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User-Friendly Fuzzy FCA
ICFCA 2013Dresden, Germany
Juraj Macko, Palacky University, Olomouc, Czech Republic
User-Friendly Fuzzy FCA
is oxymoron.Isn’t it?
User-Friendly Fuzzy FCA
FCA has strong mathematical foundations.FCA has a very clear meaning for users.
Fuzzy FCA has strong mathematical foundations.Fuzzy FCA does NOT have a clear meaning for users.
Residuated
lattice
Fuzzy set
Fuzzy relation
Arrow operators
User-Friendly Fuzzy FCA
Fuzzy concept lattice
Godel
LukasiewiczUser R.I.P.!
Half - redcircles
Measuring cup
Object
Liquid colorAttribute
User-Friendly Fuzzy FCA
Cyan Magenta Yellow Key (black)
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User-Friendly Fuzzy FCA
User-Friendly Fuzzy FCAx1 x2 x3 x4Universum X = All objects = {x1, x2, x3, x4,}
Set A = Objects which we care about = {x1, x4,}
x1 x2 x3 x4
x1 x2 x3 x4
Set A = for each object from X we specify either “We care about”
or “We do not care about” We care about = We care about in degree 1
We do not care about = We care about in degree 0
Fuzzy Set A(x) = for each object from X we specify How much we care about
(in which degree)
FCA: What you share, iF you Care About.
Part of the object,
which we care about.
Part of the object,
which we care about.
Actual amount of
RED in object
User-Friendly Fuzzy FCA
Part of the object,
which we care about.
=Is it TRUE, that the part, which we care about is
RED?
<
User-Friendly Fuzzy FCA
Yes, it is TRUE.truth degree = 1
=Is it TRUE, that the part, which we care about is
RED?
>
User-Friendly Fuzzy FCA
No, it is not TRUE.
truth degree = 0
It is TRUE in truth degree…How much is TRUE, that
the part, which we care about is RED?
>
?
?
?
?
?
User-Friendly Fuzzy FCA
How much is TRUE, that the part, which we care
about is RED?
Material truth =
actual amount of color.
User-Friendly Fuzzy FCA
>It is TRUE
in truth degree 2/5.
X
X
X
XX
X
How much is TRUE, that the part, which we care
about is RED?
It is TRUE in truth degree 1.
It is TRUE in truth degree 1 - 1/5 = 4/5
It is TRUE in truth degree 1 - 2/5 = 3/5
Similarity truth=
How similar it is comparing to
TRUE (truth degree 1)
Error-like truth=
How much is left to be TRUE
(truth degree 1)
User-Friendly Fuzzy FCA
>It is TRUE
in truth degree 1 - 3/5 = 2/5
How much is TRUE, that the part, which we care
about is RED?
Proportional=
what you see /
what you care about
User-Friendly Fuzzy FCA
>It is TRUE
in truth degree 2/3.
Mr. Godel
Mr. Lukasiewicz
Mr. Product
User-Friendly Fuzzy FCA
Mr. Godel
Mr. Lukasiewicz
Mr. Product1
User-Friendly Fuzzy FCA
Mr. Godel
Mr. Lukasiewicz
Mr. Product0
Fuzzy negation
User-Friendly Fuzzy FCA
A(x) B(y)I(x,y)
How much we care about the
object
How much is TRUE, that the part, which we care
about is RED?
Sharing=
Taking minimal truth degree
B(y)A(x)I(x,y)
How much we care about the
attribute.Has an object the amount of RED, which
we care about?Has an object the
required amount of RED?
Sharing=
Taking minimal truth degree
maximality
User-Friendly Fuzzy FCA
Original statement:The teeth are white.
Stronger statement:The teeth are snow-white.The teeth are very white.
It is very true, that the teeth are white.
Hedge – thruth stressing
1. The original statement: We care about the object.
2. The statement with the identity hedge (a*=a): We normally care about the object.
3. The statement with the globalization hedge (a*=0, a<1): We fully care about the object.
User-Friendly Fuzzy FCA
GodelScale
LukasiewiczScale
Scaling
User-Friendly Fuzzy FCA
GodelScale
LukasiewiczScale
GodelScale
LukasiewiczScale
GodelScale Lukasiewicz
Scale
User-Friendly Fuzzy FCA
Data source: Belohlavek R., FRS
Scaled fuzzy contextLukasiewicz
FCA with measures
User-Friendly Fuzzy FCA
Any real application?
OLAP CubeDB: Group by
Fuzzy OLAP CubeDB: Fuzzy group by
Fuzzy FCA with measures
User-Friendly Fuzzy FCA
Fuzzy OLAP application?
Data source: Belohlavek R., FRSNASA
User-Friendly Fuzzy FCA
User-Friendly Fuzzy FCA
Oxymoron?
C M Y K
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