Theoria et Historia Scientiarum
VI-2 (2002), 15-37
Knowledge in Memory Evolutive Systems
by
Andrée C. Ehresmann and Jean-Paul Vanbremeersch
Faculté de Mathématique et Informatique
33 rue Saint-Leu 80039 Amiens. France
ehres@u-picardie.fr
1.
Introduction
Let us give the following examples:
1.
A spring keeps the 'memory' of its shape; when it is pulled, it elongates but afterwards
it comes back to its initial shape.
2.
A thermostat measures the temperature and 'knows' at which temperature it must
stop the heating.
3.
An expert system has knowledge with respect to a particular domain, and can answer
questions on this domain.
4.
A robot has sensorial organs to recognize some features of its environment, and
several strategies (built-in or learnt) to react in an appropriate manner.
5.
Bacteria have a metabolic activity, reproduce and are able to repair damaged
DNA during the replication. These activities are autonomously controlled, using
the genetic program.
6.
A population of bacteria can adapt to changes in its environment (resistance to
antibiotics), thanks to natural selection which favors bacteria with an appropriate mutation.
7.
An animal with a nervous system gathers information on its environment by its
sensorial organs, and on its internal state (hunger, pain,...), and reacts by
innate (holding reflex of a baby) or learnt behaviors.
8. Higher animals develop a semantics, generating a primary consciousness which may modulate their action depending on several parameters. And they exchange information through communication (alarm sounds, education of the youngsters, cultural differences among communities of chimpanzees (Whiten et al., 1999)).
9.
Human language leads to more efficient representations and communication; conceptual
knowledge is developed and transmitted, generating culture.
a) Several types of knowledge
1. What is the common basis of these examples? Each
of them exemplifies some kind of knowledge, in so far as this term is accepted
with a large meaning covering the opposite pairs:
·
cognition
relying on natural processes of treatment of information (physical, biological,
social, …) / intentional knowledge emerging from it,
·
innate
(built in or inherited) / learnt,
·
pragmatic
(know-how, skills,…) / conceptual,
·
automatic
/ deliberate,
·
comprehensive
/ specialized,
·
attributed
to the agent by an external observer / explicit for the agent,
·
distributed
between several agents / controlled by a unique agent.
2. More explicitly: some knowledge is inherent to
the system, either if it has been implemented in it by construction (examples
1, 2, 3, partially 4) or, for a living organism, if it is inherited or the consequence
of natural selection (examples 5, 6). On the contrary, it can be acquired by
learning, education or culture (examples 7, 8, 9).
In examples 4, 5, 7, the behavior depends on
practical knowledge (built-in, innate or learnt strategy), to react in an appropriate
manner to some external situations. It remains implicit for the agent, to which
it is attributed only by an external observer. In the examples 8, 9 (and partially
in examples 4 and 7) the agent controls intentionally part of its behavior,
though some part remains hidden. We know how to pick up an object, but without
knowing how our muscles do it.
In examples 8 and 9, there is also some conceptual
knowledge, relying on a semantics; an intentional agent develops it combining
simpler knowledge by several means, up to logical operations. For man, language
helps the agent to interpret its own knowledge. It is communicated by
imitation, education, directly or through some material support (book, file,
CD,…).
A particular agent has its own more or less specialized knowledge (examples 3, 4, 5, 7, 8), but comprehensive knowledge is generally distributed among the members of a group (examples 5, 8, 9), each one having only a partial access to it (no mathematician can grasp the whole of mathematics). Distributed knowledge can remain implicit for the group, such as the usual unconscious social comportment analyzed by Goffman (1973).
b) Formation and interpretation of knowledge
1. The knowledge of a (not necessarily living)
system consists of internal representations in concrete relation with some
features of the environment or some activities. They are innate, or constructed
to memorize information received by the system under the form of an internal
change (strengthening of a synapse, change of probability between various
conducts, sign, record, ...). The information will be retained only if it is
stable enough (e.g., repeated several times) or significant: we continuously
get new sensory information on the objects around us, but we discard most of
them. Learning transforms a perceptual configuration more or less briefly
activated by an external event into a stable internal representation; this
'record' takes on its own identity by consolidation, and may be recalled later
on by the same or by an approximate perceptual configuration.
