1. An Introduction to Constraint Programming in JChoco

6. Constraint satisfaction Constraint satisfaction problem (CSP) is a triple where: –V is set of variables –Each X in V has set of values, D_X Usually assume finite domain {true,false}, {red,blue,green}, [0,10], … –C is set of constraints Goal: find assignment of values to variables to satisfy all the constraints

7. Di = {1,2,3,4,5} V1 V3 V2 V4 V4  V3 V1  V4  1 V4 + V2 = 5 V1  V2 V2  V3  6 AR33 figure 18, page 35 What can you infer?

8. Di = {1,2,3,4,5} V1 V3 V2 V4 V4  V3 V1  V4  1 V4 + V2 = 5 V1  V2 V2  V3  6 D1 = {1,2} D2 = {2,3} D3 = {4,5} D4 = {2,3} Do you agree? Was that easy?

9. A program that models csp6

10. With print statements and search

11. Packages used

12. Can throw an exception

13. Create a problem

16. Create variables

19. Post constraints

20. Establish AC

21. Magic code to get all solutions

22. Code to count solutions

23. output/trace Problem with no V or C

24. variables

25. Variables and constraints

26. Made AC

27. All solutions

28. 2 colour K 3 The difference allDifferent makes Just how weak/powerful is AC processing?

29. Prop0.java Run it Make it AC

30. allDifferent Prop1.java Run it

31. X + Y + Z = 20 X,Y,Z  {1..10} See Test.java How many solutions? AF2 assessed exercise One of 5 questions

32. Test.java Count number of solutions

35. As a decision problem: is there a glomb ruler with n ticks with length no more than m units?

37. How many n digit numbers are there where the number must contain at least 0ne number 3 and at least one number 5? n = 6 330095 501633 333335 120583 555355 … See ThreesAndFives.java AF2 assessed exercise

39. Magic square An example of how to represent a problem An idea from Chris Beck

40. put a number in each square each number is different a number is in the range 1 to 16 the sum of a column is the same as a sum of a row the same as the sum of a main diagonal

41. put a number in each square each number is different a number is in the range 1 to 16 the sum of a column is the same as a sum of a row the same as the sum of a main diagonal 1st stab Use sum(x) = sum(y) where x and y are - different rows - different columns - different diagonals Every element of the array is different - represent as a clique of not equals How does it go? For propagation and search?

42. put a number in each square each number is different a number is in the range 1 to 16 the sum of a column is the same as a sum of a row the same as the sum of a main diagonal 2nd stab But what is k? How does model perform?

44. Magic square on the web http://mathforum.org/alejandre/magic.square.html http://mathworld.wolfram.com/MagicSquare.html

53. Thanks to Chris Beck

54. Thanks to Bouygues