Concepts, States, and Systems
Karl Erich Wolff
CASYS´99
Third International Conference on Computing Anticipatory Systems
Liège, Belgium, August 9 –14, 1999
Concepts, States, and Systems
Karl Erich Wolff
University of Applied Sciences Darmstadt, Department of Mathematics
Schöfferstr. 3, D-64295 Darmstadt, Germany
ERNSTSCHRÖDERCENTER FOR CONCEPTUAL KNOWLEDGE PROCESSING
Research Group Concept Analysis at the Technical University of Darmstadt
Fax: +49 6257-69361, E-mail: wolff@mathematik.tu-darmstadt.de
Abstract. Mathematical System Theory is extended to Conceptual System Theory using Formal Concept Analysis (Wille 1982).
States are defined as formal concepts and 'points of time' are generalized to 'time granules', interpreted as 'pieces' of time needed for
the realization of measurements. As a generalization of classical time systems we define conceptual time systems, their state spaces
and phase spaces. Time dependent relations among the parts of a conceptual time system are introduced in 'relational conceptual
time systems'. Applications in psychology and industry, including 'conceptual films' are mentioned.
Keywords : Formal Concept Analysis, Mathematical System Theory, Granularity, Time, States, Applied Systems Analysis
1 INTRODUCTION: SYSTEMS AND GRANULARITY
The purpose of this paper is to clarify the notion of 'time' and 'state' in General Systems Theory (in the sense of
Bertalanffy (1950, 1969)). Mathematical System Theory, as described for example by Zadeh, Desoer (1963), Zadeh (1964),
Klir (1969), Kalman, Arbib, Falb (1969), Mesarovic, Takahara (1975) and Pichler (1975) is extended to Conceptual System
Theory using tools from Conceptual Knowledge Processing (Wille (1982), Ganter, Wille (1996, 1999)).
In this paper we start with the description of 'real-world' systems from the viewpoint of a pragmatic observer who writes
down his observations in some data table. If the system has been observed long enough, the observer may find some
regularities in his data. Then he might be interested in compressing the knowledge in the usually large data table, writing
down some dependencies between measured variables in the comfortable form of 'laws' about the system, which are fulfilled
in the data only up to some 'granularity'. Clearly, also the data are written down only up to some granularity; for example for
the measurement of a duration we might decide to use 'seconds', knowing that in the intended data evaluation we will choose
a coarser granularity, say 'minutes'.
The investigation of systems in physics, chemistry, biology and other branches of science led to the development of
Mathematical System Theory, where the definition of a general dynamical system (see for example Mesarovic, Takahara
(1975), p. 21) is written down as in classical mechanics as a 'model' of an observed 'real-world' system without any
theoretical description of 'granularity' and without any discussion of the differences between that model and the observed
data. Therefore, the relation between the 'pure mathematical laws' and the 'dirty' measurements can not be treated
theoretically.
Formal Concept Analysis (Wille 1982) is based on a set theoretical definition of the concept of 'concept'. It led to a new
representation of data tables in the form of 'concept lattices'. These concept lattices are complete lattices (Wille (1982),
Birkhoff (1967), Davey, Priestley (1990)) and a basic tool for the formal representation of granularity. While most
applications of Formal Concept Analysis deal with finite lattices, many important infinite lattices also are concept lattices: for
example the complete lattice of all real numbers including  and -, and the orthomodular lattice of all closed subspaces of a
Hilbert space (Ganter, Wille 1996 or 1999). Indeed, any complete lattice, and therefore any finite lattice, is isomorphic to a
concept lattice.
That leads to the very promising situation that concept lattices can be used as well for the description of coarse
granularities as for the description of smooth continuities.
1
The use of 'states' in Mathematical System Theory led the author – after a discussion with R. E. Kalman - to the
conviction that states of an arbitrary system should be described as formal concepts of a suitable formal context associated
with the system. That is demonstrated in this paper.
We start the development of Conceptual System Theory with a pragmatically meaningful description of measurements
and the introduction of a theoretical formalization of 'granularity'. At first we introduce granularity of time.
Clearly Conceptual System Theory should have a general description of time. Though in most of the usual descriptions
time is represented as a linearly ordered set, for example a subset of the set of real numbers, I like to use a much more general
description, which starts from 'time granules' interpreted as 'pieces of time' needed for a meaningful measurement of some
'variables', 'events' or 'signals'. In colloquial language typical time granules are 'week', 'day' and 'morning'. For example,
'morning' may be used in the sense of the 'time piece' 'between' sunrise and noon. This time granule 'morning' is long enough
for many measurements, for example, for the measurements of the lengths of the long-jumps of a few pupils, but it may be
not long enough for the measurement of the maximum length of the long-jumps of all pupils of a school.
Therefore, we start with a set G of 'time granules' and describe each measurement as a function f: G  V, which assigns
to each time granule g  G a value f(g)  V, interpreted as the result of the measurement f at time granule g or as a 'missing
value'. On the set V of measurement values we also wish to use a granularity, which will be described by a 'conceptual scale',
the basic tool in Conceptual Knowledge Processing for the description of granularity.
Since some of the measurements of a 'real-world' system are 'time measurements' using 'clocks' we introduce an ordered