2. Using approximate knowledge can help to react
quickly to various situations, and later lead to a finer adaptation to them
through the change of some parameters.
But this plasticity is counteracted by the risk of
errors coming from insufficient information, an inadequate analysis of the
situation (optical illusions), or modifications in the context. For example, in
classical (Pavlov, 1927) or operant (Skinner 1938) conditioning, the conditions
of the stimulus/response experiment can be changed, confusing the animal. For
conceptual knowledge, errors can also result from a wrong interpretation of the
concepts used; most errors done by students in Mathematics are of this type:
they interpret a mathematical concept (say, a derivative) from a curtailed
representation such as a learnt formula which has not been integrated as an
object on which other processes can operate.
3. Memorized data will be called 'knowledge' only if they are interpreted as such either by the 'knowing' agent or by an external observer able to attribute this knowledge to the agent. Thus knowledge is a ternary relation: knowledge of something attributed to some agent by an interpreter (possibly the knowing agent itself). Here agent and interpreter can be living organisms, groups or machines.
The attribution of knowledge to the agent by the
(internal or external) interpreter is based on:
·
direct
observation of the behavior,
·
(partial)
reading of the memory of the agent,
·
material
traces produced by the agent (books, files, CD,…),
·
inquiry
to the agent (second degree: the agent must already interpret his own
knowledge).
In each case the attribution can be false since the
interpreter has only an external and partial view of the agent's memory (even
if it is a conscious agent itself). For instance, the principle of charity
(Quine 1960, Dennett 1990) attributes an action to a rational comportment; but
the agent can act for other reasons unknown to the interpreter, or voluntarily
induce him into error. Two different interpreters (e.g., one being the agent
itself) can attribute different knowledge to the same agent: a teacher can
judge that a pupil does not know the lesson which the pupil thought he knew.
Moreover the interpretation always depends on a
specific context, because knowledge evolves. For instance, ancient Greeks
'knew' that the earth is flat, though it is not. Even knowledge relying on a
consensual authority and obtained by accurate methods, say the scientific
method, can be falsified later on; think of the changes of paradigms in science
(Kuhn, 1972), or the different perception of past history depending on the context.
c) Memory Evolutive Systems
1. For a system to have knowledge, it must at least
have some of the following capacities:
·
to
gather information via modules linked to the exterior (receptors), to respond
by adapted strategies (effectors), and to evaluate the result of these
strategies, at least locally (locate fractures in some modules),
·
to
record information which are stable enough or repeat themselves, as well as its
strategies and their result, so that it develops a memory formed by their
internal representations and can modify it to account for its successive experiences,
·
to
recall records in the memory, connect them and operate on them, e.g. to learn
new strategies formed by combining already known strategies,
·
to
interpret its knowledge, or to be observed by an external interpreter,
·
possibly
to classify the known objects according to a semantics; this allows for
conceptual knowledge and its development by logical processes (disjunction,
conjunction, negation), by induction (generalization), deduction (formation of
chains of links, as in mathematical proofs), or abduction (find the causes of a
situation),
·
to
communicate with other systems (e.g. members of a same group) to develop common
knowledge, up to culture.
2. A model of such systems will be given by the Memory Evolutive Systems (MES) which we
have developed since 15 years; it is a mathematical model, based on the theory
of categories introduced by Eilenberg and Mac Lane in 1945 (cf. Mac Lane 1970
for the main definitions). In Section 2, we describe this model in a concrete
setting, and refer to our preceding papers (a list is given in our Internet
site, Ehresmann & Vanbremeersch 1999) for a more theoretical approach,
which is only alluded to here (in paragraphs beginning by #). Section 3
analyses the processes of acquisition, consolidation and cohesion of knowledge,
distinguishing the different types of knowledge. In Section 4, we show how more complex
systems may develop a conceptual knowledge and interpret it, leading to its
diffusion on a larger scale.
