<s>
In	O
information	B-Application
science	O
,	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
(	O
FCA	O
)	O
is	O
a	O
principled	O
way	O
of	O
deriving	O
a	O
concept	O
hierarchy	O
or	O
formal	O
ontology	O
from	O
a	O
collection	O
of	O
objects	O
and	O
their	O
properties	O
.	O
</s>
<s>
Formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
finds	O
practical	O
application	O
in	O
fields	O
including	O
data	B-Application
mining	I-Application
,	O
text	B-Algorithm
mining	I-Algorithm
,	O
machine	O
learning	O
,	O
knowledge	O
management	O
,	O
semantic	B-Application
web	O
,	O
software	O
development	O
,	O
chemistry	O
and	O
biology	O
.	O
</s>
<s>
The	O
original	O
motivation	O
of	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
was	O
the	O
search	O
for	O
real-world	O
meaning	O
of	O
mathematical	O
order	O
theory	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
formalizes	O
the	O
semantic	B-Application
notions	O
of	O
extension	O
and	O
intension	O
.	O
</s>
<s>
The	O
concept	B-Algorithm
lattice	I-Algorithm
can	O
be	O
graphically	O
visualized	O
as	O
a	O
"	O
line	O
diagram	O
"	O
,	O
which	O
then	O
may	O
be	O
helpful	O
for	O
understanding	O
the	O
data	O
.	O
</s>
<s>
Then	O
the	O
mathematical	O
theory	O
of	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
may	O
be	O
helpful	O
,	O
e.g.	O
,	O
for	O
decomposing	O
the	O
lattice	O
into	O
smaller	O
pieces	O
without	O
information	B-Application
loss	O
,	O
or	O
for	O
embedding	O
it	O
into	O
another	O
structure	O
which	O
is	O
easier	O
to	O
interpret	O
.	O
</s>
<s>
In	O
his	O
article	O
"	O
Restructuring	O
Lattice	O
Theory	O
"	O
(	O
1982	O
)	O
,	O
initiating	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
as	O
a	O
mathematical	O
discipline	O
,	O
Wille	O
starts	O
from	O
a	O
discontent	O
with	O
the	O
current	O
lattice	O
theory	O
and	O
pure	O
mathematics	O
in	O
general	O
:	O
The	O
production	O
of	O
theoretical	O
results	O
—	O
often	O
achieved	O
by	O
"	O
elaborate	O
mental	O
gymnastics	O
"	O
—	O
were	O
impressive	O
,	O
but	O
the	O
connections	O
between	O
neighboring	O
domains	O
,	O
even	O
parts	O
of	O
a	O
theory	O
were	O
getting	O
weaker	O
.	O
</s>
<s>
Hence	O
,	O
by	O
its	O
origins	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
aims	O
at	O
interdisciplinarity	O
and	O
democratic	O
control	O
of	O
research	O
.	O
</s>
<s>
Then	O
—	O
and	O
later	O
in	O
model	O
theory	O
—	O
a	O
concept	O
as	O
unary	O
predicate	B-Algorithm
had	O
been	O
reduced	O
to	O
its	O
extent	O
.	O
</s>
<s>
Hence	O
,	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
is	O
oriented	O
towards	O
the	O
categories	O
extension	O
and	O
intension	O
of	O
linguistics	O
and	O
classical	O
conceptual	O
logic	O
.	O
</s>
<s>
Formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
aims	O
at	O
the	O
clarity	O
of	O
concepts	O
according	O
to	O
Charles	O
S	O
.	O
Peirce	O
's	O
pragmatic	O
maxim	O
by	O
unfolding	O
observable	O
,	O
elementary	O
properties	O
of	O
the	O
subsumed	O
objects	O
.	O
</s>
<s>
In	O
his	O
late	O
philosophy	O
,	O
Peirce	O
assumed	O
that	O
logical	O