set (T,) such that the value m(g) of each time measurement m at a time granule g is an element of T. For example, we
could measure in the time granule 'morning', that a sprint of a pupil lasted 10 seconds. Other time measurements might
describe the beginning and the end of some time granules or just some bounds for some events or time granules.
The general formalization of a system in Conceptual System Theory is described in the definition of a 'conceptual time
system', consisting of a 'time part' and an 'event part'. The time part is interpreted as a description of measurements
concerning 'time' using 'clocks' and the event part as the other measurements at some 'real-world' system. The formal
connection of the time part and the event part is the set G of time granules, which serve as the common domain where both
the functions representing the time measurements and those representing the event measurements are defined.
The dissection of a conceptual time system into a time part and an event part is introduced to define the 'phase space' of a
conceptual time system. Generalizing the classical approach in Mathematical System Theory, where a 'phase' is understood as
a pair (t, s(t)), where s(t) is the 'state' of the system at time t, we introduce the phase space of a conceptual time system as a
certain 'composition' of the 'time part' and the 'event part' using the 'states' of a conceptual time system. The 'states' are
defined as 'formal concepts' in the sense of Formal Concept Analysis. That leads to a 'natural' understanding of states with
respect to the given formal description of the system. It leads also to a clear conceptual differentiation among three roles as to
how states are used in Mathematical System Theory: namely (i) states in the sense of 'global states' (cf. Mesarovic, Takahara
(1975)), (ii) states in the sense that the set of all states yields a partition of the time set such that the given system is at each
point of time in exactly one state, and (iii) states as elements of a hierarchy, for example in the sense that "the 'state of
reading' is a sub-state of 'the state of living' ".
2 FORMAL CONCEPT ANALYSIS
Formal Concept Analysis (FCA) was introduced by Wille (1982) and developed in the research group Concept Analysis
of the mathematical department at the Technical University Darmstadt (Germany). It is now the mathematical basis for
research in Conceptual Knowledge Processing. For the mathematical foundations the interested reader is referred to Ganter,
Wille (1996 or 1999). Two elementary introductions were written by Wolff (1994) and Wille (1997). From the latter we
quote:
Formal Concept Analysis is based on the philosophical understanding that a concept is constituted by two parts: its
extension which consists of all objects belonging to the concept, and its intension which comprises all attributes shared by
those objects. For formalizing this understanding it is necessary to specify the objects and attributes which shall be
considered in fulfilling a certain task. Therefore, Formal Concept Analysis starts with the definition of a formal context ... .
2.1 Formal Contexts And Concept Lattices
The following definition of a formal context was motivated by the observation that the specific meaning of concepts in
human thinking and communication is always determined by contexts. For the description of contexts we use the most simple
verbal utterances which state that an object has an attribute. Therefore a formal context is defined as a triple (G,M,I) of sets
where I is a binary relation between G and M, i.e., I  GM. The set G is called the set of objects (Gegenstände), the set M
is called the set of attributes (Merkmale) and the statement that the pair (g,m)  I is read "object g has attribute m".
2
Why is this simple definition important for a formal theory on concepts? It is important since it is possible to define for
each formal context K a very meaningful conceptual hierarchy (B(K),), whose elements, the formal concepts of K,
represent units of thought consisting of two parts, the extension and the intension, just as it is understood in philosophical
investigations dating back to Arnauld, Nicole (1685).
A formal concept of K = (G,M,I) is defined as a pair (A,B) where A  G, B  M and A = B and B = A where A is the
set of common attributes of A, formally described as A := {m  M | gA g I m } and B is the set of common objects of
B, B := {g  G | mB g I m }. A is called the extent and B the intent of (A,B).
The set of all formal concepts of K is denoted by B(K). The conceptual hierarchy among concepts is defined by set
inclusion: For (A1 , B1 ), (A2 , B2 )  B(K) let (A1 , B1 )  (A2 , B2 ) :  A1  A2 (which is equivalent to B2  B1 ).
An important role is played by the object concepts (g) := ({g} , {g} ) for g  G and dually the attribute concepts
(m) := ({m} , {m} ) for m  M.
The pair (B(K),) is an ordered set, i.e.,  is reflexive, antisymmetric, and transitive on B(K). It has some important
properties:
 (B(K),) is a complete lattice, called the concept lattice of K, and any complete lattice is isomorphic to a concept
lattice,
 (B(K),) contains the whole information of K, i.e., K can be reconstructed from B(K),
 If B(K) is finite it can be drawn as a line diagram in the plane, such that K can be reconstructed.
Line diagrams of concept lattices can be drawn automatically by computer programs (Wille 1984,1989) and serve as an
important communication tool for the representation of multidimensional data (Wolff 1996).
It is clear that binary relations and therefore formal contexts are used in nearly all branches of mathematics and in many
applications, therefore Formal Concept Analysis is very useful in many situations, even if the formal contexts are not finite.
One of the most famous infinite examples is the context (Q, Q, Q) of the rational numbers Q with the usual rational ordering
Q. The concept lattice B(Q, Q, Q) is isomorphic to the complete lattice of all real numbers including  and - with the
