(ed.), (Kluwer Academic Publishers, Dordrecht), p. 949-957.

__ABSTRACT__. It is argued that in order to tackle a complex problem
domain the first thing to do is to construct a well-structured problem
formulation, i.e. a "representation". Representations are analysed as systems
of distinctions, hierarchically organized towards securing the survival
of an agent with respect to his situation. A preliminary variation-selection
model is proposed for the generation of new distinctions. A research project
for building a general model of representation construction is outlined,
combining theoretical, computational and empirical-psychological approaches.

**1. Introduction**

Until now the theory of problem-solving (e.g. Newell and Simon, 1972)
has mainly emphasized the *search *of solutions within a problem space.
From this viewpoint, problem-solving capability (i.e. intelligence) should
be seen as the possession of adequate *heuristics*, which allow to
make the search more efficient. This view presupposes that the search space
is explicit and well-structured, i.e. that at each decision point there
is a well-defined set of operators (or problem states to be generated)
from which the most promising can be chosen according to some heuristic
rule.

This approach is typically applied in "game" situations (e.g. chess)
and "toy" problems (e.g. the tower of Hanoi problem), where the possible
"moves" are fixed by predetermined rules. As we all know, games and toys
are primarily used by children (and adults) to learn about the world by
*playing*, i.e. by performing simulations of actions so that the result
of these actions can be explored without being confronted with the complexities
and dangers of the real world. In this sense we have learned a lot about
problem-solving and about intelligence by constructing computer models
of how to play games or how to manipulate toys.

However, this does not mean that we are able to build models of how
to cope with the real world. Recently, the awareness has been growing in
AI that, in order to get insight into practical intelligence, we need to
build autonomous agents (e.g. robots) capable of directly interacting with
a real environment. However, in order to experiment efficiently with such
systems we should simultaneously develop a *theory* of problem-solving
in complex, ill-structured environments. The present paper proposes the
outline of a research project, aimed at the construction of such a theory.

First, we should attempt to define an "ill-structured problem environment". It can be characterized, first, by the presence of a "problem", i.e. a situation which is to be changed in some way; second, by the absence of the structure needed for efficient search : well-defined goal(s), problem-states, operators, constraints, heuristic criteria ... In a more radical formulation : when confronted with such a problem, we know that something is to be done, we do not want the situation to evolve on its own, but we do not know what to change, how to change it, or what the result of the change should be like.

Some examples of ill-structured problems may show the practical
applicability of the theory we are looking for. The management of a large
socio-economic system (e.g. a firm, an organization or a state) is clearly
a very complex problem (cfr. Dörner & Reither, 1978; Dörner
et al., 1983) : it is in general not at all clear which goals are to be
pursued, which means are available, or which information is relevant. The
availability of communication and information technology will in general
only increase the complexity of decision-making. Indeed, existing computer
systems are only capable of solving well-structured problems. In this way
they will merely increase the available information and hence the possibilities
for choice, without reducing the ill-structuredness of the situation. Another
example is research : the development of scientific theories is clearly
an ill-structured problem domain. The *discovery *of new concepts
or models is basically a process of building simple structures out of the
available data, which are often inconsistent, ambiguous, vague and changeful.

Clearly, the first thing to be done in order to solve an ill-structured
problem is to *formulate* it in a well-structured way, i.e. to describe
explicitly the initial situation which is to be changed, the goal which
is to be achieved, the problem-space which is to explored, the operators
which are to be used, ... Such a well-structured formulation is traditionally
called a *representation* of the problem (cfr. Amarel, 1967; Burghgraeve,
1976; Korf, 1980; Heylighen, 1986). A representation of how to build such
representations may then be called a *metarepresentation* (Heylighen,
1987a,b). Once we know how to construct (and transform) representations
of ill-structured problem domains, we can simply apply the existing knowledge
about search through problem spaces in order to be able to solve all types
of problems. We will now propose a conceptual framework for the analysis
of representations.