thinking	O
aims	O
at	O
perceiving	O
reality	O
,	O
by	O
the	O
triade	O
concept	O
,	O
judgement	B-Application
and	O
conclusion	O
.	O
</s>
<s>
The	O
data	O
in	O
the	O
example	O
is	O
taken	O
from	O
a	O
semantic	B-Application
field	O
study	O
,	O
where	O
different	O
kinds	O
of	O
bodies	O
of	O
water	O
were	O
systematically	O
categorized	O
by	O
their	O
attributes	O
.	O
</s>
<s>
The	O
data	O
table	O
represents	O
a	O
formal	O
context	O
,	O
the	O
line	O
diagram	O
next	O
to	O
it	O
shows	O
its	O
concept	B-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
Accordingly	O
,	O
puddle	B-Application
has	O
exactly	O
the	O
characteristics	O
temporary	O
,	O
stagnant	O
and	O
natural	O
.	O
</s>
<s>
In	O
this	O
diagram	O
the	O
concept	O
immediately	O
to	O
the	O
left	O
of	O
the	O
label	O
reservoir	O
has	O
the	O
intent	O
stagnant	O
and	O
natural	O
and	O
the	O
extent	O
puddle	B-Application
,	O
maar	O
,	O
lake	O
,	O
pond	O
,	O
tarn	O
,	O
pool	O
,	O
lagoon	B-Application
,	O
and	O
sea	O
.	O
</s>
<s>
This	O
is	O
why	O
in	O
French	O
a	O
concept	B-Algorithm
lattice	I-Algorithm
is	O
sometimes	O
called	O
a	O
treillis	O
de	O
Galois	O
(	O
Galois	B-Algorithm
lattice	I-Algorithm
)	O
.	O
</s>
<s>
Conversely	O
,	O
it	O
can	O
be	O
shown	O
that	O
every	O
complete	O
lattice	O
is	O
the	O
concept	B-Algorithm
lattice	I-Algorithm
of	O
some	O
formal	O
context	O
(	O
up	O
to	O
isomorphism	O
)	O
.	O
</s>
<s>
Formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
handles	O
such	O
data	O
by	O
transforming	O
them	O
into	O
the	O
basic	O
type	O
of	O
a	O
(	O
"	O
one-valued	O
"	O
)	O
formal	O
context	O
.	O
</s>
<s>
For	O
possible	O
negations	O
of	O
formal	O
concepts	O
see	O
the	O
section	O
concept	B-Algorithm
algebras	I-Algorithm
below	O
.	O
</s>
<s>
For	O
each	O
finite	O
formal	O
context	O
,	O
the	O
set	O
of	O
all	O
valid	O
implications	O
has	O
a	O
canonical	O
basis	O
,	O
an	O
irredundant	O
set	O
of	O
implications	O
from	O
which	O
all	O
valid	O
implications	O
can	O
be	O
derived	O
by	O
the	O
natural	O
inference	O
(	O
Armstrong	B-Application
rules	I-Application
)	O
.	O
</s>
<s>
Formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
has	O
elaborate	O
mathematical	O
foundations	O
,	O
making	O
the	O
field	O
versatile	O
.	O
</s>
<s>
They	O
also	O
reveal	O
structural	O
information	B-Application
and	O
can	O
be	O
used	O
for	O
determining	O
,	O
e.g.	O
,	O
the	O
congruence	O
relations	O
of	O
the	O
lattice	O
.	O
</s>
<s>
Triadic	O
concept	B-Algorithm
analysis	I-Algorithm
replaces	O
the	O
binary	O
incidence	O
relation	O
between	O
objects	O
and	O
attributes	O
by	O
a	O
ternary	O
relation	O
between	O
objects	O
,	O
attributes	O
,	O
and	O
conditions	O
.	O
</s>
<s>
Although	O
triadic	O
concepts	O
can	O
be	O
defined	O
in	O
analogy	O
to	O
the	O
formal	O
concepts	O
above	O
,	O
the	O
theory	O
of	O
the	O
trilattices	O
formed	O
by	O
them	O
is	O
much	O
less	O
developed	O
than	O
that	O
of	O
concept	B-Algorithm
lattices	I-Algorithm
,	O
and	O
seems	O
to	O
be	O
difficult	O
.	O
</s>
<s>
Fuzzy	O
concept	B-Algorithm
analysis	I-Algorithm
:	O