usual ordering on this set. This conceptual construction of the real numbers shows that Formal Concept Analysis covers not
only finite structures.
Since each complete lattice is isomorphic to a concept lattice, and complete lattices, closure systems and closure operators
are mathematically equivalent, Formal Concept Analysis enriches the application of these theories by a strong
communicational component, which stems from the contextual meaning of the objects and attributes and the rich possibilities
for visualizing multidimensional data by line diagrams of concept lattices.
2.2 Conceptual Scaling
The word 'scaling' is understood here in the sense of 'embedding something in a certain (usually well-known) structure',
called a scale: for example, embedding some objects according to the values of measurements of their temperature into a
temperature scale. Another example is the embedding of conference talks in the time schedule, which is usually a direct
product of two time chains, one for hours and one for days. More generally, in conceptual scaling objects or values are
embedded in the concept lattice of some context, called a conceptual scale.
Conceptual Scaling Theory was developed by Ganter and Wille (1989). The general process in conceptual scaling starts
with the representation of knowledge in a data table with arbitrary values and possibly missing values. These data tables are
formally described by many-valued contexts (G,M,W,I), where G is a set of 'objects', M is a set of 'many-valued attributes', W
is a set of 'values' and I is a ternary relation, I  GMW, such that for any g  G, m M there is at most one value w
satisfying (g,m,w)  I. Therefore, a many-valued attribute m can be understood as a (partial) function and we write
m(g) = w iff (g,m,w)  I. A many-valued attribute m is called complete if it is a function. (G,M,W,I) is called complete if
each m  M is complete.
The central granularity-choosing process in conceptual scaling theory is the construction of a formal context
Sm = (Wm, Mm, Im) for each mM such that Wm  mG := {m(g) | gG }. Such formal contexts, called conceptual scales,
represent a contextual language about the set of values of m. Usually one chooses Wm as the set of all 'possible' values of m
with respect to some purpose. Each attribute n Mm is called a scale attribute. The set n = {w | w Im n } is the extent of the
attribute concept of n in the scale Sm. Hence, the choice of a scale induces a selection of subsets of W m - describing the
granularity of the contextual language about the possible values. The set of all intersections of these subsets constitutes just
the closure system of all extents of the concept lattice of Sm.
The granularity of the language about the possible values of m induces in a natural way a granularity on the set G of
objects of the given many-valued context, since each object g is mapped via m onto its value m(g) and m(g) is mapped via the
object concept mapping m of Sm onto m(m(g)): g  m(g)  m(m(g)).
Hence the set of all object concepts of Sm plays the role of a frame within which each object of G can be embedded.
3
For two attributes m, m´  M each object g is mapped onto the corresponding pair:
g  (m(g), m´(g))  ( m(m(g)), m´(m´(g)) )  B(Sm )  B(Sm´ ).
The standard scaling procedure, called plain scaling, constructs from a conceptual system ((G,M,W,I), (Sm | m  M)),
consisting of a many-valued context (G,M,W,I) and a scale family (Sm | m  M) the derived context, denoted by
K := (G, {(m,n) | m  M, n  Mm }, J), where g J (m,n) iff m(g) I m n
(g  G, m  M, n  Mm ).
The concept lattice B(K) can be (supremum-)embedded into the direct product of the concept lattices of the scales (Wille
1982, Ganter, Wille 1996 or 1999). That leads to a very useful visualization of multidimensional data in so-called nested line
diagrams which is implemented in the program TOSCANA (Vogt, Wille 1994).
Plain scaling can be expressed in terms of infomorphisms (Barwise, Seligman, 1997) and vice versa, which was shown by
Wolff (1999a).
Finally we mention that Fuzzy Theory introduced by Zadeh (1965) also developed some notion of a scale, namely the
linguistic variables (Zadeh 1975). It was shown by Wolff (1998) that Fuzzy Theory can be extended (by replacing the unit
interval in the definition of the membership function by an arbitrary ordered set (L,)) to so-called L-Fuzzy Theory, which
allows for developing analogously to Formal Concept Analysis a Fuzzy Scaling Theory which is equivalent to Conceptual
Scaling Theory.
3 GRANULATED TIME
3.1 Time Granules Replace Time Points
In the following the formal representation of time will be discussed carefully. The standard representation of time as a
subset T of the set R of all real numbers, structured by the induced order or other structures like addition, topology, or metric,
seems to be very meaningful at first sight. But if we wish to describe events by time functions we have some difficulties,
namely to justify pragmatically, that, for example, the velocity of a particle at a certain point t of time has a certain value. In
practice we can measure the velocity of a particle only as the mean velocity during a certain time interval. To assign a value
of a measurement not to a point of time but to an interval, which is now viewed as a whole new concept, is also practiced in
colloquial language, for example, by saying "I was reading the whole afternoon" and then qualifying "For some minutes of
this afternoon I did not read". Therefore one should introduce a theory of 'granulated time', which describes what is usually
done in practice, namely, using a granularity of time, which may be, for example, a set G of subsets of the time set. The
elements of G are called 'time granules'. For example, for the purpose of describing historical events we mainly use the union
of the set of years, the set of months, and the set of days. For pre-historical events we like to take coarser time granules. The
