%
% Quasigroup problem in MiniZinc.
%
% This model is a translation of the EssencePrime model quasiGroup3Idempotent.eprime
% from the Minion Translator examples.
% """
% The quasiGroup existence problem (CSP lib problem 3)
%
% The quasiGroup existence problem (CSP lib problem 3)
%
% An m order quasigroup  is an mxm multiplication table of integers 1..m,
% where each element occurrs exactly once in each row and column and certain
% multiplication axioms hold (in this case, we want axiom 3 to hold).
% """
%
% http://www.dcs.st-and.ac.uk/~ianm/CSPLib/prob/prob003/spec.html:
% """
% QG3.m problems are order m quasigroups for which (a*b)*(b*a) = a.
% """
%

%
% Model created by Hakan Kjellerstrand, hakank@gmail.com

% Licenced under CC-BY-4.0 : http://creativecommons.org/licenses/by/4.0/

include "globals.mzn";

int: n;
set of int: nDomain = 0..n-1;

array[nDomain, nDomain] of var nDomain: quasiGroup;
array[nDomain] of var nDomain: qgDiagonal;

% solve satisfy;
solve :: int_search([quasiGroup[row, col] | row, col in nDomain], first_fail, indomain_min, complete) satisfy;
% solve :: int_search(qgDiagonal, first_fail, indomain_min, complete) satisfy;

constraint
% accessor for diagonal
forall(i in nDomain) (
qgDiagonal[i] = quasiGroup[i,i]
)
/\
% All rows have to be different
forall(row in nDomain) (
all_different([quasiGroup[row,col] | col in nDomain])
)
/\
% All columns have to be different
forall(col in nDomain) (
all_different([quasiGroup[row,col] | row in nDomain])
)
/\
%  (j*i)*(i*j) = i
forall(i in nDomain) (
forall(j in nDomain) (
quasiGroup[quasiGroup[i,j],quasiGroup[j,i]] = i
)
)

% Idempotency
% forall i : nDomain .
%     (quasiGroup[i,i] = i),

% Implied (from Colton,Miguel 01)
% All-diff diagonal
% allDifferent(qgDiagonal) %,

% anti-Abelian
% forall i : nDomain .
%   forall j : nDomain .
%     (i != j) =>
%     (quasiGroup[i,j] != quasiGroup[j,i]),

% if (i*i)=j then (j*j) = i
% forall i : nDomain .
%  forall j : nDomain .
%    (quasiGroup[i,i]=j) => (quasiGroup[j,j]=i),

% Symmetry-breaking constraints
% forall i : nDomain .
%      quasiGroup[i,n-1] + 2 >= i

;

output
[ "\nqgDiagonal: ", show(qgDiagonal)
] ++
[ "\nquasiGroup: "
] ++
[
if col = 0 then "\n" else " " endif ++
show(quasiGroup[row, col])
| row, col in nDomain
] ++ ["\n"];

%
% data
%
n = 4; % 4 works