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Constraint Satisfaction Problems

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  1. Constraint Satisfaction Problems Introduction to AI

  2. Map Colouring • Given a map of “countries”: • Assign a colour to each country such that IF two countries share a border THEN they are given different colours • We can only use a limited number of colours • Alternatively, we must use the minimal possible number of colours

  3. 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)}

  4. 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

  5. Map Colouring: Example • Consider some “square” countries: • How many colours are needed?

  6. Example: 3 colours? Take the 3 colours to be red, green, blue How would you be sure that you have not missed out some possible 3 colouring? Need some complete search method! NO COLOUR LEFT!

  7. Example: 4 colours? WITH JUST ONE MORE COLOUR IT IS POSSIBLE NO COLOUR LEFT!

  8. Motivations • Map Colouring is a specific problem • so why care? • Map Colouring is a typical “Constraint Satisfaction Problem (CSP)” • CSPs have many uses • scheduling • timetabling • window (task pane) manager in a GUI • and many other common optimization problems with industrial applications

  9. Constraint Satisfaction Problems Must be Hot&Sour Soup No Peanuts Chicken Dish Appetizer Total Cost < $30 No Peanuts Pork Dish Vegetable Seafood Rice Not Both Spicy Not Chow Mein Constraint Network

  10. Potential Follow-up “Constraint Programming” • A different programming paradigm • “you tell it what to solve, the system decides how to solve” • a program to solve Sudoku can be only 20 lines of code! • e.g. constraints on each row that all numbers in a row are different is just forall( j in 1..9 ) alldifferent( all(i in 1..9) pos[i,j] ); • Very different from the usual paradigms: • Procedural (Fortran, Pascal, C) • Object-Oriented (C++, Java, C#) • Functional (Lisp, ML, Haskell) • Commercialised by ILOG, www.ilog.com, and others

  11. From Maps to CSPs • Will convert the map colouring into general idea of a CSP: • Assign values • such that the assigned values satisfy some constraints

  12. B A C D E F Map Colouring • Firstly add some labels

  13. B A C D E F Map Colouring: Constraints • Graphs are more common than maps – so convert to a graph • Edge means “share a border”

  14. B A C D E F Graph 3-Colouring • Node, or “variable”, must be given a value from the set of colours { r, g , b } • E.g. B := b • Edge between two nodes  only allowed pairs of values from the set{ (r,g), (r,b), (g, r), (g, b), (b, r), (b, g) }

  15. Search Methods • If want a complete search method then it is standard to use depth-first search with partial assignments • Work with Partial Assignments • Start with the empty assignment • Generate children by adding a value for one more variable • Analogy: path-finding – we started with the empty path, and then added one more segment at each time • We already did this with the map colouring at the start of the lecture!

  16. Backtracking Example

  17. Backtracking Example

  18. Backtracking Example

  19. Backtracking Example

  20. Backtracking Search Animation • Nodes are not created until they are ready to be used – to reduce memory usage A 2nd child of A is not created immediately,just remember that it is there B E D C

  21. A B C Can we 2-colour a Triangle? • Can we assign values from { r , g } with the following variables and constraints? • Obviously not! • But how can we see this using search?

  22. name of node is just the branch variable Can we 2-colour a Triangle? • Can we assign values from { r , g } to nodes A, B, and C • “Generate-and-Test”: Colour nodes in the order, A, B, C A A=g A=r All the attempts to generate a satisfying assignment fail B B=r B=g Etc, etc C C C=r C=r C=g C=g Fail Fail Fail Fail

  23. Early Failure • Suppose a partial assignment fails i.e. violates a constraint • Whatever values we eventually give to the so-far un-assigned variables the constraint will stay violated, and the solution will fail • In the backtracking search: • as soon as a constraint is violated by the current partial assignment then we can prune the search

  24. Can we 2-colour a Triangle? • Naive Backtracking Search: • Backtracking with pruning of bad partial assignments A A=g A=r B B=r B=g C Fail C=r Did not have to try values for C C=g Fail Fail

  25. Varieties of constraints • Unary constraints involve a single variable, • e.g., SA ≠ green • Binary constraints involve pairs of variables, • e.g., value(SA) ≠ value(WA) • Higher-order constraints involve 3 or more variables, • e.g., cryptarithmetic column constraints

  26. Example: Latin Squares Puzzle X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 red RT RS RC RO green GT GS GC GO blue BT BS BC BO yellow YT YS YC YO Variables Values Constraints: In each row, each column, each major diagonal, there must be no two markers of the same color or same shape.

  27. 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

  28. Graph Matching Example Find a subgraph isomorphism from R to S. R 2 1 (1,a) (1,b) (1,c) (1,d) (1,e) 4 3 (2,a) (2,b) (2,c) (2,d) (2,e) X X X S e (3,a) (3,b) (3,c) (3,d) (3,e) (3,a) (3,b) (3,c) (3,d) (3,e) X X X X X X X X X a c (4,a) (4,b) (4,c) (4,d) (4,e) X X X X b d How do we formalize this problem?

  29. Example: 4-Queens • States: 4 queens in 4 columns (44 = 256 states) • 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)