1. Constraint Satisfaction Problems
Chapter 5
Section 1 – 3
Modified by Vali Derhami
CS Constraint Satisfaction

2. Forward checking وارسی پیشرو
Idea:
Keep track of remaining legal values for unassigned variables
هنگامیکه به یک متغیر انتسابی می شود تمام مقادیری از متغیرهای انتساب نیافته دیگر که با این مقدار دهی ناسازگارند را از دامنه آن متغیرها حذف میکند
Terminate search when any variable has no legal values
CS Constraint Satisfaction

3. CS 3243 - Constraint Satisfaction
Forward checking
Idea:
Keep track of remaining legal values for unassigned variables
Terminate search when any variable has no legal values
CS Constraint Satisfaction

4. CS 3243 - Constraint Satisfaction
Forward checking
Idea:
Keep track of remaining legal values for unassigned variables
Terminate search when any variable has no legal values
CS Constraint Satisfaction

5. CS 3243 - Constraint Satisfaction
Forward checking
Idea:
Keep track of remaining legal values for unassigned variables
Terminate search when any variable has no legal values
CS Constraint Satisfaction

6. Constraint propagation
Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures:
NT and SA cannot both be blue!
Constraint propagation repeatedly enforces constraints locally
CS Constraint Satisfaction

7. Arc consistency سازگاری کمان
Simplest form of propagation makes each arc consistent
for every value dx of X there is some allowed dy
برای هر مقدار dx از متغیر X یک مقدار مجاز برای متغیر Y باشد
CS Constraint Satisfaction

8. CS 3243 - Constraint Satisfaction
Arc consistency
Simplest form of propagation makes each arc consistent
for every value dx of X there is some allowed dy
CS Constraint Satisfaction

9. CS 3243 - Constraint Satisfaction
Arc consistency
Simplest form of propagation makes each arc consistent
X Y is consistent iff
for every value dx of X there is some allowed dy
If X loses a value, neighbors of X need to be rechecked
CS Constraint Satisfaction

10. CS 3243 - Constraint Satisfaction
Arc consistency
Simplest form of propagation makes each arc consistent
X Y is consistent iff for every value dx of X there is some allowed dy
If X loses a value, neighbors of X need to be rechecked
Arc consistency detects failure earlier than forward checking Can be run as a preprocessor or after each assignment
4 Feb 2004
CS Constraint Satisfaction

11. Arc consistency algorithm AC-3
Time complexity: O(n2d3)
4 Feb 2004
CS Constraint Satisfaction

12. CS 3243 - Constraint Satisfaction
Local search for CSPs
Hill-climbing, simulated annealing typically work with "complete" states, i.e., all variables assigned
To apply to CSPs:
allow states with unsatisfied constraints
operators reassign variable values
Variable selection: randomly select any conflicted variable
Value selection by min-conflicts heuristic:
choose value that violates the fewest constraints
i.e., hill-climb with h(n) = total number of violated constraints
Violate: نقض کردن
4 Feb 2004
CS Constraint Satisfaction

13. CS 3243 - Constraint Satisfaction
Example: 4-Queens
Actions: move queen in column
Goal test: no attacks
Evaluation: h(n) = number of attacks
Given random initial state, can solve n-queens in almost constant time for arbitrary n with high probability (e.g., n = 10,000,000)
CS Constraint Satisfaction

14. CS 3243 - Constraint Satisfaction
Summary
CSPs are a special kind of problem:
states defined by values of a fixed set of variables
goal test defined by constraints on variable values
Backtracking = depth-first search with one variable assigned per node
Variable ordering and value selection heuristics help significantly
Forward checking prevents assignments that guarantee later failure
Constraint propagation (e.g., arc consistency) does additional work to constrain values and detect inconsistencies
Iterative min-conflicts is usually effective in practice
4 Feb 2004
CS Constraint Satisfaction