formal notion of an 'event' can be introduced as a function defined on the set G. Usual time functions can be obtained from
this definition by choosing the one-element-subsets of T as time granules.
In the following section we introduce 'granulated time sets', where each time granule is a subset of the time set. This first
approach will be generalized in chapter 5.
3.2 First Approach: Granulated Time Sets
For the description of time we use an ordered set (T, ). We do not assume that (T, ) is a chain (or a linearly ordered set,
which is defined to be an ordered set where any two elements s, t  T are comparable, that is s  t or t  s). Then it is also
possible to describe events happening in several (linearly) ordered time sets; an example is a flight through different time
zones, which is described on the flight ticket using departure and arrival time in different time zones.
To describe the granularity of time we just introduce a set G of 'time granules' g, each being a subset of T, interpreted as a
'piece' of time, usually an interval, large enough to admit the measurements we wish to perform at g, and small enough to
fulfill the desired time accuracy.
Definition: 'granulated time set'
Let (T, ) be an ordered set and G be a subset of the power set P(T) := {X | X  T}. Then the triple (T, , G) is called a
granulated time set on the time (T, ) with granularity G. The elements of G are called time granules.
Example:
Let (R, ) be the usual ordered set of real numbers; for s, t  R and s < t let [s,t] := {x  R | s  x  t } and
[s,t):= {xR | s  x < t}, Z be the set of integers. Then G1:= { [s, s+1] | s  Z } and G2:= { [s, s+1) | s  Z } are two different
granularities on (R, ). Clearly, G2 is a partition of R, while G1 is a covering of R, but not a partition.
4
On the set G of time granules of a granulated time set (T, , G) we may introduce several structures, for example the set
inclusion. The following construction of an ordering G on G works, if we start with an arbitrary complete lattice (T, ), for
example the usual complete lattice R  {, - } of the real numbers including  and -. Then each granule g  G has, as a
subset of T, an infimum and a supremum, denoted by inf(g) and sup(g), called the initial point and the endpoint of g.
Definition: 'the endpoint ordering on G'
Let (T, , G) be a granulated time set such that for any g  G inf(g) and sup(g) exist (in the ordered set (T, )).
For granules g, h  G we define g G h iff inf(g)  inf(h) and sup(g)  sup(h).
Clearly, the mapping f: G  TT, defined by f(g) := (inf(g), sup(g)) is order preserving from the ordered set (G, G) into
the product order (TT, ).
In the example above both granularities G1:= { [s, s+1] | s  Z } and G2:= { [s, s+1) | s  Z } have isomorphic endpoint
orderings, which are isomorphic to the usual 'discrete' ordering on Z.
3.3 Granulated Time Systems
If we observe a 'real-world' system using n measuring instruments Mi (i  {1,..,n}), and we measure at a time granule g
with all n measuring instruments Mi , then we say that we have used the n measuring instruments simultaneously at g. In this
sense the following definition can be interpreted as a description of simultaneous measurements over all time granules.
Definition: 'granulated time system'
Let (T, , G) be a granulated time set, V a set and E  F(G, V) := {f | f : G  V} the set of all functions from G into V.
Then the tuple (T, , G, V, E) is called a granulated time system on (T, , G) with event set E and value set V. Each element
e  E is called an event. For any event e and any time granule g we say that the measurement of e at g yields e(g).
Clearly, the usual time functions from T into V can be described as events on G = {{t}| t  T}.
To also describe the situation, where there are 'missing values', we could take partial functions as a formal description of
events. But it is easier to introduce a value, say "/" in V, such that the equation 'e(g) = / ' is interpreted as 'e was not measured
at g'.
Therefore, a standard interpretation of a granulated time system (T, , G, V, E) is a single 'measuring run' of the system
over its granules. Another interpretation is obtained, if we partition the set E into classes Ek (k  {1,..,K}) and interpret each
granulated time system (T, , G, Vk , Ek) as a single measuring run over all granules measured by the k-th observer, or in
another interpretation, measured at the k-th part (for example: output or input) of the system. In the following definition we
describe formally a system with a set X of 'input events' and a set Y of 'output events' and a relation S  X  Y indicating all
pairs (x,y), where the input x 'generates' or 'causes' the output y.
Definition: 'granulated input-output-system'
Let (T, , G) be a granulated time set, A and B sets, X  F(G, A), Y  F(G, B) and S  X  Y. Then
(T, , G, A, B, X, Y, S) is called a granulated input-output-system. The elements of X are called input events, the elements of
Y are called output events and the elements of S input-output-pairs.
The classical general time system (cf. Mesarovic, Takahara (1975), p.17 and Pichler (1975), p. 47) can be described as
granulated input-output-systems with granularity G = {{t}| t  T}.
Before introducing conceptual time systems in general we study an example.
4 EXAMPLE: E-MAIL MESSAGES
In this example at first we describe a small 'real-world' system by the following 'e-mail story':
This morning I found two new e-mail messages e1 and e2; both arrived last night. At first I read e1 and put it into the
waste. Later on I read e2 and saved it in a suitable directory.
For the formal representation by a conceptual time system we specify the conceptual meaning of the four mentioned time
granules by the set G := {night, morning, early, late}, the last two granules in the sense of 'early' and 'late' ('in the morning').
5
A formal description of the meaning of these granules is given by the following formal context SG, called the granule scale:
SG
night
morning
early
late
<7

