Formal Concept Analysis: Mathematical Foundations

287 Pages·1999·12.199 MB·English

by  Prof. Dr. Bernhard Ganter, Prof. Dr. Rudolf Wille (auth.)

Formal Concept Analysis
Springer
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Bernhard Ganter · Rudolf Wille
Formal
Concept Analysis
Mathematical Foundations
With 105 Figures
Springer
Prof. Dr. Bernhard GllIttr
IMtltut fIIr ~bra
Faknltil ftrr Mathematik
und Naturwi. .e nschaften
TechniKhe Universitlt Dretden.
Dre.oden, Germany
D~l062
Prot Dr. Rudnlf Wille
Arbeitlgruppe Allgemeine A1gebra
Fachbere!ch Mathematik
Thchnl.ch. Univenitit Darmstadt
D-64l89 Darmatadt, Germany
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Garrett Birkhoff
with his application-oriented view of lattice theory! and
Hartrnut von Hentig
with his critical yet constructive understanding of science2
have had a decisive influence on the genesis of Formal Concept Analysis.
1 G. Birkhoff: Lattice Theory. Amer. Math. Soc., Providence. 1st edition 1940,
2nd (revised) edition 1945, 3rd (new) edition 1967.
2 H. von Hentig: MagieI' odeI' Magistel<? tiber die Einheit del' Wissenschaft im
Verstandigungsplw;efJ. Klett, Stuttgart 1972.
Preface
Formal Concept AllalY.5is is a field of applied mathematics based on the math
ematization of concept and conceptual hierarchy. It thereby activates math
ematical thinking for conceptual data analysis and knowledge processing.
The underlying notion of "concept" evolved early in the philosophical
theory of concepts and still has effects today. For example, it has left its
mark in the German standards DIN 2:)30 and DIN 2;3:)1. In mathematics
it played a special role during the emergence of mathematical logic in the 19th
century. Subsequently, however, it had virtually no impact on mathematical
thinking. It was not until 1979 that the topic was revisited and treated more
thoroughly. Since then, through a large number of contributions, Formal
Concept Analysis has obtained such breadth that a systematic presentation
is urgently needed, but can no longer be realized in one volume.
Therefore, the present book foruse:':! on the mathematical foundations of
Formal Concept Analysis, which ran be regarded chiefly as a branch of ap
plied lattice theory. A series of examples serves to demonstrate the utility of
the lnathematical definitions and results; in particular, to show how Formal
Concept Analysis can be used for the conceptual unfolding of data contexts.
These examples do not play the role of case studies in data analysis. A
separate volume is intended for a comprehensive treatment of methods of
conceptual data and knowledge processing. The general foundations of For
mal Concept Analysis will also be treated separately.
It is perfectly possible to use Formal Concept Analysis when examining
human conceptual thinking. However, this would be an application of the
mathematical met hod and a matter for the experts in the respective sci
ence, for example psychology. The adjective "formal" in the name of the
theory has a delimiting effect: we are dealing with a mathematical field of
work, that derives it,,; comprehensibility and meaning from its connection with
well-established notions of "concept", but which does not strive to explain
conceptual thinking in turn.
The mathematical foundations of Formal Concept Analysis are treated
in seven chapters. By way of introduction, elements of mathematical order
and lattice theory which will be llsed in the following chapters have been
compiled in a chapter ":tro ". However, all difficult notation and results from
this chapter will be introduced anew later on. A reader who knows what is
undertitood by a lattice in mathematics may skip this chapter.
The first chapler describes the basic step in the formalization: An el
ementary form of the representation of data (the "cross table") is defined
mathematically ("formal rontexf'). A formal concept of such a data context
is then explained. The totality of all such concepts of a context in their hier
archy can be interpreted as a mathematical structure ("concept lattice"). It
is also possible to allow more complex data types ("many-valued contexts").
These are then reduced to the basic type by a method of interpretation called
"conceptual scaling".
VIII Preface
The second chapter examines the q lIestion of how all concepts of a data
context can be determined and represented in an easily readable diagram. In
addition, implications and dependencies between attributes are dealt with.
The third chapter supplies the basic notions of a structure theory for concept
lattices, namely part- and factor structures as well as tolerance relations. In
each case the extent to which these can be elaborated directly within the
contexts is studied.
These mathematical tools are then used in the fourth and fifth chapter, in
order to describe more complex concept lattices by means of decomposition
and construction methods. Thus, the concept lattice can be split up into
(possibly overlapping) parts, but it is also possible to use the direct product
of lattices or of contexts as a decomposition principle. A further approach
is that of substitution. In accordance with the same principles, it is possible
to construct contexts and concept lattices. As an additional construction
principle, we shall describe a method of doubling parts of a concept lattice.
The structural properties examined in mathematical lattice theory, for
example the distributive law and its generalizations or notions of dimension,
playa role in Formal Concept Analysis as well. This shall be treated in the
sixth chapter. The seventh chapter finally deals with structure-comparing
maps, examining various kinds of morphisms. Particular attention is given
to the scale measures, occuring in the context of conceptual scaling.
\Ve limit ourselves to a concise presentation of ideas for reasons of space.
Therefore, we endeavour to give a complete reference to further results and
the respective lit<'rature at the end of each chapter. However, we have only
taken into account such contributions closely connected with the topic of the
book, i.e., with the mathematical foundations of Formal Concept Analysis.
The index contains all t<,chnical terms defined in this book, and in addition
some particularly important hywords. The bibliography also serves as an
author index.
The genesis of this book has been aided by the numerous lectures and ac
tivities of the "Forschungsgruppe Begriffsanalyse" (Research Group on Con
cept Analysis) at Darmstadt University of Technology. It is difficult to state