**2. Representations as Distinction Systems.**

In order to begin our study, we must analyse the object of the study
: representation. A representation can always be considered as a system,
i.e. an organized, goal-directed whole of interrelated elements. The goal
or function of a representation is to structure the field of experience
of the intelligent *agent* using the representation, in such a way
that the agent can search efficiently for solutions when confronted with
a problematic *situation*, which is to be changed.The second question
to be asked then is : what are its elements ? The elements we are looking
for are primitive structurations of the problem environment as experienced
by the agent.

The simplest form or structure we can imagine is a *distinction*
(Spencer-Brown, 1969). A distinction can be defined as the process (or
its result) of discriminating between a class of phenomena and the complement
of that class (i.e. all the phenomena which do not fit into the class).
As such a distinction structures the universe of all experienced phenomena
in two parts. Such a part which is distinguished from its complement or
background will hereafter be called an *indication* (Spencer-Brown,
1969). If more than one distinction is applied the structure becomes more
complex, and the number of potential indications increases, depending on
the number of distinctions and the way they are interrelated.

In contrast to Spencer-Brown (1969), we will not assume any general
axioms for distinctions. In particular we will not assume that the complement
of the complement **a' **of an indication **a** is again the same
indication (law of double negation) : **(a')' = a **. This means that
we do not suppose distinctions to be symmetric : the complement or negation
**a'** of an indication **a** has not necessarily the same status
as **a**. However, if this symmetry property is assumed, together with
a related axiom about conjunctions (or disjunctions) of distinctions (the
law of idempotence, cfr. Heylighen, 1987a), it can be shown that a set
of distinctions gets a Boolean algebra structure, isomorphic to the algebra
of classes in set theory or to the algebra of propositions in logic (Spencer-Brown,
1969).

How do distinctions determine problem-solving efficiency ? Clearly, to formulate a problem you need to make a minimum number of distinctions. At least you should be able to distinguish the situation to be changed from the situation corresponding to a satisfying problem-solution. Furthermore, in order to be able to search for a solution you should distinguish different problem-states, which can be reached by distinct operators. The more general you want your representation to be, i.e. the more potential problems you want to be able to solve, the more distinctions you must make.

However, more distinctions means more states, more operators, more decision points, hence more search to be carried out. Clearly, in order to minimize search you must minimize the number of distinctions. This means that for a computationally tractable representation of a real problem domain most phenomena must remain "indistinguishable" (cfr. Hobbs, 1985). In the terminology of (Heylighen, 1987a) : different phenomena are "assimilated" to the same class, which is "distinguished" from other classes.

Assimilation and distinction necessarily go together : the number of potentially distinguishable phenomena in the universe can be considered to be infinite, the number of actual distinctions used for solving a problem in the real world must be finite. Which finite set of distinctions is selected from this infinite set will depend upon the problem domain. The problem of problem-formulation could hence provisionally be formulated as : how to determine the optimal (i.e. minimal, yet large enough to cover all relevant solution paths) set of distinctions for a particular problem domain ?

However, a representation is not just a set of distinctions, it
is a *system*. This means that we must first look for the properties
of and relations between distinctions, in order to understand how they
are organized towards the fulfillment of their function. The structural
units of a representation can be described in a hierarchy of levels, ordered
from the more "subjective", agent-determined structures, to the more "objective",
situation-determined structures (see fig. 1) :

1) a problem is determined first by the autonomous agent, whose
ultimate aim is survival; 2) in order to survive the agent must specify
more concrete goals or values, which represent classes of situations for
which the long-term survival is more probable, and which can be reached
by a sequence of actions (cfr. Heylighen, 1988); 3) to attain these goals
the system must dispose of a set of operators (also called "(production)
rules"), representing possible actions changing the situation; 4) the operators
have arguments, which may be called problem-states, and which represent
distinguished situations; 5) the states can be analysed as compound logical
propositions, consisting of primitive propositions formed according to
the **Predicate (object)** scheme; a predicate may be conceived as a
class of phenomena ; 6) an object to which this predicate is attributed
corresponds then to an instantiation of that class; 7) finally, the objects
and predicates perceived by the agent depend upon the physical stimuli
received from the external (or internal) situation of the agent, i.e. by
its environment.

