BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCESecond Semester 2007-2008EA C461 Artificial Intelligence

Lab 3: Solving N-Queens Problem in PROLOG

Document Prepared By:

Mukesh Kumar Rohil, CS & IS Group, BITS, Pilani – 333031 (Rajasthan), India rohil@bits-pilani.ac.in

To solve N-Queens problem let us first discuss solution to eight queens’ problem.

1. The eight queen’s problem: The problem here is to place eight queens on the empty chessboard in such a way that no queen attacks any other queen. A queen can perform horizontal, vertical or diagonal attack. 1.1 Solution 1: The board position of the queens is represented by their Y-coordinates (i.e.

[Y1, Y2, Y3, …, Y8] since to prevent the horizontal attack no two queens cann be in the same row. Each solution is represented as permutation of the list [1,2,3,4,5,6,7,8]. The program is listed below:

Sol

uti

on 1

% solution(Queens) if Queens is a list of Y-coordinates of eight non-attacking queens

solution(Queens) :- permutaion([1,2,3,4,5,6,7,8], Queens), safe(Queens).

permutation([], []).permutation([Head|Tail], PermList) :- permutation(Tail, PermTail), del(Head, PermList, PermTail). % Here insert Head in permuted Tail

%del(Item, List, NewList): deleting Item from List gives NewListdel(Item, [Item|List], List).del(Item, [First|List], [First|List1]) :- del(Item, List, List1).

% safe(Queens) if Queens is a list of Y-coordinates of non-attacking queenssafe([]).safe([Queen|Others]) :- safe(Others), noattack(Queen, Others,1).

noattack(_, [], _).noattack(Y, [Y1|Ylist], Xlist) :- Y1 – Y =\= Xdist, Y – Y1 =\= Xdist, Dist1 is Xdist + 1,

noattack(Y, Ylist, Dist1).

1.2 Solution 2: Each queen has to be placed on some square; that is; into some column, some row, some upward diagonal, and some downward diagonal. So we will consider a richer representation with four coordinates: x: columns, y:rows, u: upward diagonal, and v: downward diagonal. The coordinates are not independent: given x and y, u and v can be computed as u = x –y, v = x + y. The domains for all four dimensions are: Dx = [1, 2, 3, 4, 5, 6, 7, 8], Dy = [1, 2, 3, 4, 5, 6, 7, 8], Du = [-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7], and Dv = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The program is listed below:

Sol

uti

on 2

% solution(Queens) if Queens is a list of Y-coordinates of eight non-attacking queens

solution(Ylist) :- sol(Ylist, [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8], [-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] ).

sol([], [], Dy, Du, Dv).sol([Y|Ylist], [X|Dx1], Dy, Du, Dv) :- del(Y, Dy, Dy1), U is X – Y, del(U, Du, Du1), V is X + Y, del(V, Dv, Dv1), sol(Ylist, Dx1, Dy1, Du1, Dv1).

%del(Item, List, NewList): deleting Item from List gives NewListdel(Item, [Item|List], List).del(Item, [First|List], [First|List1]) :- del(Item, List, List1).

2. The N queen’s problem: The problem here is to place N queens on the empty N x N size chessboard in such a way that no queen attacks any other queen.

2.1 Solution 1: Use the solution 1 given for eight queens problem to solve N queens problem.2.2 Solution 2: Use the solution 2 given for eight queens problem to solve N queens problem.(Hint: For both you need a procedure gen(N1, N2, List) which will, for two given intergers N1 and N2, produces the list: List = [N1, N1 + 1, N1 + 2, …., N2 -1, N2]. For solution 2 you need to define solution(N,S) where N is the size of board, and S is a solution represented as a list of Y-coordinates of N queens)

3. Compare the time taken by each solution when you take 8, 16, 32 and 64 queens and complete the following table.

Queens Time taken bySolution 1 Solution 2

8163264

4. Name your files as L3_YourIDNo_N_S.pl, where YourIDNo is your 11 character BITS id.no. N is number of queens 8 or N, and S is solutions number either 1 or 2. The four files to zipped as L3_YourIDNo.zip and should emailed at rohil@bits-pilani.ac.in before 07:00 PM today.