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Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Formal Concept Analysis
Bernhard Ganter
Institut fur Algebra
TU Dresden
D-01062 Dresden
bernhard.ganter@tu-dresden.de
Description Logics Workshop Dresden 2008
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Data from a hospital
Interview data from a treatment of Anorexia nervosa
over
-sen
sitive
withdra
wn
self-c
onfiden
t
dutifu
l
cord
ial
diffi
cult
att
entive
easily
offen
ded
calm
appre
hen
sive
chatt
y
super
fici
al
sensitive
am
bitio
us
Myself × × × × × × × × × ×
My Ideal × × × × × × × ×
Father × × × × × × × × × × × ×
Mother × × × × × × × × × × ×
Sister × × × × × × × × × ×
Brother-in-law × × × × × × ×
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Data from a hospital
A biplot of the interview data
uncomplicated
valiant
thick-skinned
complicated
difficult
apprehensive
over-sensitive
sympathetic
attentive
superficial
Brother-in-law
My Ideal
Myself
Sister
Father
Mother
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Data from a hospital
The concept lattice of the interview data
Brother-in-lawMyIdeal
Mother
Sister
Father
Myself
ambitiouscordial
attentivedutiful
super ficial
self-confident
chatty
over-sensitivecalmsensitive
apprehensivedifficultwithdrawn
easily offended
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Formal definitions
Unfolding data in a concept lattice
The basic procedure of Formal Concept Analysis:
Data is represented in a very basic data type, called a formal
context.
Each formal context is transformed into a mathematicalstructure called a concept lattice. The information containedin the formal context is preserved.
The concept lattice is the basis for further data analysis. Itmay be represented graphically to support communication, orit may be investigated with algebraic methods to unravel itsstructure.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Formal definitions
Unfolding data in a concept lattice
The basic procedure of Formal Concept Analysis:
Data is represented in a very basic data type, called a formal
context.
Each formal context is transformed into a mathematicalstructure called a concept lattice. The information containedin the formal context is preserved.
The concept lattice is the basis for further data analysis. Itmay be represented graphically to support communication, orit may be investigated with algebraic methods to unravel itsstructure.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Formal definitions
Unfolding data in a concept lattice
The basic procedure of Formal Concept Analysis:
Data is represented in a very basic data type, called a formal
context.
Each formal context is transformed into a mathematicalstructure called a concept lattice. The information containedin the formal context is preserved.
The concept lattice is the basis for further data analysis. Itmay be represented graphically to support communication, orit may be investigated with algebraic methods to unravel itsstructure.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Formal definitions
Unfolding data in a concept lattice
The basic procedure of Formal Concept Analysis:
Data is represented in a very basic data type, called a formal
context.
Each formal context is transformed into a mathematicalstructure called a concept lattice. The information containedin the formal context is preserved.
The concept lattice is the basis for further data analysis. Itmay be represented graphically to support communication, orit may be investigated with algebraic methods to unravel itsstructure.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
An example about airlines . . .
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
. . . and its concept lattice
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
Formal contexts
A formal context (G , M, I ) consists of sets G , M and a binaryrelation I ⊆ G × M.
For A ⊆ G and B ⊆ M, define
A′ := {m ∈ M | g I m for all g ∈ A}
B ′ := {g ∈ G | g I m for all m ∈ B}.
The mappingsX → X ′′
are closure operators.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
Formal concepts
(A, B) is a formal concept of (G , M, I ) iff
A ⊆ G , B ⊆ M, A′ = B, A = B ′.
A is the extent and B is the intent of (A, B).
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
Concept lattice
Formal concepts can be ordered by
(A1, B1) ≤ (A2, B2) : ⇐⇒ A1 ⊆ A2.
The set B(G , M, I ) of all formal concepts of (G , M, I ), with thisorder, is a complete lattice, called the concept lattice of(G , M, I ).
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
The basic theorem
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
Why complete lattices?
Complete lattices are closely related to closure operators andthereby to closure systems (also called Moore families).
