Structuring Knowledge

Structuring Knowledge in a Network of Concepts

Francis Heylighen

PESP, Free University of Brussels

The basic evolutionary-systemic and constructive principles that have been
discussed in my two previous contributions to this volume can be directly
applied to the design of a computer support system that would help
Principia Cybernetica collaborators to develop a coherent system of
philosophical thought. In fact the same type of support system might be
applied to any complex problem domains where on the basis of a lot of
ill-structured, ambiguous and sometimes inconsistent data a more or less
simple and reliable model is to be built. The problem we are speaking about
is one of applied epistemology. A good epistemology, offering a concrete
and general theory of how knowledge develops during individual or cultural
evolution, should also be useful as a guide when a new model is practically
to be developed.
Network representations of knowledge

I start from the assumption that a lot of knowledge is already available,
in literature and in the heads of different (potential) contributors to the
project, but that that knowledge must be integrated into a coherent and
transparent model. The knowledge will be assumed to be written down in the
form of “chunks”, containing text, formulas, drawings, sound, …, whatever
media are most appropriate to express the underlying ideas. I further
suppose these chunks to be split up into distinct “ideas” or “concepts”,
such that one chunk should define not more than one concept.

Of course, these different concepts will be related and one chunk will
in general contain references to several other chunks. For example, the
chunk denoting the concept “dog” might contain the following sentence: a
dog is a carnivorous mammal, with a protruding snout. This means that the
concept dog has associations with a least the concepts mammal, carnivorous
and snout. If these concepts are also available as chunks, then we might
create a link from the dog chunk to the mammal chunk and so on. Computer
applications that allow such an easy representation and manipulation of
chunks connected by links are called hypermedia systems. The chunk with its
text and graphics can be shown in a window on the screen, and it suffices
to click on one of the links to show the next chunk to which the link is
pointing (Heylighen, 1991).

Hypermedia system are useful for storing a large amount of complex,
interrelated information (e.g. an encyclopedia) in a easy to handle way.
However, there is an inherent ambiguity involved, since it is not a priori
clear what a link is supposed to mean: any kind of association, as well
causal, as logical, as intuitive as spatial, …, might be represented by a
link. Therefore we need a better structured system if we want our networks
of concepts to support us more efficiently. By introducing different types
of chunks (nodes) and links we may turn our hypermedia system into a
semantic network: the different types of links will determine (part of) the
meaning of the concept to which they are attached. The problem with
semantic networks for knowledge representation is still that of ambiguity:
there is an unlimited number of link and node types that may seem
appropriate, and their interrelationships will in general be very unclear.
In order to limit the set of types, we need an unambiguous, fundamental
interpretation of what concepts and links in our network really stand for.
I will now propose such an interpretation with the corresponding types, and
show how it can be applied to the structuring of knowledge.

Distinction and entailment types

A concept (node) is supposed to represent a distinction: a way to separate
phenomena denoted by the concept (belonging to its class or extension),
from phenomena that do not belong to its extension. Defining a concept
means proposing a procedure for explicitly carrying out that distinction.
Definition will be assumed to be a bootstrapping operation: a concept is
always defined in terms of other concepts, that are themselves defined in
terms of other concepts, and so on. In general there is no primitive level
of meaningful concepts in terms of which all other concepts can be defined.
This is in accordance with my constructive philosophy, stating that any
foundations of a conceptual system must be empty of meaning in order to be
acceptable as basis for a complete philosophical explanation (Heylighen,
1990b).

One way to define a concept is by listing the set of concepts that it
entails together with the set of concepts entailed by it. By entailment I
mean an “if…then” relation, which is more general than the logical
(material) implication. For example, if a phenomenon is a dog, then it is
also a mammal: dog -> mammal. It means that a phenomenon denoted by the
first concept cannot be present or actual, without a phenomenon denoted by
the second one being (simultaneously) or becoming (afterwards) actual.

In order to derive fundamental types of distinctions (concepts, nodes)
and links (entailments), we will posit two basic dimensions of distinction:
stability (or time) and generality, with the corresponding values of
instantaneous – temporary – stable, and of specific – general. The
combination of these 3 x 2 values leads to 6 types of distinction (see
table).

time\generality | general | specific
—————————————————————–
stable | class | object
temporary | property | situation
instantaneous | change | event

For example, an object is a distinction that is stable (it is not supposed
to appear or disappear while we are considering it), and specific (it is
concrete, there is only of it). A property is a distinction that is general
(several phenomena may be denoted by it, it represents a common feature),
and temporary (it may appear or disappear, but normally it remains present
during a finite time interval). An event is instantaneous (it appears and
disappears within one moment), and specific (it does not denote a class of
similar phenomena, but a particular instance).

With these node types we can now derive the corresponding link types by
considering all possible combinations of two node types. There is one
constraint, however: we assume that a more invariant (stable or general)
distinction can never entail a less invariant one. Otherwise, the second
would be present each type the first one is, contradicting the hypothesis
that it is less invariant than the first one. For example, a class cannot
entail an object, a situation cannot entail an event. Yet it is possible
that concepts with the same type of invariance, (e.g. two objects) might be
connected by an entailment relation. All remaining possible combinations
can now be summarized by the following scheme (the straight arrows
represent entailment from one type to another (more invariant) one, the
circular arrows entailment from a concept of a type to a concept of the
same type):

For example when an object A entails a class B, A -> B, then A is an
Instance_of B. When an object A always entails the presence of another
object B, then B must belong to or be a part of A. When a change A entails
another change B, then A and B “covary” and hence A can be interpreted as
the cause of B. When an event A entails a situation B, then A must be
simultaneous with or preceding B in time.