2. Organization and functioning
of a system with knowledge
Here we describe the organization and
functioning of a system in which knowledge plays a role. The system can be a
living or artificial organism, a neural system, an animal group (e.g. a hive),
a human society,… It will be modeled by a Memory Evolutive System (MES).
a) Description of the system
1. The system has a given timescale, from its 'birth' to its 'death'.
Its state at a time t of its life is determined by its present organization consisting
in:
·
Its
components at this time, among which we distinguish: (i) its agents which are its individual
constituents (neurons, bees, members) and also more or less complex
associations of these constituents participating in common activities (visual
areas in a neural system, the class of workers in a hive, departments of an
enterprise); (ii) objects constituting its memory, also called records; they are internal
representations of the knowledge of the system, which encompasses known
features of the environment, strategies to deal with at any level (from
metabolic regulations up to complex skills), possible conceptual knowledge and
its material support; (iii) more or less temporary objects, acting as
information, which are the internal traces of the signals received by the
system from the environment at t.
·
Relationships (called links) between these components which allow the implicit or explicit
transfers of data, energy, constraints between agents, the reception of
information, its possible recognition by recall from the memory or its later
storage, and commands of adapted strategies to effectors. Each link operates
with a specific delay of propagation.
·
Successive
links can be composed, and the composites of two paths of links are identified
if they are functionally equivalent.
The agents, their knowledge, the information they
receive and their interactions change in time. The change is measured by the transition from the state at t to the state at a later time t',
which indicates which of the components and links existing at t are still there and what they have
become at t', as we could recognize a
particular member of a group on two successive photographs of the group. Thus a
component of the system, say an agent N, is not represented by a unique
invariant object (as in usual models), but by the sequence (Nt) of its successive states
all along its life. And the same for the links between components.
# We have so modeled the system by an Evolutive System (Ehresmann & Vanbremeersch
1987), which is defined by:
- a (finite or infinite) part of the
real numbers R representing its timescale,
- for each time t, a category1 Kt,
the state-category at t,
- for each time t' > t, a functor2
k(t,t')
(the transition from t to t')
from a sub-category of Kt to
Kt', these transitions being
transitive, i.e.,
if k(t,t')(Nt) = Nt'
is defined and t' < t", then k(t',t")(Nt') is defined iff k(t,t")(Nt) is defined, and then both
are equal.
2. All the components are not of the same complexity
level. An agent which is an association of other agents (a department of an
enterprise) is more 'complex' than these agents (its employees), and it can
itself be one of the constituents of a more complex agent (division regrouping
several departments). In the memory, a complex skill is formed by coordinating
together more elementary skills. Even a cell has a whole hierarchy of lower
level components from its atoms up (Chandler 1997).
Thus the system has a hierarchical structure: a
component N of a given level is obtained by binding together a pattern formed
by components of the next lower level with some distinguished links between
them; such a decomposition pattern represents a (not necessarily unique) internal
organization of N.
The links between two components N and N' can be
'simple' in the sense that they are obtained by binding a compatible family of
links (or 'cluster') between the lower level components of N and N'. But there
are also 'complex' links which emerge at higher levels. In Section 3, we'll
come back on the genesis of these links and their role in the interconnection
of all knowledge.
# The Evolutive
System is hierarchical, in the sense
(Ehresmann & Vanbremeersch 1987) that its components are divided into
several complexity levels, with a component of level n+1 being the colimit of a pattern3 (or inductive
limit of a diagram in the sense of Kan 1958) of linked components of level n.
And it has a
hierarchical sub-system, forming its memory.
3. The changes are essentially
attributable to what Thom (1988) calls the archetypal operations: "birth,
death, scission, collusion". Some agents will disappear while others
arrive. Some signals will be memorized and become knowledge, while others will
be discarded. New complex components are formed: agents can assemble to form a
new group having some competence, more complex skills are learnt,… Conversely,
a sub-group can dissolve.
# The transitions between the
state-categories are constructed by the process complexification with respect to a strategy (Ehresmann &
Vanbremeersch 1987). A strategy on Kt consists of a set of external elements
A to add, a set of patterns P to bind together, a set of more or less complex
components B to suppress. The complexification is a new category which is
explicitly constructed (as a special case of the construction of the prototype
of a sketch given by A. & C. Ehresmann in 1972):
- Its objects are: those of Kt except the B's, the added elements A,
and, for each pattern P to bind, a new higher level object CP which becomes its
colimit.