Closure operators are closely related to propositional Horntheories. The formulas are implications, i.e., expressions of theform
A → B,
where A and B are sets of attributes.
Combining this, we may say that
Complete lattices are the algebraic structures corresponding topropositional Horn theories.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
Why algebraic structures?
The mathematical approach is different from that of Logic.We ask different questions and thereby get other results.
We admire the theoretical and algorithmic strength of DL. Atthe same time, we can build on a rich and well-developedmathematical theory of lattices and ordered sets.
Structure theoretic aspects, such as decompositions,morphisms, representations and classifications can becombined with established visualisations.
We try to systematically build a mathematical theory, theparts of which are as simple and as general as possible.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
More examples
The boolean lattice
What are the formal concepts of (G , G , 6=)?
They are precisely the pairs
(A, G \ A) A ⊆ G .
(G , G , 6=) therefore is the standard con-text of the power set lattice of G .
a b c d
a × × ×
b × × ×
c × × ×
d × × ×
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context arithmetic
Lattice constructions and context constructions
Many lattice properties and lattice constructions have nicecounterparts on the formal context side. For example,
the dual lattice corresponds to the transposed formal contextcomplete sublattices correspond to closed subrelationscongruence relations correspond to compatible subcontexts
tolerance relations correspond to block relationsdirect products correspond to context sumstensor products correspond to context products
Galois connections correspond to bonds
and so on.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context arithmetic
Closed relations and block relations
A closed relation of (G , M, I ) is a subrelation C ⊆ I suchthat every formal concept of (G , M, C ) also is a concept of(G , M, I ).The closed relations correspond to the complete sublattices ofB(G , M, I ).
A block relation of (G , M, I ) is a superrelation R ⊇ I suchthat every extent of (G , M, R) is an extent of (G , M, I ) andevery intent of (G , M, R) is an intent of (G , M, I ).The block relations correspond to the complete tolerancerelations of B(G , M, I ).
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context arithmetic
Closed relations and block relations
A closed relation of (G , M, I ) is a subrelation C ⊆ I suchthat every formal concept of (G , M, C ) also is a concept of(G , M, I ).The closed relations correspond to the complete sublattices ofB(G , M, I ).
A block relation of (G , M, I ) is a superrelation R ⊇ I suchthat every extent of (G , M, R) is an extent of (G , M, I ) andevery intent of (G , M, R) is an intent of (G , M, I ).The block relations correspond to the complete tolerancerelations of B(G , M, I ).
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context arithmetic
The context sum
The concept lattice of the context sum
(G , M, I ) + (H, N, J)
is isomorphic to the direct product
B(G , M, I ) × B(H, N, J).
G
H J
I
M N
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context product
Context product and tensor product
The product of formal contexts (G , M, I ) and (H, N, J) is definedto be the context
(G , M, I ) × (H, N, J) := (G × H, M × N,∇),
where(g , h) ∇ (m, n) : ⇐⇒ g I m or h J n.
The concept lattice
B((G , M, I ) × (H, N, J))
of the product is the tensor product
B(G , M, I ) ⊗ B(H, N, J).
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context product
Context product and tensor product
The product of formal contexts (G , M, I ) and (H, N, J) is definedto be the context
(G , M, I ) × (H, N, J) := (G × H, M × N,∇),
where(g , h) ∇ (m, n) : ⇐⇒ g I m or h J n.
The concept lattice
B((G , M, I ) × (H, N, J))
of the product is the tensor product
B(G , M, I ) ⊗ B(H, N, J).
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context product
An example of a context product
×
× ×
××
×
×=
× × × ×× × × ×
× × × × ×× × × × ×
× × × ×× × × ×
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Context product
Bonds and closed relations of the sum
R ⊆ G × N is a bond from (G , M, I ) to(H, N, J) if and only if
I ∪ R ∪ J ∪ (H × M)
is a closed relation of the context sum
(G , M, I ) + (H, N, J)
(assuming G ∩ H = ∅ = M ∩ N).
