% I assume that the locations on the board are numbered 
% (1,1), (1,2), ...., (9,9) 
% the topleft corner is (1,1) and the rightbottom corner 
% is (9,9) 
% row is horizontal 
% column is vertical 

% cell(i,j,k) means that the number k is placed at the 
% location (i,j) 
% to run the program use the command 
%    lparse sudoku.lp instance | smodels 
% where instance is the name of the file describing the 
% problem (the website has three examples, download them 
% and save them into exp1, exp2, exp3 for example; then 
%    lparse sudoku.lp exp1 | smodels solves the first instance) 

% define that 1,..,9 are number 
% (i.e., number(1),...,number(9) are facts of the program
% 

number(1..9).

% this is to define that I,... are variables of the type 
% number 

#domain number(I;J;K1;K2;N).

% defining the small squares 
% e.g. the locations (I,J) for in1(I) and in1(I) are true 
%      represents the first square 

in1(N) :- N >= 1, N <= 3.
in2(N) :- N >= 4, N <= 6.
in3(N) :- N >= 7, N <= 9.

% for each number k (k=1,...,9) 
% there are exactly 9 atoms of the from 
% cell(I,J,k) where I is a number (between 1 and 9) 
% J is a number (between 1 and 9) 
% in the stable model (if one exists!)  

9 {cell(I1,J1,K) : number(I1): number(J1)} 9 :- number(K). 

% for each K, the above rule is equvalent to the following rule:  
% 9 {cell(1,1,1), cell(1,2,1),...,cell(9,9,1)} 9 :- number(1).    
% between the brackets are all the 81 locations of the board.

% one cell cannot have two numbers 
% neq(K1,K2) means that K1 is different than K2 

:- cell(I, J, K1), cell(I, J, K2), neq(K1, K2). 

% for each small square, each number (K) can 
% appear at least one and at most one time 

1 {cell(I1,J1,K) : in1(I1) : in1(J1)} 1 :- number(K). 

% for example, for K=1, the above rule is equivalent to 
% the rule 
% 1 {cell(1,1,1),cell(1,2,1),cell(1,3,1),
%    cell(2,1,1),cell(2,2,1),cell(2,3,1),
%    cell(3,1,1),cell(3,2,1),cell(3,3,1)} 1 :- number(1).

1 {cell(I1,J1,K) : in2(I1) : in2(J1)} 1 :- number(K).
1 {cell(I1,J1,K) : in3(I1) : in3(J1)} 1 :- number(K).
1 {cell(I1,J1,K) : in1(I1) : in2(J1)} 1 :- number(K).
1 {cell(I1,J1,K) : in1(I1) : in3(J1)} 1 :- number(K).
1 {cell(I1,J1,K) : in2(I1) : in3(J1)} 1 :- number(K).
1 {cell(I1,J1,K) : in2(I1) : in1(J1)} 1 :- number(K).
1 {cell(I1,J1,K) : in3(I1) : in2(J1)} 1 :- number(K).
1 {cell(I1,J1,K) : in3(I1) : in1(J1)} 1 :- number(K).

% each number k, appears one and only one on each row 

1 {cell(1,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(2,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(3,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(4,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(5,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(6,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(7,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(8,J1,K)  : number(J1)} 1 :- number(K).
1 {cell(9,J1,K)  : number(J1)} 1 :- number(K).

% each number k, appears one and only one on each column  

1 {cell(J1,1,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,2,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,3,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,4,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,5,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,6,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,7,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,8,K)  : number(J1)} 1 :- number(K).
1 {cell(J1,9,K)  : number(J1)} 1 :- number(K).