- There are two kinds of links from CP
to a CP'. The (P,P')-simple links
bind clusters from P to P'. A cluster
is a maximal family of links from the objects Pi of P to those of P' satisfying the condition: there is
at least one link from each Pi
to some object of P', and if there are several they are correlated by a zig-zag
of links in P'.
- The complex links from P to P' are obtained by composing a sequence of
simple links binding non-adjacent clusters.
b) Local
regulations
1. The preceding description is purely
formal and 'external': it could be given only by an observer with a complete
view of the system, and that cannot exist for complex enough systems, specially
autonomous systems (Matsuno 1989). In particular knowledge is distributed among
the agents; it is constructed through their combined action and later on takes
its own identity. For instance, the bees of a hive 'know' how to construct the
hive, but each one participates in a very fragmentary way to this process,
thanks to some instinctive strategies; the 'construction of the hive', which is
a consequence of the temporal combination of all these strategies, figures
among the knowledge of the system, but it remains hidden to the bees and can be
attributed to their society only by an external observer.
2. The system is autonomous in the sense
that it is internally controlled by the agents. While some agents cooperate,
there are also competitions or even conflicts between them. We call a CoRegulator (CR) a subsystem formed by a
small pattern of agents of the same complexity level (possibly forming a higher
level agent), acting together at a specific discrete timescale (e.g., an operon
in the genome of bacteria). The global dynamics is modulated by the competition
between a net of CRs which operate in parallel, but with different rhythms;
e.g., in an industry, a workshop has a daily cycle, while design departments
can plan over several years.
With respect to knowledge, each CR has a
partial and differential access to the collective memory which it contributes
to develop by operating a stepwise trial-and-error learning process at its own
timescale.
# The system is thus modeled by a Memory Evolutive System (MES): it is a
hierarchical evolutive system, with a hierarchical sub-system called the memory, and a net of evolutive
sub-systems with discrete timescales, its CRs
(Ehresmann & Vanbremeersch, 1991).
3. One step of a
particular CR extends between two successive dates of its timescale; it is divided
in several more or less overlapping phases (forming an epistemo-praxeological
loop in the sense of Vallée 1995), which we illustrate by a meeting of the
editorial board of a Journal:
·
In
the first phase (or actual present),
formation of the actual landscape of
the CR, which is a (more or less distorted) internal representation of the
system for the CR; it filters the partial information received by the agents
during their actual present, and plays the part of a working memory during the
step. For instance, the editors will register the various papers submitted
since their last meeting, the referees' reports on papers formerly received,
the letters sent by readers, and control if the decisions taken at their
preceding meeting have been correctly carried out.
·
Selection
on the landscape of a strategy to react in an adapted way to the context, to memorize
the stable enough information, possibly combine them into new knowledge and/or
transmit them; the choice is supported by the recall from the memory of former
similar situations. The board will select the papers to be published in the
next issue of the Journal, the referees for newly received papers, the intended
schedules; for this, they will recall preceding reactions of readers, which
referees have previously done a good job, and former delays of the printer. The
strategy can be really 'chosen' by the CR (intentional action), or imposed on
it by other CRs (e.g., the direction of the Journal), by external constraints
(excessive cost for too long papers), or represent a known 'automatic' answer
to the given situation (always the same referee for some kind of papers).
·
Commands
to effectors to realize the strategy:
the papers for the next issue are sent to the printer, the new ones to
the chosen referees. The landscape gives only a partial and a more or less
flawed view of the system and the various CRs may conflict, so the objectives
of the strategy are not necessarily fulfilled. The step can even be interrupted
by a fracture if no strategy can be
found (the editors cannot agree on which paper to publish), or if the selected
strategy cannot be effected (the printer refuses to continue printing the
Journal). Such a fracture may reveal a lack of internal coherence in the
knowledge of the system, or a wrong correspondence with 'Reality' (whence the
problem of Truth…).
·
If
the step processes without a fracture, at the next step, the result is
evaluated by comparing the anticipated landscape with the newly obtained
landscape; and the strategy is memorized with its result. At their next
meeting, the editors will verify if the issue is well printed and the reports
of referees received. They will note if the printing delays have been respected
and no complaint has arrived.