H J
G I R
M N
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Order relation bond
The sublattice corresponding to the identity bond
The concept lattice of the formal contextmade from the identity bond is isomorphicto the order relation of B(G , M, I ), withthe componentwise order. G I
G I I
M M
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Order relation bond
The order lattice of M3
0
a b c
1
0 ≤ 0
0 ≤ b 0 ≤ c
b ≤ b c ≤ c
b ≤ 1 c ≤ 1
1 ≤ 1
0 ≤ a
a ≤ a 0 ≤ 1
a ≤ 1
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Order relation bond
The order lattice of M3
0
a b c
1
0 ≤ 0
0 ≤ b 0 ≤ c
b ≤ b c ≤ c
b ≤ 1 c ≤ 1
1 ≤ 1
0 ≤ a
a ≤ a 0 ≤ 1
a ≤ 1
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Order relation bond
The identity bond as a context product
Note that
G I
G I I
M M
=×
× (G , M, I )
In other words: The order relation of a lattice is isomorphic to thetensor product of the lattice with a three-element chain.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Order relation bond
The identity bond as a context product
Note that
G I
G I I
M M
=×
× (G , M, I )
In other words: The order relation of a lattice is isomorphic to thetensor product of the lattice with a three-element chain.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Order relation bond
Free distributive lattices
The tensor product of n three-element chains is the free
distributive lattice on n generators, FCD(n), also known asthe lattice of all monotone boolean functions in n variables.
The order relation of FCD(n) is isomorphic to the tensorproduct of FCD(n) and a three-element chain. Therefore
The free distributive lattice on n + 1 generators is isomorphicto the order relation of the free distributive lattice on n
generators.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Order relation bond
Free distributive lattices FCD(1), FCD(2), FCD(3)
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
The stem base
Duquenne and Gigues have discovered that each implicationaltheory has a canonical base (with respect to the Armstrongderivation rules). We call this the stem base.
To find the stem base for a given formal context (G , M, I ) requiresthe notion of a pseudo-intent: P ⊆ M is a pseudo-intent iff
P is not an intent and
P contains the closure of every pseudo-intent that is properlycontained in P.
The stem base then is
{P → P” | P is a pseudo-intent}.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
Pairs of squares
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
But did we consider all possible cases?
How can we decide if our selection of examples is complete?
A possible strategy is to prove that every implication that holds forthese examples, holds in general.
Compute the stem base of the context of examples, and
prove that these implications hold in general,
or find counterexamples and extend the example set.
This can nicely be organised in an algorithm, called attribute
exploration.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
But did we consider all possible cases?
How can we decide if our selection of examples is complete?
A possible strategy is to prove that every implication that holds forthese examples, holds in general.
Compute the stem base of the context of examples, and
prove that these implications hold in general,
or find counterexamples and extend the example set.
This can nicely be organised in an algorithm, called attribute
exploration.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
But did we consider all possible cases?
How can we decide if our selection of examples is complete?
A possible strategy is to prove that every implication that holds forthese examples, holds in general.
Compute the stem base of the context of examples, and
prove that these implications hold in general,
or find counterexamples and extend the example set.
This can nicely be organised in an algorithm, called attribute
exploration.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
Stem base of the example set
common edge → parallel, common vertex, common segment
common segment → parallel
parallel, common vertex, common segment → common edge
overlap, common vertex → parallel, common segment,common edge
overlap, parallel, common segment → common edge, commonvertex
overlap, parallel, common vertex → common segment,common edge
disjoint, common vertex → ⊥
disjoint, parallel, common segment → ⊥
disjoint, overlap → ⊥
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
Two of the implications do not hold in general
common edge → parallel, common vertex, common segment
common segment → parallel
parallel, common vertex, common segment → common edge
overlap, common vertex → parallel, common segment,common edge
overlap, parallel, common segment → common edge, commonvertex
overlap, parallel, common vertex → common segment,common edge
disjoint, common vertex → ⊥
disjoint, parallel, common segment → ⊥
disjoint, overlap → ⊥
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
Conterexamples for the two implications
overlap, common vertex → parallel, common segment,common edge
Counterexample:
overlap, parallel, common segment → common edge, commonvertex
Counterexample:
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
Conterexamples for the two implications
overlap, common vertex → parallel, common segment,common edge
Counterexample:
overlap, parallel, common segment → common edge, commonvertex
Counterexample:
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
The base
A better choice of examples
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Outline
1 Concept latticesData from a hospitalFormal definitionsMore examples
2 Elements of structure theoryContext arithmeticContext productOrder relation bond
3 Attribute logicThe base
4 Rough SetsTraditional and generalised
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
ZdzisÃlaw Pawlak’s rough sets
An approximation space (U,∼) is a set U with an equivalencerelation ∼ on U.