Each of these units can be interpreted as a particular type of distinction : 1) the distinction between survival (i.e. maintenance of the identity or self-environment boundary, cfr. Heylighen, 1988) and destruction of the agent; 2) the distinction between "better" situations and "worse" situations. In the General Problem Solver (GPS; Newell & Simon, 1972) these distinctions are called "differences" between goal and non-goal states. The number of such "differences" can then be used as a basic evaluation criterion for states, allowing "hill-climbing" search heuristics; 3) the distinction between the situation before and after the operator has been applied. Clearly, if these situations cannot be distinguished, the operator is meaningless. In GPS these "distinctions" are coupled to the previous ones by a matrix, connecting a list of operators to a list of differences by specifying whether a particular operators is able to reduce a particular difference between the initial state and the state to be attained; 4) propositions describing potential situations form a Boolean algebra which can be interpreted as an algebra of distinctions (Spencer-Brown, 1969). The basic distinction here is that between a proposition and its negation; 5) as we already pointed out, a predicate corresponds to a class, and a class is the result of a distinction between phenomena; 6) an object on the other hand, arises when a stable "form" or "system" is distinguished from its "background" or "environment"; 7) sensory stimuli, finally, are the result of a differential excitation of elmentary receptors (e.g. nerve cells), creating a distinction between "activated" and "non-activated" receptors.

**Figure 1 : **hierarchy of representation levels and their corresponding
distinction levels

**3. The Representation of Change.**

The simplest way to represent the changes which are to be brought about in order to solve a problem is by keeping all distinctions invariant, and varying only the indications. An indication should be conceived as that part (of all the parts of the universe of experience which are distinguished) that is "indicated", "activated" or "actualized". For example, an intelligent agent can conceive a large number (possibly infinite) of situations, internally represented as "problem states", but at a specific instant in time only one of these situations will be "actual". All the other ones are "potential", they are not yet indicated. An evolution in time, i.e. a sequence of actual situations, will then be represented as a sequence of indications, actualizing one state and then moving on to the next one (Heylighen, 1987a).

During such an evolution only "states" change, all "structures" are
conserved. This means that the set of objects, the set of predicates, the
state space, the set of operators, the goals, values or heuristics, ...,
all remain invariant. A representation functioning according to this scheme
may be called *classical* (Heylighen, 1987a,b). A prototypical example
of such a representation is the physical theory of classical mechanics.
It is characterized by properties such as determinism, rationality, reversibility,
absoluteness of space and time, causality, sequentiality, ... On the level
(4) of states, the requirement of distinction conservation signifies that
an allowed operator will necessarily map distinct states ("causes") onto
distinct states ("effects"), and "equal" states (i.e. states which are
not distinguished, which are assimilated) onto "equal" states.

This requirement is clearly not satisfied by the paradigm of search, used for describing heuristic problem-solving. Indeed, it is assumed that in general there is no determined function or algorithm, leading deterministically from the initial state to the goal state. Distinct states have to be explored starting from the same initial state, and distinct initial states can lead to the same goal state. In this sense search representations are similar to the physical representations used in thermodynamics, where state trajectories with distinct starting points may come together (equifinality), or where a trajectory starting from the same initial state can branch, leading to distinct end points (bifurcation)(cfr. Heylighen,1987a).

The evolution of states in the search representation can be summarized
by the *generate-and-test* principle : distinct states are generated
from an initial state ; the one (or more) states which are evaluated positively
after a test, are retained and used as starting points for a subsequent
exploration step. This principle is equivalent to the principles of *trial-and-error*
and *variation-and-selection*. In each case there is a first phase
of variation, i.e. the generation of a variety of new, distinct states
("trials"), followed by a second phase of selective retention, during which
all states that are evaluated negatively ("errors") are eliminated. Such
a principle could be viewed as a general description of processes in which
certain distinctions are destroyed (because distinct initial states can
generate the same variations) while other distinctions are generated (because
the same initial state can generate distinct, selectively retained variations).

However, although a search representation is non-classical (i.e. non-distinction- conserving) at the level of states, it is still classical at all other levels : during a heuristic search process all objects, predicates, operators, heuristics, and goals, determining the representation structure, are kept invariant. Yet there exist representations which are non-classical at those other levels. For example, the theory of quantum mechanics is characterized by the non-conservation of distinctions at the levels of predicates (Heylighen, 1987a), whereas relativistic quantum field theories describe the non-conservation of object distinctions (annihilation and creation of elementary particles).

