% =================================================================
% == ==
% == An Introduction to ARTIFICIAL INTELLIGENCE ==
% == Janet Finlay and Alan Dix ==
% == UCL Press, 1996 ==
% == ==
% =================================================================
% == ==
% == chapter 2, pages 39-40: fuzzy set theory ==
% == ==
% == Prolog example, Alan Dix, August 1996 ==
% == ==
% =================================================================
% fuzzy_and/fuzzy_or/fuzzy_not do the fuzzy logic
% equivalents of and/or/not
fuzzy_and(P,Q,PandQ) :- min(P,Q,PandQ).
fuzzy_or(P,Q,PorQ) :- max(P,Q,PorQ).
fuzzy_not(P,NotP) :- NotP is 1-P.
% fuzzy sets are encoded similar ordinary sets (that is as lists)
% except the list contains (value,degree) pairs.
fuzzy_member(X,S,Degree) :-
member((X,Degree),S).
fuzzy_member(X,S,0) :-
\+ member((X,Degree),S).
% the support of a fuzzy set is the (ordinary) set of all
% values which have a non-zero degree of membership
fuzzy_support([],[]).
fuzzy_support([(X,0)|Fs],Ss) :-
fuzzy_support(Fs,Ss).
fuzzy_support([(X,Degree)|Fs],[X|Ss]) :-
Degree \= 0,
fuzzy_support(Fs,Ss).
% The fuzzy union of two fuzzy sets F1 and F2 is the fuzzy sets
% whose support is the union of the supports of F1 and F2
% and the degree of membership of each element is the
% fuzzy 'or' of the degree of membership of F1 and F2
% This corresponds to ordinary logic where X is a member
% of the union of S1 and S2 if it is a member of S1 *or*
% it is also a member of S2. In fuzzy logic this Boolean
% or is replaced by its fuzzy equivalent.
fuzzy_union(F1,F2,F) :-
fuzzy_support(F1,S1),
fuzzy_support(F2,S2),
union(S1,S2,Ss),
do_union(F1,F2,Ss,F).
do_union(F1,F2,[],[]).
do_union(F1,F2,[X|Ss],[(X,Degree)|F]):-
fuzzy_member(X,F1,D1),
fuzzy_member(X,F2,D2),
fuzzy_or(D1,D2,Degree ),
do_union(F1,F2,Ss,F).
% fuzzy intersection of F1 and F2 is similar except that its support
% is the intersection of their supports, and the degree of membership
% is the fuzzy and of the degree of membership of F1 and F2
fuzzy_intersect(F1,F2,F) :-
fuzzy_support(F1,S1),
fuzzy_support(F2,S2),
intersect(S1,S2,Ss),
do_intersect(F1,F2,Ss,F).
do_intersect(F1,F2,[],[]).
do_intersect(F1,F2,[X|Ss],[(X,Degree)|F]):-
fuzzy_member(X,F1,D1),
fuzzy_member(X,F2,D2),
fuzzy_and(D1,D2,Degree),
do_intersect(S1,S2,Ss,F).
% RUNNING THIS CODE
%
% Use the example in the book:
%> fuzzy_and(0.9,0.6,Degree).
%
% Try:
%> fuzzy_support(
%+ [(porsche944,0.9),(bmw316,0.5),(vauxhallNova12,0.1)], % FastCar
%+ Supp ).
%
% and:
%> fuzzy_intersect(
%+ [(porsche944,0.9),(bmw316,0.5),(vauxhallNova12,0.1)], % Fast Car
%+ [(porsche944,0.6),(rollsRoyce,0.8)], % Pretentious Car
%+ Result ).
%
% EXAMPLES
%
%> fuzzy_and(0.9,0.6,Degree).
%
%> fuzzy_support(
%+ [(porsche944,0.9),(bmw316,0.5),(vauxhallNova12,0.1)], % FastCar
%+ Supp ).
%
%> fuzzy_intersect(
%+ [(porsche944,0.9),(bmw316,0.5),(vauxhallNova12,0.1)], % Fast Car
%+ [(porsche944,0.6),(rollsRoyce,0.8)], % Pretentious Car
%+ Result ).