The advantage of this scheme is that most of the intuitive and often
used semantic categories (objects, classes, causality, whole-part
relations, temporal precedence, etc.) can be directly constructed from it,
in a simple and uniform format. Complementarily, given some of those
everyday categories, we can use the scheme to reduce them to simple
entailment links between nodes of specific types. In fact the types
themselves can be represented as nodes, and each node of a particular type
will have an entailment link to that ‘type’-node. This allows us to reduce
a complicated set of semantic categories to an extremely simple formal
strcuture.

Knowledge structuring

Given that structure, consisting of a list of nodes and entailme nt links
between them, we can now start to formally analyse the network. Define the
input and output sets of a node:

Input: I(x) = { y | y -> x} = “extension” of concept x

Output : O(x) = { y | x -> y } = “intension” of concept x

The meaning (definition, distinction) of x can be interpreted as determined
by the disjunction of its input elements, and the conjunction of its output
elements. Our previous remark about definitions can now be reformulated as
the following bootstrapping axiom (Heylighen, 1990ab):

two nodes are distinct if and only if their input and output sets are
distinct:

x =\ y <=> I(x) =\ I (y), O(x) =\ O(y)

However, such a complete definition assumes that all concepts allowing to
distinguish between x and y are present in the network. In practice, the
network of concepts we are building by writing down our knowledge in the
form of connected chunks, will be incomplete in some respects, redundant in
other respects. Instead of using the axiom as a static description of how a
complete network should be structured, we can use it as a procedure to find
ways to make the network more adequate, by adding missing concepts, or by
deleting redundant ones. We can distinguish the following two main
techniques (cf. Heylighen, 1991; Bakker, 1987; Stokman & de Vries, 1988):

Node identification

When input and output sets of two nodes x and y are identic or similar, the
computer support system may propose the user to either identify (merge) the
two nodes, and replace them by one single node, or to add new nodes or
links that would more clearly differentiate between x and y. An algorithm
may test the identity or inclusion of the input and output sets, and
according to the results, propose the following possibilities to the user:

1) I (x) = I (y):

a) O (x) = O (y) => Identify (or distinguish) x and y

b) O (x) [proper_subset_of] O (y) => Identify x and y, or distinguish I
(x) from I (y)

2) I(x) [proper_subset_of] I(y):

a) O(x) = O(y) => Identify x and y, or distinguish O(x) from O(y)

b) O(x) [proper_subset_of] O(y) => Identify x and y

c) O(y) [proper_subset_of] O(x) => Connect x to y, x -> y

Node integration

When a cluster of nodes have a common set of “external” input or output
nodes (that is to say nodes that do not belong to the cluster), then from
the point of view of those external nodes, the nodes inside the cluster are
indistinguishable. Hence the nodes, though not strictly indistinguishable
according to the bootstrapping axiom, behave indistinguishably from a
certain viewpoint.

From that point of view, the cluster may be called closed (Heylighen,
1990a) and it might therefore be replaced by a single “integrated” node.
The integrated node “summarizes” the cluster nodes on a more abstract
level, and may hence simplify the conceptual model. Similar to the case of
node identification, the external indistinguishability of clustered nodes
may be spurious, and this should prompt the user to add additional
distinguishing links and nodes.

There are different types of closure, with different meanings and formal
properties, depending upon which sets of external input or output nodes are
common among the cluster, for example: transitive closure, equivalence,
cyclical closure, … If the closure is only approximative (the cluster
nodes have several external neighbours in common, but these do not form a
complete set of any specific type), then this method is similar to the one
called “conceptual clustering” in machine learning, where the boundaries
between clustered and non-clustered nodes become fuzzy, and depend on the
treshold chosen for the number of common neighbours.

In conclusion, the present set of concepts and techniques, when implemented
on a computer through a suitable intuitive interface, should enable an
individual or group of users to elicit and structure their knowledge about
a domain under the form of a network of concepts connected by entailment
links, and support them to minimize the redundancy, complexity and
incompleteness of their model.

The introduction of new nodes and links by the user corresponds to a
form of variation by recombination of concepts. The recognition of a closed
cluster of nodes by the system corresponds to the selection of a
distinction that is more stable or invariant than the distinctions between
the internal concepts of the cluster (Heylighen, 1990a), with closure as
fundamental selection criterion. The elicitation and structuring of
concepts in this manner hence follows the general evolutionary mechanism
that was postulated in my previous papers about evolutionary philosophy.

References

Bakker R.R. (1987): Knowledge Graphs: representation and structuring of
scientific knowledge, (Ph.D. Thesis, Dep. of Applied Mathematics,
University of Twente, Netherlands).

Heylighen F. (1990): “A Structural Language for the Foundations of
Physics”, International Journal of General Systems 18, p. 93-112 .

Heylighen F. (1990): “Relational Closure: a mathematical concept for
distinction-making and complexity analysis”, in: Cybernetics and Systems
‘90, R. Trappl (ed.), (World Science, Singapore), p. 335-342.

Heylighen F. (1991): “Design of a Hypermedia Interface Translating between
Associative and Formal Representations”, International Journal of
Man-Machine Studies.

Stokman F.N. & de Vries P.H. (1988): “Structuring Knowledge in a Graph”,
in: Human-Computer Interaction, Psychonomic Aspects, G.C. van der Veer &
G.J. Mulder (eds.), (Springer, Berlin).