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Constraint Satisfaction Patrick Prosser

Constraint Satisfaction Patrick Prosser. An Example, Exam Timetabling. Someone timetables the exams We have a number of courses to examine how many? Dept has 36 Faculty? University? There are constraints if a student S takes courses Cx and Cy Cx and Cy cannot be at same time!

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Constraint Satisfaction Patrick Prosser

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  1. Constraint Satisfaction Patrick Prosser

  2. An Example, Exam Timetabling • Someone timetables the exams • We have a number of courses to examine • how many? • Dept has 36 • Faculty? • University? • There are constraints • if a student S takes courses Cx and Cy • Cx and Cy cannot be at same time! • If Cy and Cz have no students in common • they can go in room R1 if there is space • Temporal and resource constraints

  3. An Example, Exam Timetabling • Represent as graph colouring • vertices are courses • colours are time • vertices have weight (room requirements) • edge connects vertices of diff colour • How complex is this • if we have n vertices and k times • an n-digit number to the base k? • How would you solve this • backtracking search? • Greedy? • Something else • GA? • SA, TS, GLS, HC, ...

  4. An Example, Exam Timetabling • How does the person solve this? • Is that person intelligent? • Is there always a solution? • If there isn’t, do we want to know why? • Do you think they can work out “why”?

  5. A CSP • A csp has • n variables • each has a domain of (m) values • constraints define compatible tuples of values • n-ary, binary • find an assignment • of values to variables • that satisfies the constraints • or show none exists • O(mn)

  6. An example 1 2 3 4 5 6 7 4 Make a crossword puzzle! Given the above grid and a dictionary, fill it. Then go get the clues (not my problem)

  7. An example 1 2 3 4 5 6 7 1A 4D 4 4A 2D 1A 1 across 4D 4 down 2D 2 down 4A 4 across 7D 7 down 7D Variables

  8. An example 1 2 3 4 5 6 7 1A 4D 4 4A 2D 1A-4D: 4th of 1A equals 1st of 4D 1A-2D: 2nd of 1A equals 1st of 2D 2D-4A: 4th of 2D equals 2nd of4D 4D-4A: 4th of 4A equals 4th of 4D 4A-7D: 7th of 4A equals 2nd of 7D 7D Constraints

  9. An example 1 2 3 4 5 6 7 1A 4D 4 4A 2D 1A: any 6 letter word 4A: any 8 letter word 4D: any 5 letter word 2D: any 7 letter word 7D: any 3 letter word 7D Domains (also unary constraints!)

  10. An example 1 2 3 4 5 6 7 1A 4D 4 4A 2D Find an assignment of values to variables, from their domains, such that the constraints are satisfied (or show that no assignment exists) 7D A CSP!

  11. An example 1 2 3 4 5 6 7 1A 4D 4 4A 2D Choose a variable Assign it a value Check compatibility If not compatible try a new value If no values remain re-assign previous variable 7D Good old fashioned BT!

  12. Questions? 1 2 3 4 5 6 7 1A 4D 4 4A 2D What variable should I choose? 7D What value should I choose? What reasoning can I do when making an assignment? What reasoning can I do on a dead end? Decisions, decisions!

  13. Where’s the AI?

  14. Scene Labelling David Waltz, MIT, 1975

  15. In a trihedral world, these are the only scenarios

  16. Walk along the edge in this direction and the object is on the right, and to your left is open space + - This is an outside edge, with the object on both sides This is an inside edge Label an edge as follows

  17. We now have the following cases - - + + - + - - + - + - - + - - - + + + - +

  18. + + + - - + + + + - - - + The edges are the variables, labels their domains, meeting points are the constraints A consistent labelling is an interpretation

  19. Another example: n-queens • Place n non-attacking queens on an n x n chess board • representations? That was BT There is a polynomial solution circa 1800

  20. It’s all just depth first search, right?

  21. BT Thrashes! past variable v[h] past conflict with v[h] current variable v[i] future future variable v[j]

  22. Another example: n-queens • Forward Checking

  23. Forward Checking 1 2 3 NOTE: arrows go forward! 4 5 6 7 8 9

  24. How to improve search • use a heuristic • variable and/or value ordering • dynamic or static • Fail First? • More inferencing at each search state • old trade off, knowledge versus search • maintain consistency

  25. Consistency • arc consistency • what’s that then? • Propagate supports • deduce illegal values • polynomial at each search node • AC can be specialised for special constraints • MAC • the heart of constraint programming

  26. Give us a demo?

  27. Not just binary csp’s! • N-ary • but can be mapped to binary • why bother with n-ary? • allDiff, sum, permutation, marriage • Not just arc consistency • path • inverse • restricted • singleton • When, what?

  28. Applications • scheduling • timetabeling • frequency allocation • transportation • design • layouts • packing • ...

  29. Toolkits • ILOG • solver, scheduler, dispatcher • chip • choco • OZ • Eclipse • Jsolver • Screamer • CSPLab

  30. Research Direction • reactivity and explanation • retraction in particular • better heuristics • new search algorithms • complete, quasi-complete, local • new levels of consistency • new specialised constraints • effects of representation • understanding the problem • its structure and why it is hard

  31. stop

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