>9
<10




>11


<12




FIGURE 1. The granule scale: a formal context describing the time granules
The attributes, like '<7', can be understood in the usual ordering of numbers, but we do not assume a special structure on
the set of attributes of the granule scale in general.
The concept lattice of this granule scale is represented by the following line diagram:
FIGURE 2. The concept lattice of the granule scale
Reading the example: The granule 'early' has the attributes '>9', '<10', '<12'. The object concept of the granule 'morning' is
the pair ({early, morning, late}, {>9, <12}). It is a superconcept of the object concept of 'early'.
After having described the granules we now have to specify the events we are interested in. At first we wish to protocol
the locations of the e-mail messages: for each time granule g and each e-mail e we 'measure' the 'place' where the e-mail e
was stored at the time granule g. An e-mail may be at three 'places': it may be 'new' ('n'), and then, after being treated, it is
either in the 'waste' ('w') or in a suitable 'directory' ('d'). Secondly, we describe the two actions a1 (putting e-mail e1 into the
waste) and a2 (saving e2 into a suitable directory). We use the value set V := {n, w, d, /, 0, 1}. Let E := {e1, e2, a1, a2}. Then
the following table represents the many-valued 'granule-event context'.
night
morning
early
late
e1
n
w
w
w
e2
n
/
n
d
a1
0
1
1
0
a2
0
1
0
1
FIGURE 3. The many-valued granule-event context
Reading the example: E-mail e2 was 'new' in the night, it was 'new' 'early' in the morning, and it was in its 'directory' 'late'
in the morning. Since e2 was at different places during the morning, we do not assign one of the values n, w, d. Instead we
assign the value '/' in the meaning of a 'missing value'. The action a1 happens 'early' in the 'morning'; we therefore assign a '1'
to 'early' and to 'morning'. Action a1 does not happen in the 'night' and 'late'.
The reader should recognize the different and meaningful ways of ascribing a value to the granule 'morning' as a 'supergranule' of 'early' in the cases e2 and a1.
6
Now we introduce scales for the many-valued attributes in E. For e1 and e2 we choose the following scale Se, which is
just a modified nominal scale extended by an attribute 'treated' ('t') which indicates that an e-mail at the places 'w' or 'd' is
'treated'.
Se
n
w
d
/
n

w
d
t




FIGURE 4. The event scale Se
FIGURE 5. The concept lattice of the event scale Se
For the many-valued attributes a1 and a2 we choose the following 'action scale' Sa.
Sa
0
1
1

FIGURE 6. The action scale Sa
The concept lattice of Sa is just a chain of two concepts, namely the object concept of 0 as top element and the object
concept of 1 as bottom element, which is also the attribute concept of 1.
Using these scales we get the conceptual system C := ((G, E, V, I), (Se1 , Se2 , Sa1 , Sa2)), where Se1 = Se2 = Se and Sa1 =
Sa2 = Sa .The usual plain scaling yields the derived context KC. In the following table we describe the apposition SG | KC of
the granule scale SG (with attribute set MG) and KC to represent also the meaning of the granules in the usual time ordering.
(For the formal definition of the apposition of two contexts the reader is referred to Ganter, Wille (1996 or 1999).)
SG | KC
night
morning
early
late
<7

