final

README.md

@@ -1,38 +1,181 @@
-# Sudoku Solver (Prolog)
+# INTRODUCTION
 
-Run by using the command
+A **Latin square** of order _n_ is an _n × n_ array in which each cell contains a single symbol
 
-    problem(1,Rows), sudoku(Rows), maplist(portray_clause, Rows).
+from a set _S_ with _n_ elements, such that each symbol occurs exactly once in each row
+
+and exactly once in each column.
+
+_Example:_
+
+![](sudoku-screenshots/latin.png)
+a Latin square of order 3
+
+A **Sudoku Latin square** is a Latin square of order 9 on the symbol set {1,..., 9} that
+
+is partitioned into 3 _×_ 3 squares, and each square contains all symbols.
+![](sudoku-screenshots/1.png)
+
+This can also be represented as
+![](sudoku-screenshots/2.png)
+
+This can be further represented as a list of variables:
+![](sudoku-screenshots/3.png)
+
+And we represent the whole sudoku latin square as a list of lists.
+![](sudoku-screenshots/4.png)
+
+So, in total, we have a list of rows.
 
 
-More problems- 
+# CLP(FD) CONSTRAINTS
 
-    puzzle(1, [[_,4,_,9,_,_,_,5,_],
-               [2,_,_,_,_,_,_,4,_],
-               [1,9,_,_,8,_,7,_,_],
-               [5,_,_,_,_,_,1,_,_],
-               [_,_,7,_,6,_,_,_,3],
-               [_,_,_,_,3,_,8,9,_],
-               [_,8,_,3,4,_,_,6,_], 
-               [3,_,_,2,_,8,_,_,_],
-               [_,_,_,_,_,_,_,_,_]]).
+The library clpfd or Constraint Logic Programming over Finite Domains contains a lot
 
-    puzzle(2, [[_,_,9,5,_,_,_,3,7],
-               [1,3,7,9,_,_,_,5,2],
-               [2,_,_,_,_,3,6,9,_],
-               [3,5,2,_,1,_,_,_,6],
-               [_,_,_,4,5,2,3,_,_],
-               [_,8,1,_,3,_,2,_,_],
-               [6,_,3,_,4,_,8,_,9],
-               [5,2,_,_,_,1,_,6,_],
-               [_,_,_,3,_,7,_,_,_]]).
+of built-in predicates that are useful for solving the Sudoku puzzle easily.
 