Extensive	O
work	O
has	O
been	O
done	O
on	O
a	O
fuzzy	O
version	O
of	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Concept	B-Algorithm
algebras	I-Algorithm
:	O
Modelling	O
negation	O
of	O
formal	O
concepts	O
is	O
somewhat	O
problematic	O
because	O
the	O
complement	O
of	O
a	O
formal	O
concept	O
(	O
A	O
,	O
B	O
)	O
is	O
in	O
general	O
not	O
a	O
concept	O
.	O
</s>
<s>
However	O
,	O
since	O
the	O
concept	B-Algorithm
lattice	I-Algorithm
is	O
complete	O
one	O
can	O
consider	O
the	O
join	O
(	O
A	O
,	O
B	O
)	O
Δ	O
of	O
all	O
concepts	O
(	O
C	O
,	O
D	O
)	O
that	O
satisfy	O
;	O
or	O
dually	O
the	O
meet	O
(	O
A	O
,	O
B	O
)	O
𝛁	O
of	O
all	O
concepts	O
satisfying	O
.	O
</s>
<s>
The	O
concept	B-Algorithm
lattice	I-Algorithm
equipped	O
with	O
the	O
two	O
additional	O
operations	O
Δ	O
and	O
𝛁	O
is	O
known	O
as	O
the	O
concept	B-Algorithm
algebra	I-Algorithm
of	O
a	O
context	O
.	O
</s>
<s>
Concept	B-Algorithm
algebras	I-Algorithm
generalize	O
power	O
sets	O
.	O
</s>
<s>
Weak	O
negation	O
on	O
a	O
concept	B-Algorithm
lattice	I-Algorithm
L	O
is	O
a	O
weak	O
complementation	O
,	O
i.e.	O
</s>
<s>
A	O
(	O
bounded	O
)	O
lattice	O
such	O
as	O
a	O
concept	B-Algorithm
algebra	I-Algorithm
,	O
which	O
is	O
equipped	O
with	O
a	O
weak	O
complementation	O
and	O
a	O
dual	O
weak	O
complementation	O
,	O
is	O
called	O
a	O
weakly	O
dicomplemented	O
lattice	O
.	O
</s>
<s>
Temporal	O
concept	B-Algorithm
analysis	I-Algorithm
(	O
TCA	O
)	O
is	O
an	O
extension	O
of	O
Formal	B-Algorithm
Concept	I-Algorithm
Analysis	I-Algorithm
(	O
FCA	O
)	O
aiming	O
at	O
a	O
conceptual	O
description	O
of	O
temporal	O
phenomena	O
.	O
</s>
<s>
It	O
provides	O
animations	O
in	O
concept	B-Algorithm
lattices	I-Algorithm
obtained	O
from	O
data	O
about	O
changing	O
objects	O
.	O
</s>
<s>
In	O
this	O
simple	O
case	O
,	O
a	O
typical	O
visualization	O
of	O
a	O
temporal	O
system	O
is	O
a	O
line	O
diagram	O
of	O
the	O
concept	B-Algorithm
lattice	I-Algorithm
of	O
the	O
view	O
into	O
which	O
trajectories	O
of	O
temporal	O
objects	O
are	O
embedded	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
simple	O
and	O
fast	O
algorithms	O
for	O
generating	O
formal	O
concepts	O
and	O
for	O
constructing	O
and	O
navigating	O
concept	B-Algorithm
lattices	I-Algorithm
.	O
</s>
<s>
Concept	B-Algorithm
lattices	I-Algorithm
with	O
a	O
few	O
million	O
elements	O
can	O
be	O
handled	O
without	O
problems	O
.	O
</s>
<s>
The	O
main	O
purpose	O
of	O
these	O
tools	O
varies	O
from	O
formal	O
context	O
creation	O
to	O
formal	O
concept	B-General_Concept
mining	I-General_Concept
and	O
generating	O
the	O
concepts	O
lattice	O
of	O
a	O
given	O
formal	O
context	O
and	O
the	O
corresponding	O
implications	O
and	O
association	B-Algorithm
rules	I-Algorithm
.	O
</s>
<s>
The	O
mathematical	O
and	O
algorithmic	O
results	O
of	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
may	O
thus	O
be	O
used	O
for	O
the	O
theory	O
of	O
maximal	O
bicliques	O
.	O
</s>
<s>
The	O
notion	O
of	O
bipartite	O
dimension	O
(	O
of	O
the	O
complemented	O
bipartite	O
graph	O
)	O
translates	O