>9



MG
<10 >11 <12







e1
n

w



e2
d
t



n

w
d
t



a1
1
a2
1




FIGURE 7. The phase context of the conceptual time system E-MAIL
7
Clearly, the attributes of SG can be understood as special events. That will be discussed in detail in section 6.4, where the
phase space of a conceptual time system is introduced in general. Then the context SG | KC will be called the phase context
and its concept lattice the phase space; it is represented by the following line diagram.
FIGURE 8. The phase space of the conceptual time system E-MAIL
Reading the example: E-mail e1 was 'new' in the 'night', which is the only time granule '<7'; e1 was 'treated' and in the
'waste' during all time granules '>9'. The action a1 happened in the time granules 'early' and 'morning', but not in the time
granule 'late'. E-mail e1 was never (in not any time granule) in the 'directory', e2 was never in the 'waste'. Each time granule
lies in the range '<12'. (In all line diagrams we use "a1" and "a2" instead of "a11" and "a21".)
5 CONCEPTUAL TIME SYSTEMS
In the following we generalize granulated time system (T, , G, V, E) to conceptual time systems. As in the previous
example we separate the set G of time granules from the time set and start with an arbitrary set G. The elements of G are
called 'time granules', now not necessarily subsets of a time set. The meaning of the time granules is represented by some
'time measurements', formally described by a conceptual system T := ((G, M, W, IT), (Sm | m  M)), which generalizes the
ordered time set (T,  ).
Definition: 'conceptual time system'
Let G be an arbitrary set and T := ((G, M, W, IT), (Sm | m  M)) and C := ((G, E, V, I), (Se | e  E )) conceptual systems
(on the same object set G). Then the pair (T, C) is called a conceptual time system on G.
T is called the time part and C the event part of (T, C).
This definition prepares us for the general introduction of states and phases of conceptual time systems. Clearly, a pair of
two conceptual systems on the same object set can be described also as a single conceptual system. Compared with the
classical one-dimensional time as a subset of the set R of the real numbers, the time part T represents not only the
measurement of time using one or several clocks, but also the conceptual frame describing the meaning of the time granules
with respect to the measurements, which seems to me to be a pragmatically meaningful general description of time.
In the E-MAIL example the time part T can be easily described such that the granule scale is the derived context KT.
8
6 STATES
6.1 What Is A State?
What is a state of a system? This problem is treated by many authors. Zadeh (1964, p.40) writes in his paper 'The Concept
of State in System Theory':
To define the notion of state in a way which would make it applicable to all systems is a difficult, perhaps impossible, task.
In this chapter, our modest objective is to sketch an approach that seems to be more natural as well as more general than
those employed heretofore, but still falls short of complete generality.
In the following chapter our objective is to sketch an approach that seems to be still more natural as well as more general
than those employed heretofore. It is developed in the complete generality of conceptual time systems.
6.2 States In Conceptual Time Systems
Among the many meanings of 'state' we are interested here in that meaning which is described in dictionaries as a
'condition in which a thing is, mode of existence as determined by circumstances'. That meaning is mainly used in connection
with an observation of a thing (or 'system'), which is stable during some range of time, 'stable' clearly only relative to some of
its properties. For example, we say, that 'the patient is now again in a healthy state'.
In a conceptual time system (T, C) we would like to say 'The system is at granule g in state s(g)'. The central idea in the
introduction of states in conceptual time systems is that a state s(g) should be described by the event values e(g) – but these
values should be considered in the conceptual frame described by the event scales Se = (We, Me, Ie). Two values v, w  We
are called e-equivalent iff they have the same object concept in Se. If the measurements at two different granules g and h yield
e-equivalent values for each event e, then the state s(g) should be the same as the state s(h). That shows, that the states of
(T, C) should be defined as the object concepts of the derived context KC. On the time part we should look at the object
concepts of KT.
Definition: 'state space and granule space of a conceptual time system'
Let (T, C) be a conceptual time system and KT and KC the derived contexts of T and C. For each time granule g we
define the state s(g) of (T, C ) at granule g by s(g) := C(g) := the object concept of g in KC and the granule concept t(g)
of (T, C ) at granule g by t(g) := T(g) := the object concept of g in KT.
The set S(T, C):= {s(g) | g  G } is called the state space of (T, C), the set G(T, C):= {t(g) | g  G } is called the granule
space of (T, C).
This definition yields the 'partition meaning' of states, namely, that the set G of time granules is partitioned by the states,
or, equivalently, that a system is at each time granule in exactly one state.
This definition also yields the statement that the state space is an ordered set with the usual conceptual ordering on it.
6.3 States In Subsystems And General States
In this section we clarify the relation between the states in a conceptual system and the states in its subsystems.
Definition: 'subsystem of a conceptual system'
Let C = ((G, E, V, I), (Se | e  E )) be a conceptual system; for an arbitrary subset R  E let IR := {(g,e,v)  I | e  R}.
Then C(R) := ((G, R, V, IR), (Se | e  R )) is called the R-part of C or a subsystem of C.
Analogously we define the 'R-part of a formal context':
Definition: 'R-part of a formal context'
Let K := (G, M, I) be a formal context, and R  M, and IR := {(g,m)  I | m  R}. Then the subcontext
K(R) := (G, R, IR) is called the R-part of K.
Let C = ((G, E, V, I), (Se | e  E )) be a conceptual system, KC := (G, MC , J) the derived context of C and R  E. Let
R := {(e,n)  MC | e  R}, then KC(R) := KC(R) is called the R-part of KC.
If (T, C) is a conceptual time system on G and g  G and R  MC, then the object concept of g in KC(R) is called the
state of g in KC(R) and is denoted by sR(g).
9
To clarify the relation between the states in KC and the states in the R-part KC(R) we study the example in Fig. 7 and
choose R := {e1, e2}. The concept lattice of the {e1, e2}- part is drawn in the following figure.
FIGURE 9. The concept lattice of the {e1, e2}- part
In Fig. 9, the 'morning'-state, which is the object concept of 'morning', has the extent {early, morning, late}. Clearly, this
extent of KC(R) is an extent also in KC (Fig. 8), namely of the attribute concept of '>9'. In general, each extent of an R-part
K(R) of a context K is also an extent of the whole context K. That can be used to embed the concept lattice of K(R) into the