-    puzzle(3, [[_,5,_,1,_,_,_,_,_],
-               [2,_,_,5,_,_,6,_,_],
-               [1,_,_,_,8,_,2,_,_],
-               [_,8,_,4,3,_,_,_,_],
-               [_,_,_,_,_,_,_,4,_],
-               [_,_,_,_,_,7,9,3,2],
-               [_,4,_,6,7,_,_,_,_],
-               [_,7,_,_,_,_,_,1,9],
-               [9,_,_,_,_,8,_,_,_]]).
+**(ins)/2** states the domains of variables
+
+    in some systems: fd_domain/3, domain/
+
+**all_distinct/**
+
+    or alternatively: all_different/1, fd_all_different.
+
+    describes a list of different integers
+
+_Examples_ :
+![](sudoku-screenshots/s1.png)
+
+Constraint Propagation:
+![](sudoku-screenshots/s2.png)
+
+# CONSISTENCY TECHNIQUES
+
+**all_distinct/1** uses powerful methods from graph theory to prune the search space:
+
+Using the previous example, the following graph states that X can be either 1 or 2, Y
+
+can be either 1 or 2 and Z can be either 1, 2 or 3.
+![](sudoku-screenshots/g1.png)
+
+This is a value graph for a set of constraints.
+
+Prolog automatically reasons about this value graph to detect whether there can still
+
+be a solution and also to find out which assignment can not occur in a solution.
+![](sudoku-screenshots/g2.png) ![](sudoku-screenshots/g3.png)
+
+# PROGRAM
+
+```
+:- use_module(library(clpfd)). % Including clpfd library.
+
+% Defining a sudoku latin square:
+
+sudoku (Rows) :-
+
+    % Rows must be a list of length 9.
+    length (Rows, 9),
+
+    %Each of the rows must also be a list of 9.
+    maplist(same_length(Rows), Rows),
+
+    % The concatenation of all elements of this list is the list Vs.
+    append(Rows, Vs),
+    Vs ins 1..9,
+
+    % We can define the all the elements of each of the rows
+    % must be purely distinct.
+    % We can do the following:
+    % Rows = [Rs1|_], all_distinct(Rs1),
+    % Rows = [_,Rs2|_], all_distinct(Rs2),
+    % ...
+    % But there is a simpler way to do this.
+
+    maplist(all_distinct, Rows),
+    % This states that all_distinct must hold for each of the rows.
+
+    % Doing the same with columns.
+    transpose(Rows, Columns),
+    maplist(all_distinct, Columns),
+
+    % Defining the sub-squares.
+    Rows = [As,Bs,Cs,Ds,Es,Fs,Gs,Hs,Is],
+    square(As, Bs, Cs),
+    square(Ds, Es, Fs),
+    square(Gs, Hs, Is).
+
+square([], [], []). % This predicate uses 3 rows.
+
+square([N1,N2,N3|Ns1],
+       [N4,N5,N6|Ns2],
+       [N7,N8,N9|Ns3]) :-
+        all_distinct([N1,N2,N3,N4,N5,N6,N7,N8,N9]),
+        square(Ns1, Ns2, Ns3).
+
+
+% Defining different problems.
+
+problem(1, [[_,_,_,_,_,4,_,_,2],
+            [_,6,_,2,_,_,_,3,_],
+            [_,8,_,_,_,3,5,_,9],
+            [_,4,_,_,_,_,1,_,_],
+            [1,_,_,7,_,5,_,_,_],
+            [5,_,3,_,_,_,_,_,_],
+            [_,9,_,3,_,_,_,_,_],
+            [_,_,4,_,6,1,_,_,_],
+            [_,_,5,_,_,_,7,_,_]]).
+
+problem(2, [[_,_,9,5,_,_,_,3,7],
+            [1,3,7,9,_,_,_,5,2],
+            [2,_,_,_,_,3,6,9,_],
+            [3,5,2,_,1,_,_,_,6],
+            [_,_,_,4,5,2,3,_,_],
+            [_,8,1,_,3,_,2,_,_],
+            [6,_,3,_,4,_,8,_,9],
+            [5,2,_,_,_,1,_,6,_],
+            [_,_,_,3,_,7,_,_,_]]).
+
+problem(3, [[_,5,_,1,_,_,_,_,_],
+            [2,_,_,5,_,_,6,_,_],
+            [1,_,_,_,8,_,2,_,_],
+            [_,8,_,4,3,_,_,_,_],
+            [_,_,_,_,_,_,_,4,_],
+            [_,_,_,_,_,7,9,3,2],
+            [_,4,_,6,7,_,_,_,_],
+            [_,7,_,_,_,_,_,1,9],
+            [9,_,_,_,_,8,_,_,_]]).
+```
+
+# SOLUTION
+
+We can get the solution using the following queries:
+```
+problem(1,Rows), sudoku(Rows), maplist(portray_clause, Rows).
+```
+![](sudoku-screenshots/p1.png)
+```
+problem(2,Rows), sudoku(Rows), maplist(portray_clause, Rows).
+```
+![](sudoku-screenshots/p2.png)
+```
+problem(3,Rows), sudoku(Rows), maplist(portray_clause, Rows).
+```
+
+# ANOTHER EXAMPLE
+
+Following is the step by step method of how prolog solves the Sudoku puzzle using
+intelligent constraint propagation.
+![](sudoku-screenshots/01.png) ![](sudoku-screenshots/02.png)
+![](sudoku-screenshots/03.png) ![](sudoku-screenshots/04.png)
+![](sudoku-screenshots/05.png) ![](sudoku-screenshots/06.png)
+
+Inconsistent values are indicated by small dots. Black dots represent “obvious”
+propagation, whereas, blue dots represent “intelligent” propagation.
+![](sudoku-screenshots/07.png) ![](sudoku-screenshots/08.png)
+
+# REFERENCES
+
+- https://metalevel.at/sudoku
+- https://www.swi-prolog.org/man/clpfd.html

db.pl

@@ -15,7 +15,9 @@ sudoku(Rows) :-
         square(Gs, Hs, Is).
 
 square([], [], []).
-square([N1,N2,N3|Ns1], [N4,N5,N6|Ns2], [N7,N8,N9|Ns3]) :-
+square([N1,N2,N3|Ns1],
+       [N4,N5,N6|Ns2],
+       [N7,N8,N9|Ns3]) :-
         all_distinct([N1,N2,N3,N4,N5,N6,N7,N8,N9]),
         square(Ns1, Ns2, Ns3).
 
@@ -27,4 +29,24 @@ problem(1, [[_,_,_,_,_,4,_,_,2],
             [5,_,3,_,_,_,_,_,_],
             [_,9,_,3,_,_,_,_,_],
             [_,_,4,_,6,1,_,_,_],
-            [_,_,5,_,_,_,7,_,_]]).
+            [_,_,5,_,_,_,7,_,_]]).
+
+problem(2, [[_,_,9,5,_,_,_,3,7],
+            [1,3,7,9,_,_,_,5,2],
+            [2,_,_,_,_,3,6,9,_],
+            [3,5,2,_,1,_,_,_,6],
+            [_,_,_,4,5,2,3,_,_],
+            [_,8,1,_,3,_,2,_,_],
+            [6,_,3,_,4,_,8,_,9],
+            [5,2,_,_,_,1,_,6,_],
+            [_,_,_,3,_,7,_,_,_]]).
+
+problem(3, [[_,5,_,1,_,_,_,_,_],
+            [2,_,_,5,_,_,6,_,_],
+            [1,_,_,_,8,_,2,_,_],
+            [_,8,_,4,3,_,_,_,_],
+            [_,_,_,_,_,_,_,4,_],
+            [_,_,_,_,_,7,9,3,2],
+            [_,4,_,6,7,_,_,_,_],
+            [_,7,_,_,_,_,_,1,9],
+            [9,_,_,_,_,8,_,_,_]]).