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