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CS461: Artificial Intelligence. Lecture 6: Constraint Satisfaction Problems. Outline. Constraint Satisfaction Problems (CSP) Backtracking search for CSPs Local search for CSPs. Constraint satisfaction problems (CSPs). Standard search problem

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Lecture 6: Constraint Satisfaction Problems


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    1. CS461: Artificial Intelligence Computer Science Department Lecture 6: Constraint Satisfaction Problems

    2. Outline • Constraint Satisfaction Problems (CSP) • Backtracking search for CSPs • Local search for CSPs Computer Science Department

    3. Constraint satisfaction problems (CSPs) • Standard search problem • state is a "black box“ – any data structure that supports successor function, heuristic function, and goal test • CSP: • state is defined by variablesXi with values from domainDi • goal test is a set of constraints specifying allowable combinations of values for subsets of variables • Simple example of a formal representation language • Allows useful general-purpose algorithms with more power than standard search algorithms Computer Science Department

    4. Example: Map-Coloring • VariablesWA, NT, Q, NSW, V, SA, T • DomainsDi = {red,green,blue} • Constraints: adjacent regions must have different colors • e.g., WA ≠ NT, or (WA,NT) in {(red,green),(red,blue),(green,red), (green,blue),(blue,red),(blue,green)} Computer Science Department

    5. Example: Map-Coloring • Solutions are complete and consistent assignments, e.g., WA = red, NT = green,Q = red,NSW = green,V = red,SA = blue,T = green Computer Science Department

    6. Constraint graph • Binary CSP: each constraint relates two variables • Constraint graph: nodes are variables, arcs are constraints Computer Science Department

    7. Varieties of CSPs • Discrete variables • finite domains: • n variables, domain size d  O(dn) complete assignments • e.g., Boolean CSPs, incl.~Boolean satisfiability (NP-complete) • infinite domains: • integers, strings, etc. • e.g., job scheduling, variables are start/end days for each job • need a constraint language, e.g., StartJob1 + 5 ≤ StartJob3 • Continuous variables • e.g., start/end times for Hubble Space Telescope observations • linear constraints solvable in polynomial time by linear programming Computer Science Department

    8. Varieties of constraints • Unary constraints involve a single variable, • e.g., SA ≠ green • Binary constraints involve pairs of variables, • e.g., SA ≠ WA • Higher-order constraints involve 3 or more variables, • e.g., cryptarithmetic column constraints Computer Science Department

    9. Real-world CSPs • Assignment problems • e.g., who teaches what class • Timetabling problems • e.g., which class is offered when and where? • Transportation scheduling • Factory scheduling • Notice that many real-world problems involve real-valued variables Computer Science Department

    10. Standard search formulation (incremental) Let's start with the straightforward approach, then fix it States are defined by the values assigned so far • Initial state: the empty assignment { } • Successor function: assign a value to an unassigned variable that does not conflict with current assignment  fail if no legal assignments • Goal test: the current assignment is complete • This is the same for all CSPs • Every solution appears at depth n with n variables use depth-first search • Path is irrelevant, so can also use complete-state formulation • b = (n - l )d at depth l, hence n! · dn leaves Computer Science Department

    11. NxD WA WA WA NT T [NxD]x[(N-1)xD] WA NT WA NT WA NT NT WA Standard search formulation N layers Equal! N! x DN There are N! x DN nodes in the tree but only DN distinct states?? Computer Science Department

    12. Backtracking search • Variable assignments are commutative}, i.e., [ WA = red then NT = green ] same as [ NT = green then WA = red ] • Only need to consider assignments to a single variable at each node • Depth-first search for CSPs with single-variable assignments is called backtracking search • Backtracking search is the basic uninformed algorithm for CSPs • Can solve n-queens for n ≈ 25 Computer Science Department

    13. D WA WA WA WA NT D2 WA NT WA NT DN Backtracking search Computer Science Department

    14. function BACKTRACKING-SEARCH (csp) returns a solution, or failure return RECURSIVE-BACKTRACKING({}, csp) function RECURSIVE-BACKTRACKING(assignment, csp) returns a solution, or failure ifassignment is complete then returnassignment var SELECT-UNASSIGNED-VARIABLE(VARIABLES[csp], assignment, csp) for eachvalue in ORDER-DOMAIN-VALUES(var, assignment, csp) do ifvalue is consistent with assignment according to CONSTRAINTS[csp] then add {var=value} to assignment result RECURSIVE-BACKTRACKING(assignment, csp) ifresult != failurethen returnresult remove {var = value} from assignment return failure ☜ BACKTRACKING OCCURS HERE!! Backtracking search Computer Science Department

    15. Backtracking example Computer Science Department

    16. Backtracking example Computer Science Department

    17. Backtracking example Computer Science Department

    18. Backtracking example Computer Science Department

    19. Improving backtracking efficiency • General-purpose methods can give huge gains in speed: • Which variable should be assigned next? • In what order should its values be tried? • Can we detect inevitable failure early? Computer Science Department

    20. Most constrained variable • Most constrained variable: choose the variable with the fewest legal values • a.k.a. minimum remaining values (MRV) heuristic Computer Science Department

    21. BackpropagationMRVminimum remaining values • choose the variable with the fewest legal values Computer Science Department

    22. Backpropagation - MRV [R,B,G] [R,B,G] [R] [R,B,G] [R,B,G] Computer Science Department

    23. Backpropagation - MRV [R,B,G] [R,B,G] [R] [R,B,G] [R,B,G] Computer Science Department

    24. Backpropagation - MRV [R,B,G] [R,B,G] [R] [R,B,G] [R,B,G] Computer Science Department

    25. Backpropagation - MRV [R,B,G] [R,B,G] [R] [R,B,G] [R,B,G] Computer Science Department

    26. Backpropagation - MRV [R,B,G] [R,B,G] [R] [R,B,G] [R,B,G] Computer Science Department

    27. Backpropagation - MRV [R,B,G] [R,B,G] [R] [R,B,G] [R,B,G] Solution !!! Computer Science Department

    28. Most constraining variable • Tie-breaker among most constrained variables • Most constraining variable: • choose the variable with the most constraints on remaining variables (most edges in graph) Computer Science Department

    29. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    30. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    31. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    32. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    33. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs Dead End 3 arcs [R,B,G] [R,B,G] Computer Science Department

    34. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    35. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    36. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    37. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs Dead End 3 arcs [R,B,G] [R,B,G] Computer Science Department

    38. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    39. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    40. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    41. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    42. Backpropagation - MCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Solution !!! Computer Science Department

    43. Least constraining value • Given a variable, choose the least constraining value: • the one that rules out the fewest values in the remaining variables • Combining these heuristics makes 1000 queens feasible Computer Science Department

    44. Back-propagation LCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    45. Back-propagation LCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    46. Back-propagation LCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    47. Back-propagation LCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    48. Back-propagation LCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department

    49. Back-propagation LCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Dead End Computer Science Department

    50. Back-propagation LCV [R,B,G] [R,B,G] 4 arcs 2 arcs 2 arcs [R] 3 arcs 3 arcs [R,B,G] [R,B,G] Computer Science Department