concept lattice of K by an injective order-preserving mapping R, called the part embedding, which maps each concept (A,B)
of K(R) onto the concept R((A,B)) := (A,A) of K with the same extent, but possibly a larger intent A in K. The part
embedding is the main connection between the subsystem KC(R) and the system KC. It also demonstrates the difficulties that
arise, if one does not distinguish between the states of a system and the states of a subsystem. One of these difficulties is that
the part embedding R does not preserve the property that a concept is an object concept; R is not state preserving. In the
example mentioned above, the 'morning'-state is mapped onto the attribute concept of '>9' – and that is not an object concept
in Fig. 8.
That leads to the introduction of 'general states' in the sense of the following definition. The notion of general states and
their conceptual ordering is a mathematical description of the 'order meaning' of states, which is intended in phrases like "the
'state of reading' is a sub-state of 'the state of living' ".
Definition: 'general time granules and general states'
Let (T, C) be a conceptual time system and KT and KC the derived contexts of (T, C).
Each concept (A,B) of KT is called a general time granule of (T, C).
The concept lattice B(KT) is called the time space of (T, C).
For (A,B), (C,D)  B(KT) we say that (A,B) is a time-sub-concept of (C,D) iff (A,B)  (C,D) in B(KT).
Each concept (A,B) of KC is called a general state of (T, C).
The concept lattice B(KC) is called the general state space of (T, C).
For (A,B), (C,D)  B(KC) we say that (A,B) is a sub-state of (C,D) iff (A,B)  (C,D) in B(KC).
Figure 9 represents the general state space of the {e1, e2}- part of the conceptual time system E-MAIL.
The term 'general state' is meaningful, since each general state (A,B) of (T, C ) with non-empty extent is a state of KC(B);
more precisely, (A,B) = sB(g) for all g  A. Hence, the part embedding B from the concept lattice of KC(B) into the concept
lattice of KC maps the state sB(g) onto the general state (A,B), which is not necessarily a state of (T, C ), but (A,B) = sB(g) is
a state of KC(B).
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6.4 Phase Spaces
In classical Mathematical System Theory a phase of a system is a pair (t, s) of a point t of time and a state s. This classical
division is represented in the notion of a conceptual time system (T, C), which allows us to introduce the notion of the phase
space of an arbitrary conceptual time system.
Definition: 'phase space of a conceptual time system'
Let (T, C) be a conceptual time system on G and KT and KC the derived contexts of T and C.
The apposition KT|KC of the derived contexts is called the phase context of (T, C).
The direct product B(KT)  B(KC) of the two concept lattices of KT and KC is called the general phase space of (T, C ).
Each element of B(KT)  B(KC) is called a general phase of (T, C).
The concept lattice B(KT|KC) is called the phase space of (T, C).
For any general time granule (A,B)  B(KT) and any general state (C,D)  B(KC) we say, that the system (T, C) is at the
general time granule (A,B) in the general state (C,D) iff A  C (which is equivalent to T(A,B)  C(C,D) in B(KT|KC),
where T and C are the part embeddings of KT and KC into KT|KC).
For any granule g  G the pair (t(g), s(g))  B(KT)  B(KC) is called a phase of (T, C) and we say, that (T, C) is in the
phase (t(g), s(g)) or (T, C) is at time granule g in the state s(g) iff T(t(g))  C(s(g)) in B(KT|KC).
To visualize a small example of a general phase space, we choose a subsystem of the E-MAIL example, namely the
{a1, a2}-part of KC. The following line diagram shows the state space of the {a1, a2}-part of KC.
FIGURE 10. The state space of the {a1, a2}-part of KC
Reading the example: In the 'morning' both actions a1 and a2 happen. In the 'early' morning only a1 happens. In the night
neither a1 nor a2 happens.
The line diagram in Fig.11 shows the general phase space for C({a1,a2}), the {a1, a2}-part of KC. The direct product of
B(SG) (see Fig. 2) and the lattice in Fig. 10 is drawn as a nested line diagram, where each point in Fig. 2 is "blown up" and a
copy of the state space in Fig. 10 is inserted. For an introduction to nested line diagrams the reader is referred to Ganter,
Wille (1996 or 1999).
Before explaining Fig. 11 with its mathematical background, we give a rough picture of its meaning: In the "blown-up
point" labeled '> 9' and 'morning' of the first factor SG we can see, what happened in the 'morning', namely action a1 and a2;
action a1 happened indeed in the 'early morning', and action a2 happened in the 'late morning'. In the 'night' granule the only
black circle at the top of the state space diagram indicates that neither a1 nor a2 happened in the 'night'.
Hence, for each of the four time states its "blown-up point" represents a 'photo' which shows all events of this time
granule. Looking at these photos in the sequence 'night', 'early', 'late' we can imagine a 'conceptual film'. The even shorter
film with only two photos, namely 'night' and 'morning', is a conceptual film, which in the second half is in speeded-up
motion.
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FIGURE 11. The general phase space of the {a1, a2}-part of KC with SG as granule scale
To prepare the mathematical background of the phase space representation in nested line diagrams we compare Fig.11
with the usual line diagram of the phase context of SG | KC({a1,a2}) in Fig.12:
FIGURE 12. The phase space of the {a1, a2}-part of KC with SG as granule scale
The line diagram in Fig.12 is very similar to the structure of the black points in Fig. 11 – indeed, the concept lattice of
SG | KC({a1,a2}) can be embedded in the direct product B(SG)  B(KC({a1,a2})); more generally, if a formal context is divided into
two (or more) parts, its concept lattice can be embedded (supremum-preserving) in the direct product of the concept lattices
of the parts (Ganter, Wille (1996 or 1999), p.77). Therefore, the phase space can be embedded (supremum-preserving) in the
general phase space.
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7 RELATIONAL CONCEPTUAL TIME SYSTEMS
In this chapter the notion of granulated input-output-systems (cf. section 3.3) and therefore also the notion of classical
general time systems are generalized. Instead of a single binary relation S  X  Y, interpreted as the 'cause relation' between
input X and output Y, we introduce a set  of relations of arbitrary arity on the set of parts of a conceptual time system.
These parts may be interpreted as subsystems, describing, for example, persons in a family. Therefore we like to introduce
arbitrary relations among the parts. Clearly, in many practical problems these relations are time dependent. That is modeled in
the following definition of a relational conceptual time system.
Definition: 'relational conceptual time system'
Let (T, C) be a conceptual time system on G, and KT and KC the derived contexts of T and C. We denote the phase