The representation of all possible problem-formulation processes we are looking for, however, would be characterized by the possibility of creating or eliminating distinctions at all levels of a problem representation. Indeed, in order to solve an arbitrary ill-structured problem, we should be able to generate a completely new system of distinctions, containing as well goals, operators, problem-states, predicates as objects. Hence we must generalize the variation-selection dynamics in order to describe change at all levels of a representation.

**4. Towards a Dynamics of Distinctions.**

Suppose now that we would have a workable generate-and-test scheme for constructing and ameliorating representations. Then we could apply this scheme to itself in order to make it more efficient. This is a peculiar characteristic of metarepresentations : a metarepresentation allows to manipulate representations, yet it is itself a representation ; hence it can manipulate itself (cfr. Pitrat, 1986; Lenat, 1983; Newell, Shaw & Simon, 1960). This argument can also be used to explain why it is meaningless to look for a meta-metarepresentation, a meta-meta- metarepresentation, etc. Indeed, such higher level representations could only be used to reason about representations (from whatever level) and hence would not be in any way more powerful than the (second level) metarepresentation. By using this self-representing feature of metarepresentations we can now speed up the process of developing a metarepresentation through a bootstrapping procedure.

Indeed, one way to approach the construction of a metarepresentation,
i.e. a representation of all possible ways of generating a problem representation,
is by applying the general analysis of representation systems we have made
(fig. 1) at the metalevel. The elements of the lowest level of our metarepresentation,
the *meta-objects*, correspond to distinctions. The next level, the
*meta-predicates*, is formed by the different classes according to
which the distinctions can be categorized : the class of goal distinctions,
the class of operator distinctions, ... , and by the (as yet not clearly
defined) relations between distinctions belonging to these classes. The
*meta-states* correspond then to systems of distinctions, structured
by the meta-predicates. For the *meta-operators*, we need production
rules which would be able to generate new structured sets of distinctions.
The highest level of the metarepresentation, the *meta-goals* and
*meta-values*, would then correspond to criteria for evaluating whether
a particular meta-state, i.e. a distinction system, would be more or less
adequate for representing a particular problem domain, which is defined
by the relation between the autonomous agent and its environmental situation.

This description provides us with a first, relatively well-defined formulation of the problem of problem-formulation. However, in order to make the problem really well-structured, so that its resolution can be approached by simple search, we need a further explicitation of the different meta-structures. The least explicit levels until now are the two highest levels, which determine the dynamics of distinction systems.

In our definition of a problem representation, we have assumed that
a problem is determined by two factors external to the representation itself
: the autonomous agent, and the situation as he experiences it through
his sense organs. This means that two sets of distinctions are given a
priori : the survival-destruction distinction for the agent, and the set
of distinguished stimuli (fig. 1). This last set is normally very large,
and can be used as a basis for generating new distinctions by the combination
(intersection, union, complement) of classes of stimuli. Roughly speaking,
if the set of distinguishable stimuli has cardinal number N, the set of
distinctions which can be generated corresponds to the power set of the
set of stimuli, and thus has cardinal number *2 ^{N}*. This
number is so large that for all practical purposes it can be considered
to be infinite. (in fact, if we consider operator distinctions as functions
mapping classes of stimuli to classes of stimuli, then the number of possible
operators is even larger).

It is clear then that the variety of potential distinctions to
be generated is extremely large. The difficulty resides in the selection
of the most adequate distinctions. Ultimately this adequacy is determined
by the capability of the distinction system to ensure the survival of the
agent, i.e. to ensure that the agent will not be eliminated by *natural
selection*. Hence there is an a priori selection criterion, determined
by the relation between the agent and its environment. This criterion is
based on the stability, homeostasis, capability of counteracting destructive
perturbations, ... , of the agent. However, this criterion can only be
applied in a very indirect way to the selection of representational distinctions.