A subset A ⊆ U is rough if it is not a union of equivalence classes.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
Approximations
A rough set A is represented (approximately) by a pair(R(A), R(A)), giving the
lower approximation R(A)
and the
upper approximation R(A)
of A.
In this way, a interval arithmetic for sets is established.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
The lower approximation
R(A)
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
The upper approximation
R(A)
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
The algebra of approximating pairs
The algebra of approximating pairs (alsocalled rough sets) is well studied. It canbe obtained from the formal context shownhere. U 6∼
U 6∼ 6∼
U U
Note this formal context is the context product of theindiscernibility relation (u, u, 6∼) with the context of a threeelement chain. The lattice of rough sets therefore is the lattice ofall intervals of the power set of indiscernibility equivalence classes.It is a Stone lattice.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
Operator pairs for generalised approximation
Meanwhile, rough sets have been generalised. Approximations areconsidered that do not rely on equivalence relations.Natural assumptions to keep are that
R is a kernel operator, and
R is a closure operator.
The family of approximating pairs
(R(A), R(A))
then generates (but not necessarily forms!) a sublattice of thedirect product of the lattice of kernels and the lattice of closures.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
The approximation lattice
The lattice generated by such generalised approximations can becharacterised in general: With some technical cheating, the formalcontext is as given in the diagram, where
(G , M, I ) is a formal context whose extent-complements are the kernel sets,
the extents of (G , N, J) are the closures,and
m ⊥ n : ⇐⇒ m′ ∪ n′ = G .
G J
M I−1 ⊥
G N
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
A natural generalisation
From the general characterisation it can be derived that there is aunique generalisation having most of the nice properties of theclassical case. It occurs when the indiscernibility equivalencerelation is generalised to an indiscernibility quasi-order.
For an indiscernibility quasi-order the al-gebra of approximations gets a strikinglysimple form.
It is the context product of (G , G , 6≥) withthe context of the three element chain.
The concept lattice therefore is isomorphicto the order relation of the lattice of quasi-order ideals.
G 6≥
G 6≥ 6≥
G G
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
A non-symmetric indiscernibility?
We have arrived at a seemingly counter-intuitive conclusion: Themathematically most promising generalisation of the rough setapproach is obtained by replacing the indiscernibility relation by anot necessarily symmetric quasi-order.Does non-symmetric indiscernibility make any sense?
Yes, it does! The intuition that an attribute-based indiscernibilityshould be symmetric tacitely assumes that attributes are scalednominally, that is, with mutually exclusive attribute values.The generalised version is natural when subsumption betweenattribute values is permitted.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
A non-symmetric indiscernibility?
We have arrived at a seemingly counter-intuitive conclusion: Themathematically most promising generalisation of the rough setapproach is obtained by replacing the indiscernibility relation by anot necessarily symmetric quasi-order.Does non-symmetric indiscernibility make any sense?
Yes, it does! The intuition that an attribute-based indiscernibilityshould be symmetric tacitely assumes that attributes are scalednominally, that is, with mutually exclusive attribute values.The generalised version is natural when subsumption betweenattribute values is permitted.
Bernhard Ganter, Dresden University FCA@DL2008
Outline Concept lattices Elements of structure theory Attribute logic Rough Sets
Traditional and generalised
Thank you for your attention!
Bernhard Ganter, Dresden University FCA@DL2008
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