context KT|KC by K := (G, M, J); let P be a subset of the power set of M, P  {X | X  M}.
For any integer k  1 any subset R  P k is called a k-ary relation on P. Any function : G  {R | R  P k} is called a Gdependent k-ary relation on P. A function  is called a G-dependent relation on P iff  is a G-dependent k-ary relation
on P for some k  1. Let  be a set of G-dependent relations on P.
Then the quadruple (T, C, P , ) is called a relational conceptual time system on G.
Clearly, in this general definition the connection between the conceptual time system (T, C) and the set  of Gdependent relations on P is very loose, even more so, if the functions    are constant on G. In some applications that is
appropriate, in others one has to describe further connections.
Granulated input-output systems can be described as the special case that in a relational conceptual time system
(T, C, P , ) on G the set  has only one element  which is constant, say (g) = S for all g  G, and S is a binary relation
on P, i.e., S  P 2.
That is demonstrated in the E-MAIL example: we wish to express the 'cause' relation such that the action a1 causes the
event e1, and the action a2 causes the event e2. To describe this cause relation in KC we take the corresponding subsets of the
set of attributes of KC: {a11} causes {e1n, e1w, e1d, e1t}, and {a21} causes {e2n, e2w, e2d, e2t}. In this example the cause
relation S = {({a11}, {e1n, e1w, e1d, e1t}), ({a21}, {e2n, e2w, e2d, e2t})} is indeed functional. Therefore, we can choose a
one-element set C of global states (in the sense of Mesarovic, Takahara (1975), p. 12). This example shows that the states,
defined as object concepts of KC, are in general quite different from the global states, which are, by definition, just an
auxiliary set to decompose a relation S into functions.
8 APPLICATIONS OF CONCEPTUAL SYSTEM THEORY
In this section I mention some applications of Conceptual System Theory in psychology and in industry. Indeed, the
combination of carrying out the following projects in practice and also working in the mathematical foundations of Formal
Concept Analysis led to the development of Conceptual System Theory.
8.1 Conceptual Time Systems In Psychology
In many projects in cooperation with Spangenberg (Sigmund Freud Institute Frankfurt) the development of useful
graphical representations of the therapeutic process for patients was a central research topic. Therefore we had to study
processes with a granularity in time and also a granularity in the values (like "very sensitive") of the questionnaires. A typical
problem was the conceptual representation of the development of a patient over a period of two years using data in four
questionnaires. In most of these investigations the data were given in the form of a repertory grid. For more details the reader
is referred to Spangenberg, Wolff (1991,1993).
8.2 Conceptual Time Systems In Industry
Conceptual time systems were applied during my cooperation with a computer firm in the production of wafers based on
process data of a reactor. As in many other optimization and control processes the dependencies of the multidimensional
quality data from the multidimensional input data was studied over the control period.
A similar problem arose in the representation of thousands of error messages in a large industrial plant. Here I developed
the first conceptual film, which showed in a sequence of 30 line diagrams a nearly constant error message flow; but in the
middle of that time a burst of 'unusual' errors occurred, which from the single line diagrams we would not have detected as
'unusual'.
13
As an application of conceptual time systems in the chemical industry I mention a quality control problem in a distillation
column. The representation of the data in an (order theoretical) 4-dimensional state space, drawn in the plane, was the basic
graphical tool for the discussion of the four most important variables in the process (Wolff 1995).
9 CONCLUSION AND FUTURE PERSPECTIVES
This paper presents just the basic foundations of Conceptual System Theory. This combination of Mathematical System
Theory with Conceptual Knowledge Processing (based on the mathematical theory of Formal Concept Analysis) leads to a
'soft' introduction of granularity, which can be modeled by the user from coarse granularity to the continuity of real numbers.
The basic granularity tool is Conceptual Scaling Theory.
Conceptual System Theory introduces a generalization of the notion of time, such that in conceptual time systems
multidimensional meaningful time structures can be used depending on the purpose of the investigation.
The central result is the precise mathematical formulation of the concept of 'state' and the concept of 'phase' in terms of
formal concepts of the derived context of the many-valued context which consists just of the measurements used for the
observation of a 'real-world' system with respect to the chosen scales representing the desired granularity.
An important step is the investigation of the relation between the states of a system and the states of a subsystem, which is
described by the 'part embedding' of the subsystem in the whole system. That leads to a clear understanding of two main roles
of states, namely the 'partition role', which means that a system is at each time granule in exactly one state and the
'hierarchical role', which means, that the states of a system are members of a conceptual hierarchy.
As far as I can see, the third sense of the term 'state' in classical Mathematical System Theory, namely in the sense of
global states, is not related to the states defined as object concepts of the formal context KC.
The connection between Conceptual System Theory and the 'model oriented' classical Mathematical System Theory
requires further research. The evaluation of finite data tables usually leads to dependencies among the measured variables (or
events). Such dependencies, like Ohm´s law, require not only the lattice structure, but also the distribution of the time
granules over all states of the system. That leads to the construction of models representing dependencies, for example in the
form of equations. Coming from the side of the models, the classical system theorists developed a system theory in general
without granularity. Both sides, the 'data table side' and the 'model side' are connected by the pragmatic decisions of the
model builder, not by a mathematical construction yielding the model automatically from the data table.
For the near future, many ideas, conceptions, methods, and theorems in classical Mathematical System Theory need to be
discussed in Conceptual System Theory. Some of the main future topics are the conceptual definitions of basic concepts in
Mathematical System Theory, like transitions, and deterministic systems, as well as causality concepts like anticipatory and
non-anticipatory systems (Mesarovic, Takahara (1975)).
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