in detail which kind of support was due to whom. Therefore, we can here
only express our gratitude to all those who contributed to the work presented
in this book.
Two years after the German edition, this English translation has been
finished. In its content there are only a few minor changes. Although there
is ongoing active work in the field, the mathematical foundations of Formal
Concept Analysis have been stable over the last years.
The authors are extremely grateful to Cornelia Franzke for her precise
and cooperative work when translating the book. They would also like to
thank K.A. Baker, P. Eklund and R.J. Cole, M.F. Janowitz, and D. Petroff
for their careful proofreading.
Contents
O. Order-theoretic Foundations
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Ordered Sets 1
001 0000000000000000000000000000000000000000000
Complete Lattices
002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5
Closure Operators
003 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8
0.4 Galois Connections
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11
Hints and References
005 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15
1. Concept Lattices of Contexts
00000000000000000000000000000 17
Context and Concept
1.1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17
Context and Concept Lattice
1.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23
Ul Many-valued Contexts
00 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 36
Context Constructions and Standard Scales
1.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 46
l.fi Hints and References
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 58
2. Determination and Representation
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 63
All Concepts of a Context
201 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 63
Diagranls
202 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 68
Implications between Attributes 79
20:3 00 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0
Dependencieti between Attributes
2.4 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 91
Hints and References
2 005 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 94
3. Parts and Factors
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97
Subcontexts
301 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97
Complete Congruences
:\02 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 000 0 0 00 0 0 0 0 0 0 0 0 104
;U Closed Sub relations
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 112
Block R.elations and Tolerances
3.4 0 0 0 0 0 0 0 0 0000 0 0 0 0 0 0 0 0 000 0 0 0 0 119
;:Ui Hints and References
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 127
4. Decompositions of Concept Lattices
0 0 0 0 0000 0000 00 0 0 0 0 0 0 0 0 129
Sub direct Decompositions
401 0 0 0 0 0 0 0 0 00000 000 0 0 0 0 0 0 0 0 000 0 0 0 0 129
Atlas-decompositions
402 0 0 0 0 0 0 0 0 0 0 0 0 0 0000 000 00000 00 0 000 0 0 0 0 1:\6
Substitution
403 0000000000000000000000000000000000000000000 150
Tensorial Decompositions
4.4 00 0 0 0 0 0 0 0 0 0000 000 000 00 000 000 0 0 0 0 163
Hints and References
405 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 180
x (
~ontE'llt"
5. Constructions of Concept Lattices ........................ 18:1
5.1 Subdirect Product Constructions ......................... 184
5.2 Gluings ............................................... 19:~
.5.3 Local Doubling ......................................... 198
5.4 Tensorial Constructions ................................. 205
5.5 Hints and References .................................... 216
6. Properties of Concept Lattices .. .......................... 219
6.1 Distributivity .......................................... 219
6.2 Semimodularity and Modularity .......................... 224
6.:~ Semidistributivity and Local Distributivity ................ 228
6.4 Dimension ............................................. 236
6 ..5 Hints and References .................................... 243
7. Context Comparison and Conceptual Measurability ...... 245
7.1 Automorphisms of Contexts ............................. 246
7.2 Morphisms and Bonds .................................. 252
7.:~ Scale Measures ......................................... 258
7.4 Measurability Theorems ................................. 263
7.5 Hints and References .................................... 269
References .. .................................................. 271
Index .. ....................................................... 281
o.
Order-theoretic Foundations
Formal Concept Analysis is based on mathematical order theory, in partic
ular on the theory of complete lattices. The reader is not required to be
familiar with these areas. The mathematical foundations are surveyed in
this chapter. However, we limit ourselves to the most important facts, as
there is no room for a comprehensive introduction to order theory. For this
purpose, we refer to the bibliography listed at the end of this chapter. In
general, the reader is supposed to have experience with mathematical texts:
we use the technical language of mathematics, in particular of set theory,
without further explanation.
In the first section we will introduce ordered sets, in the second complete
lattices. These two sections constitute the basis for the following chapters.
On the other hand, the third section, dealing with closure systems, and the
fourth on Galois connections may be skipped at a first reading. Much of
what they contain will be introduced again later under a different name. The
second half of this chapter shows how the basic notions of Formal Concept
Analysis have their roots in order and lattice theory. In this connection, we
follow, in most aspects, the "classical" representation by Garrett Birkhoff.
0.1 Ordered Sets
Definition 1. A binary relation R between two sets M and N is a set of
pairs (m, n) with m E M and n E N, i.e., a subset of the set M x N of all
=
such pairs. Instead of (m, n) E R we often write mRn. If N M, we speak
of a binary relation on the set M. R-1 denotes the inverse relation to
R, that is the relation between Nand M with nR-1m :{:} mRn. 0
Definition 2. A binary relation R on a set M is called an order relation
(or shortly an order), if it satisfies the following conditions for all elements
x, y, z E ill:
1. xRx (reflexivity)
2. xRy and x i- y:::::} not yRx (antisymmetry)
3. xRy and yRz :::::} xRz (transitivity)
B. Ganter et al., Formal Concept Analysis
© Springer-Verlag Berlin Heidelberg 1999