G53CLP Constraint Logic Programming

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1 G53CLP Constraint Logic Programming Dr Rong Qu Modeling CSPs Case Study I

2 Constraint Programming... represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it. Eugene C. Freuder of 26

3 Constraint Programming The computer solves it A number of CP software tools available One can use the tools effectively after some training Necessary settings We ve seen foundations of constraint satisfaction Many software tools are based on this theory and algorithms 3 of 26

4 Constraint Programming The user states the problem Not the way we write/code the problem using languages of tools How we model the problem effectively in formulations In practice Model the problem understood by computers Code in specific languages Own codes for specific problems 4 of 26

5 Solving CSP How can we state the problem? Languages of different tools are different Model is the same 5 of 26

6 Solving CSP To improve the efficiency of CP solving Improve the algorithms Specialised domain information Higher level (complex) constraints (alldifferent) Different models for the problem Add redundant constraints More fundamental way Number of variables Number of constraints 6 of 26

7 Solving CSP software tools specific coding integration analyse coding problem model testing 7 of 26

8 Solving CSP... makes stating combinatorial problems easy in terms of a constraint satisfaction problem, however, it is more complicated to design solvable models. Roman Barták of 26

9 Solving CSP Primary role of constraint programmers Model the problem Translating high level constraints in the problem low level constraints supported by CP solvers 9 of 26

10 These Two Lectures Case study on AI puzzle n-queen Some in-class exercises Interactive Questions Discussions You can see the implementation/coding of these models 10 of 26

11 The n-queen Problem Any solution? How many solutions? 11 of 26

12 The n-queen Problem 12 of 26

13 The n-queen Problem Any solution? How many solutions? Some demos 13 of 26

14 The n-queen Problem Some questions How about n is (much) larger? Is the solution you have the only one? Number of solutions for n-queen* n: n: ,680 14,200 73, ,596 2,279,184 from wikipedia 14 of 26

15 The n-queen Problem Why modeling? Large computational time We have large problems We have problems of different sizes We have different problem (but can be modeled the same, see later) 15 of 26

16 The n-queen Problem model 1 Variables x i,j 1, if there is a queen in row i, column j; 0, otherwise Domain x i,j = {0, 1} 16 of 26

17 The n-queen Problem model 1 Constraints 1. One queen each row n j=1 x i,j = 1, i = 1,, n 2. One queen each column n i=1 x i,j = 1, j = 1,, n 17 of 26

18 The n-queen Problem model 1 Constraints 3. One queen diagonally Explicitly record all compatible values between each pair of variables Or Use function to represent the relationship between variables 18 of 26

19 The n-queen Problem model 2 Variables x 1, x 2,, x n : position of queens on the chessboard Domain {0 n 2-1}: tile index of each queen placed 19 of 26

20 The n-queen Problem model 2 Constraint R = x i / n + 1 C = x i mod n + 1 Given R 1, R 2 and C 1, C 2 of two queens positions 1. One queen each row R 1 R 2 2. One queen each column C 1 C 2 20 of 26

21 The n-queen Problem model 2 Constraint R = x i / n + 1 C = x i mod n + 1 Given R 1, R 2 and C 1, C 2 of two queens positions 3. One queen diagonally R 1 R 2 C 1 C 2 R 1 R 2 C 2 C 1 21 of 26

22 The n-queen Problem model 3 Variables x 1, x 2,, x n : rows of the chessboard Domain {1 n}: column index of each queen placed 22 of 26

23 The n-queen Problem model 3 Constraint 1. One queen each column for all i, j = {1,, n}, x i x j 2. One queen diagonally x i - x j i - j & x i - x j j - i 3. One queen each row? Row constraint is in the formulation 23 of 26

24 The n-queen Problem models The search space contains all possible assignments For the above three models module 3: {x 1,, x n }, {1,, n} module 2: {x 1,, x n }, (0,, (n 2 1)} module 1: {x i,j }, {0, 1} model variables domain search space n=4 n = 8 n = 10 n = 20 3 n n n! (or n n ) 24 4 E4 3.6 E6 2.4 E18 2 n n 2 (n 2 ) n 6.6 E4 2.8 E14 1 E E52 1 n (n2 ) 6.6 E E E E of 26

25 The n-queen Problem models Fewer number of variables with small domains of variables are preferable Smaller search space Problems solved more quickly Fewer number of variables with large domains of variables are preferable model variables domain search space n=4 n = 8 n = 10 n = 20 3 n n n! 24 4 E4 3.6 E6 2.4 E18 2 n n 2 (n 2 ) n 6.6 E4 2.8 E14 1 E E52 1 n (n2 ) 6.6 E E E E of 26

26 Summary Modeling in constraint programming Case study on 8-queen problem Comparison on 3 models Model in constraint programming Few number of variables Small domain 26 of 26

Constraint Satisfaction Problems: Formulation

Constraint Satisfaction Problems: Formulation Slides adapted from: 6.0 Tomas Lozano Perez and AIMA Stuart Russell & Peter Norvig Brian C. Williams 6.0- September 9 th, 00 Reading Assignments: Much of the