We hence need "mediating" criteria, which would connect the representational
distinctions to the survival-destruction and physical stimuli distinctions.
One way to define such a "vicarious" selection criterion (cfr. Campbell,
1974), is by replacing *stability with respect to the physical environment
*by *stability with respect to the representational environment of
already given distinctions*. A distinction could be considered stable
in this sense if it would be conserved by all representational operators,
functions or relations, i.e. if it would be *continuously* connected
to the other distinctions (continuity can indeed be defined as the conservation
of topological distinctions, cfr. Heylighen, 1987a).

This mediating selection can be further structured by arranging the vicarious selectors in a hierarchy (cfr. Campbell, 1974), related to the distinction hierarchy (fig. 1), so that each potential new distinction would be selected first by the distinctions of the two contiguous levels. For example, a new goal distinction would be positively selected if : a) if reaching this goal would imply survival (level above); b) if this goal could be reached by a sequence of already given operators (level below). A predicate distinction would be selected if : a) it would allow to distinguish already distinct objects (level below); b) if it would allow to generate existing state propositions. A state distinction would be selected if : a) it would be the result of applying an existing operator to an already existing state; b) if it would correspond to a new combination of existing predicates. It is clear that these selection criteria have to be worked out further, and in particular that their mutual relation has to be specified.

**5. A Transdisciplinary Research Project.**

To tackle such a broad problem like the development of a metarepresentation it seems advisable to integrate methods from different disciplines, thus defining a transdisciplinary research project : 1) a conceptual-philosophical approach; 2) a formal-computational approach; 3) an experimental-psychological approach. (Such a project, involving researchers from different disciplines, is to be started at the Free University of Brussels (VUB). Although the present paper attempts to develop a general conceptual framework for this project, the views espoused here engage only the author ; they do not represent the group as a whole.)

The first approach would involve a theoretical analysis of the different concepts related to representation development : e.g. agent, situation, goal, object, ... This analysis could be based on existing theories, such as systems theory, theories of self-organization, epistemology, psychological theories of cognitive development, artificial intelligence models of learning and discovery, cognitive modelling, frameworks for representation in logic, mathematics and physics, ... The result of this analysis would be to provide fundamental definitions of the concepts in terms of more primitive concepts (e.g. the concept of distinction). For example, a proposition, or Boolean distinction, might be defined as a distinction which obeys the two axioms proposed by Spencer-Brown (1969) (cfr. sect. 2). The procedural interpretation of these (declarative) definitions would correspond to a program for constructing a more complex concept (e.g. an algebra of propositions) out of more primitive concepts.

This brings us to the second approach : the set of definitions can be
read as a formal system, consisting of "facts" and "rules". (A provisional
formalism for the deduction of classical and non-classical representations
from a general distinction-based framework can be found in Heylighen, 1987a).
These facts and rules could then be translated in a logic programming language
(e.g. PROLOG), and implemented on a computer. The resulting program would
correspond to an expert system, or rather to a *meta-expert system*,
which would allow to infer higher-order distinction systems (i.e. representations)
from ill-structured sets of primitive distinctions. The previous approach
could hereby be considered as the preliminary phase of "knowledge acquisition"
or "cognitive systems analysis", which collects and structures the "expert
knowledge" to be implemented. A possible application of such a program
might be the processing of existing expert systems, where a relatively
unstructured list of facts (predicative propositions) and rules would be
given as input, and where the output would consist of a well-organized,
transparent and efficient, quasi-classical representation.

The function of the third approach would be to provide impulses and feedback for the general project, by comparing the theoretical-computational models with observations of real-world phenomena. In order to achieve this, a psychological experiment would be set up, in which experimental subjects would be asked to tackle an ill-structured problem (e.g. a simulation of the administration of a town (cfr. Dörner et al., 1983) or the management of a third world country (cfr. Dörner & Reither, 1978)). They would be provided with a set of primitive distinctions determining the perceived data ("stimuli"), and the fundamental distinction to be maintained ("survival", e.g. of the town organization). Through direct observation, protocol analysis, Kelly grid techniques and interviews the observer would study how the subjects would structure the problem, which goals, operators, states, predicates, objects, ... they would distinguish, and how these representations would evolve under the influence of learning, feedback, and reflection. This process and its results could then be compared with the expectations derived from the theoretical and computer models, allowing to correct them if necessary.

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