search tree. E.g., in Australia map problem root might start with SA and consider assignments SA = red, SA = green, SA = blue, but not any other assignments. Results in expected O(d n ) leaves. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 15

16 Backtracking Search for CSPs A DFS that chooses values for one variable at a time and backtracks when a variable has no legal values left to assign is called backtracking search. Function: Backtracking-Search Receives: csp constraint satisfaction problem Returns: solution (set of assignments) or failure 1. Return Backtrack ({ }, csp) Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 16

17 Backtrack Algorithm Receives: assignment set of assigned variables; csp constraint satisfaction problem Returns: solution or failure 1. If assignment is complete then return assignment 2. var = Select-Unassigned-Variable (csp) 3. For each value in Order-Domain-Values (var, assignment, csp) 3.1 If value is consistent with assignment Add {var = value} to assignment inferences = Inference(csp, var, value) If inferences failure then Add inferences to assignment result = Backtrack (assignment, csp) If result failure then return result 3.2 Remove {var = value} from assignment 4. Return failure Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 17

18 Backtracking Search for CSPs Part of the search tree for Australia map problem. Note that because CSP representation is standardized, do not need to supply domain-specific initial state, action function, transition model, or goal test. Also keeps and modifies only one state. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 18

19 Backtracking Search for CSPs To improve performance, do not need any domain-specific knowledge. Instead, address following questions: Which variable should be assigned next (Select- Unassigned-Variable) and what order should its values be tried (Order-Domain-Values)? What inferences should be performed at each step in the search (Inference) When the search arrives at an assignment that violates a constraint, can the search avoid repeating this failure? (See Section 6.3.3) Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 19

20 Variable and Value Ordering Simplest strategy for Select-Unassigned-Variable is to choose in some static order. However, seldom results in most efficient search. E.g. in example, after WA and NT are assigned, there is only one value for SA, so a better choice for assignment than Q. And after assigning SA, forces choices for Q, NSW, and V. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 20

21 Minimum-Remaining-Values Minimum-remaining-values (MRV) heuristic choses the variable with the fewest "legal" values. Also called "most constrained variable" or "fail-first" heuristic. In particular, if some variable X has no legal values left, MRV will choose X and the search will fail immediately rather than later, avoiding pointless searches through other variables. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 21

22 Degree Heuristic MRV heuristic does not help in choosing the first variable assignment for the Australia map problem, since all domains have 3 colors. Degree heuristic attempts to reduce the branching factor on future choices by selecting the variable that is involved in the largest number of constraints on other unassigned variables. E.g., SA Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 22

23 Least-Constraining-Value Once a variable is selected, must decide on the order in which to examine its values. Least-constraining-value heuristic prefers the value that rules out the fewest choices for the neighboring variables. In general, this leaves the maximum flexibility for later variable assignments. Note value ordering does not matter if we are trying to produce all possible solutions. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 23

24 Interleaving Search and Inference AC-3 and other algorithms can be used to infer reductions in the domain of variables before beginning search. After every variable assignment, there is a new opportunity to infer domain reductions. Simplest form is called forward checking. Whenever a variable X is assigned, establish arc consistency for it. I.e., for each unassigned Y connected to it, remove from Y's domain any values inconsistent with the value chosen for X. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 24

25 Interleaving Search and Inference Table shows the progress of backtracking search with forward checking on Australia map problem. After WA and Q are assigned, NT and SA have a single value After V = blue, domain of SA is empty, thus this partial assignment is inconsistent and will backtrack Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 25

26 Interleaving Search and Inference Forward checking does not detect all consistencies. E.g., after Q is assigned, NT and SA are forced to be blue, but this is inconsistent. The MAC (Maintaining Arc Consistency) algorithm runs AC-3 with an initial queue of arcs from newly assigned variable X i to all of its neighbors. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 26

27 Local Search for CSPs Local search algorithms are effective in solving many CSPs, using a complete state formulation. Initial state assigns a value to every variable, and the search changes the value of one variable at a time. Typically, the initial state violates several constraints, and the point of the local search is to eliminate the violated constraints. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 27

28 Min-Conflicts Min-conflicts heuristic is used to chose the new value for a variable. E.g., in 8-queens, number of conflicts is the number of attacking queens. At each step, move queen to square with minimum conflicts. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 28

29 Min-Conflicts Min-conflicts is very effective for many CSPs. For n-queens, if the initial placement of queens is not counted, runtime is independent of problem size. Solves million-queens problems in an average of 50 steps (after initial assignment). Applied to Hubble Space Telescope scheduling, reduced time to schedule a week of observations from 3 weeks (!) to around 10 minutes. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 29

30 Structure of Problems In order to solve large complex problems, need to be able to decompose into subproblems. Completely independent subproblems are nice but rare. A constraint graph that is a tree (any two variables are connected by only one path) can be solved in time linear to the number of variables. Details can be found in Section 6.5 Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 30

31 Structure of Problems Cutset conditioning takes advantage of the tree algorithm by assigning some variables so that the remaining unassigned variables form a tree. Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 31

32 Structure of Problems Alternately, can construct a tree decomposition of constraint graph into a set of connected subproblems. Solve each subproblem independently, then combine solutions, treating each subproblem as a "mega-variable". Monday, February 20 CS 430 Artificial Intelligence - Lecture 18 32