Minzinc Constraint Question - Sudoku similar

I'm new to Minizinc, and I want to solve the following problem.

I have a given matrix of 6x6 cells. Rows are numbered from 1..6, columns from A..B. All cells shall be filled with digits from 1..9 by certain rules:

All 9 digits occur excatly 4 times in the matrix.
Direct neighbours in row or column cannot have the same value.
All cells in a row and all cells in a column shall have different values.
There are some exceptions from (3): a) Row 2 has two "6". b) Row 3 has two "3". c) Row 4 has two "5". d) Row 5 has three "7". e) Row 6 has two "8". f) Column B has two "3". g) Column D has two "8". h) Column F has two "9".
The sum of all digits of row 1 is >= 38.
The sum of all digits of column E is = 21.
Row 2 does not contain a "1".
Row 4 does not contain a "4".
Row 5 does not contain a "2".
Row 6 does not contain a "3".
Column C does not contain a "2".
Column B has only even digits, except for the two "3".
Column A has a ascending or descending sequence, e.g. "234567" or "987654".

The most headache for me when constructing the model in Minzinc is the "contradictionary" constraints:

All digits in a single row or column must be different.
But sometimes this rule has exceptions.

How can this realized?

Thanks in advance and best regards. Guenther

Solution

I got curious how to model this problem in a concise way with Clingo ASP as an alternative to minizinc. It’s not Constraint Programming, but is related and quite declarative, so it may give you some ideas:

(actually, minizinc supports the related solver via flatzingo as well.)

% Define rows and columns
row(1..6).
col(a;b;c;d;e;f).

% Define digits
digit(1..9).

% Each cell has exactly one digit
1 { value(R, C, D) : digit(D) } 1 :- row(R), col(C).

% Each digit appears exactly 4 times in the grid
:- digit(D), #count { R, C : value(R, C, D) } != 4.

% No two adjacent cells in a row or column can have the same digit
:- value(R1, C, D), value(R2, C, D), R1 = R2 - 1.
:- value(R, C1, D), value(R, C2, D), adjacent_col(C1, C2).

% Define adjacency in columns
adjacent_col(a, b). adjacent_col(b, c). adjacent_col(c, d).
adjacent_col(d, e). adjacent_col(e, f).
adjacent_col(X, Y) :- adjacent_col(Y, X).

% Row uniqueness with exceptions
:- row(R), digit(D), #count { C : value(R, C, D) } > 1, not allowed_repeat(R, D).

% Column uniqueness with exceptions
:- col(C), digit(D), #count { R : value(R, C, D) } > 1, not allowed_repeat(C, D).

% Define exceptions for uniqueness constraints
allowed_repeat(2, 6).
allowed_repeat(3, 3).
allowed_repeat(4, 5).
allowed_repeat(5, 7).
allowed_repeat(6, 8).
allowed_repeat(b, 3).
allowed_repeat(d, 8).
allowed_repeat(f, 9).

% Enforce the exact count for the exceptions
:- #count { C : value(2, C, 6) } != 2.
:- #count { C : value(3, C, 3) } != 2.
:- #count { C : value(4, C, 5) } != 2.
:- #count { C : value(5, C, 7) } != 3.
:- #count { C : value(6, C, 8) } != 2.
:- #count { R : value(R, b, 3) } != 2.
:- #count { R : value(R, d, 8) } != 2.
:- #count { R : value(R, f, 9) } != 2.

% Sum constraints row 1, col E
:- #sum { D : value(1, C, D), col(C) } < 38.
:- #sum { D : value(R, e, D), row(R) } != 21.

% Forbidden digits in specific rows and columns
:- value(2, C, 1).
:- value(4, C, 4).
:- value(5, C, 2).
:- value(6, C, 3).
:- value(R, c, 2).

% Column B only has even digits except for two "3"
:- value(R, b, D), D != 2, D != 4, D != 6, D != 8, D != 3.

% Column A must be an ascending or descending sequence
% (in conjunction w/ all different)
:- value(R1, a, D1), value(R2, a, D2), R2 = R1 + 1,
   D2 != D1 + 1, D2 != D1 - 1.

#show value/3.

Output: (copy and paste the code to run it online)

clingo version 5.7.2 (6bd7584d)
Reading from stdin
Solving...
Answer: 1
value(6,a,8) value(5,a,7) value(4,a,6) value(3,a,5) value(2,a,4) value(1,a,3) value(3,b,3) value(5,b,3) value(4,e,1) value(6,e,2) value(3,e,3) value(5,e,4) value(1,e,5) value(2,e,6) value(1,c,6) value(1,d,7) value(1,b,8) value(1,f,9) value(6,f,9) value(2,d,8) value(6,d,8) value(5,c,7) value(5,f,7) value(4,c,5) value(4,f,5) value(2,b,6) value(4,b,2) value(6,b,4) value(6,c,1) value(3,c,4) value(2,c,9) value(5,d,1) value(3,d,2) value(4,d,9) value(3,f,1) value(2,f,2)
SATISFIABLE

Models       : 1+
Calls        : 1
Time         : 0.158s (Solving: 0.02s 1st Model: 0.02s Unsat: 0.00s)
CPU Time     : 0.000s

The unique solution as a matrix:

R\C	A	B	C	D	E	F
1	3	8	6	7	5	9
2	4	6	9	8	6	2
3	5	3	4	2	3	1
4	6	2	5	9	1	5
5	7	3	7	1	4	7
6	8	4	1	8	2	9

Update:

I added the other solution from OP to the model to verify in both directions:

value(1,a,3). value(1,b,6). value(1,c,7). value(1,d,8). value(1,e,5). value(1,f,9).
value(2,a,4). value(2,b,8). value(2,c,6). value(2,d,9). value(2,e,6). value(2,f,2).
value(3,a,5). value(3,b,3). value(3,c,4). value(3,d,1). value(3,e,3). value(3,f,9).
value(4,a,6). value(4,b,2). value(4,c,5). value(4,d,2). value(4,e,1). value(4,f,5).
value(5,a,7). value(5,b,3). value(5,c,1). value(5,d,7). value(5,e,4). value(5,f,7).
value(6,a,8). value(6,b,4). value(6,c,9). value(6,d,8). value(6,e,2). value(6,f,1).

… and this is the output:

clingo version 5.7.2 (6bd7584d)
Reading from stdin
Solving...
UNSATISFIABLE

Models       : 0
Calls        : 1
Time         : 0.099s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.000s

Hint:

Have a look at row 4!