to	O
that	O
of	O
Ferrers	O
dimension	O
(	O
of	O
the	O
formal	O
context	O
)	O
and	O
of	O
order	O
dimension	O
(	O
of	O
the	O
concept	B-Algorithm
lattice	I-Algorithm
)	O
and	O
has	O
applications	O
e.g.	O
</s>
<s>
Given	O
an	O
object-attribute	O
numerical	O
data-table	O
,	O
the	O
goal	O
of	O
biclustering	B-Algorithm
is	O
to	O
group	O
together	O
some	O
objects	O
having	O
similar	O
values	O
of	O
some	O
attributes	O
.	O
</s>
<s>
A	O
bicluster	B-Algorithm
in	O
a	O
binary	O
object-attribute	O
data-table	O
is	O
a	O
pair	O
(	O
A	O
,	O
B	O
)	O
consisting	O
of	O
an	O
inclusion-maximal	O
set	O
of	O
objects	O
A	O
and	O
an	O
inclusion-maximal	O
set	O
of	O
attributes	O
B	O
such	O
that	O
almost	O
all	O
objects	O
from	O
A	O
have	O
almost	O
all	O
attributes	O
from	O
B	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Of	O
course	O
,	O
formal	O
concepts	O
can	O
be	O
considered	O
as	O
"	O
rigid	O
"	O
biclusters	B-Algorithm
where	O
all	O
objects	O
have	O
all	O
attributes	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Hence	O
,	O
it	O
is	O
not	O
surprising	O
that	O
some	O
bicluster	B-Algorithm
definitions	O
coming	O
from	O
practice	O
are	O
just	O
definitions	O
of	O
a	O
formal	O
concept	O
.	O
</s>
<s>
A	O
bicluster	B-Algorithm
of	O
similar	O
values	O
in	O
a	O
numerical	O
object-attribute	O
data-table	O
is	O
usually	O
defined	O
as	O
a	O
pair	O
consisting	O
of	O
an	O
inclusion-maximal	O
set	O
of	O
objects	O
and	O
an	O
inclusion-maximal	O
set	O
of	O
attributes	O
having	O
similar	O
values	O
for	O
the	O
objects	O
.	O
</s>
<s>
In	O
it	O
was	O
shown	O
that	O
biclusters	B-Algorithm
of	O
similar	O
values	O
correspond	O
to	O
triconcepts	O
of	O
a	O
triadic	O
context	O
where	O
the	O
third	O
dimension	O
is	O
given	O
by	O
a	O
scale	O
that	O
represents	O
numerical	O
attribute	O
values	O
by	O
binary	O
attributes	O
.	O
</s>
<s>
This	O
reduction	O
allows	O
one	O
to	O
use	O
standard	O
definitions	O
and	O
algorithms	O
from	O
multidimensional	O
concept	B-Algorithm
analysis	I-Algorithm
for	O
computing	O
multidimensional	O
clusters	O
.	O
</s>
<s>
The	O
formal	B-Algorithm
concept	I-Algorithm
analysis	I-Algorithm
can	O
be	O
used	O
as	O
a	O
qualitative	O
method	O
for	O
data	O
analysis	O
.	O
</s>
<s>
Including	O
the	O
fields	O
of	O
:	O
medicine	O
and	O
cell	O
biology	O
,	O
genetics	O
,	O
ecology	O
,	O
software	B-General_Concept
engineering	I-General_Concept
,	O
ontology	O
,	O
information	B-Application
and	O
library	O
sciences	O
,	O
office	O
administration	O
,	O
law	O
,	O
linguistics	O
,	O
political	O
science	O
.	O
</s>
<s>
described	O
in	O
:	O
Formal	B-Algorithm
Concept	I-Algorithm
Analysis	I-Algorithm
.	O
</s>
<s>
Foundations	O
and	O
Applications	O
,	O
conference	O
papers	O
at	O
regular	O
conferences	O
such	O
as	O
:	O
International	O
Conference	O
on	O
Formal	B-Algorithm
Concept	I-Algorithm
Analysis	I-Algorithm
(	O
ICFCA	O
)	O
,	O
Concept	B-Algorithm
Lattices	I-Algorithm
and	O
their	O
Applications	O
(	O
CLA	O
)	O
,	O
or	O
International	O
Conference	O
on	O
Conceptual	O
Structures	O
(	O
ICCS	O
)	O
.	O
</s>