#
The actual landscape L at t is the category whose objects are the
perspectives for the agents of the CR of the components B of the system of a
near complexity level; a perspective
of B is a cluster of links from B to the pattern formed by the CR during its
actual present. There is a distortion
functor from L to the system. The anticipated landscape for the end of the step
should be the complexification of L with respect to the selected strategy, and
it is compared to the actual landscape effectively obtained at the next step by
a comparison functor.
c) Global dynamics
1. The strategies selected by the various CRs at a
given time are not realized on their landscapes but relayed to the system where
they are not always compatible. Indeed, the CRs share common resources, have
differing perspectives on the distributed knowledge and there are direct and
indirect interactions between them, so that conflicts may occur between their
strategies. The editorial policy may clash with the economic constraints of the
publishers of the Journal.
An equilibration process will arise between the
strategies, called the interplay among
the strategies of the various CRs. It is not a centrally controlled
process, but a dynamical modulation between the different relayed strategies.
It depends on the respective 'weights' of the strategies and of the CRs (the
editors can argue with the publishers). A main role is played by the structural
temporal constraints of the CR: a paper or report not received at the date of
the meeting cannot be examined, the printing of the issue can be delayed if the
printer has too many other works to do.
If the constraints of a CR cannot be satisfied, a
fracture occurs in its landscape and, if it is not quickly repaired, there is a
dyschrony: the regular publication schedule cannot be resumed before several
issues. However fractures can have a creative role, by imposing a complete
overview of the situation. If the printer cannot respect the delays, a new
printer can be chosen, perhaps making also a better job.
# The structural temporal constraints of a CR at t connect its period (mean length of a step) d(t) to the mean propagation delays u(t) of the information it receives and to the mean stability spans
(cf. section 3, b) v(t) of the
components intervening in its landscape and the strategy: For almost all t (i.e.
except on a set of measure 0), we must have:
u(t) << d(t) << v(t).
2. Thus a dialectics via functional loops is
generated between CRs which are heterogeneous
with respect to their complexity level and/or their period. A series of
fast changes by lower CRs is only perceived as a whole and with a delay by a
higher CR, and may cause a fracture in it; to repair the fracture this CR may
impose new strategies on the lower CRs, and the process goes on. If the editors
progressively modify the contents of the Journal, the publishers will perceive
the change only after a delay, but then it can displease them, and they may
react by dismissing some editors.
This dialectics shapes the evolution of the system
and differentiates it from 'simple' physical systems (Rosen 1986). It leads to
the formation of more and more complex strategies and to the development of a
coherent corpus of knowledge. For instance, it explains how culture can be
transmitted, or temporally refrained, and how it affects the comportment of the
individuals who receive it.
Let us give a personal example of development of a
mathematical theory illustrating this dialectics. In the late sixties, a small
group of young research students working with Charles and Andrée Ehresmann has
developed the theory of sketches; they had frequent mutual exchanges, so that
they had adopted particular concepts and even notations. But they had almost no
contacts with the main stream of categoricians, in particular in the States,
and their work, not well published,
remained unknown. At the first conference in 1970 where their results were
exposed, the 'establishment' could not understand them, because they were far
from the current problems (topos and triple theory) and the notations were
unusual. This cold reception caused a fracture to the young students. But it
had also a beneficial effect: contacts were established, specially thanks to
the organization of "Journées Théorie et Application des Catégories"
in Paris and Amiens, and of international conferences in Amiens in 1973, 1975
and 1980. These meetings allowed to harmonize the notations and better explain
the motivations. The consequence has been a diffusion of the theory of
sketches, which has been widely adopted in the eighties, with important
applications in Computer Science (Barr & Wells 1984, Gray 1989, Walters
1991).
3. Development and plasticity of knowledge
The knowledge of the system is represented by the
content of its memory. It is based on a kernel of innate knowledge which is
later developed by learning. Its role is essential to recognize objects or
configurations already met and to respond in a more adapted way.
a) Acquisition of knowledge
One of the objectives of the system will be the
formation of records and links to memorize new situations and strategies. It
will be achieved through the strategies of the CRs and the interplay among
them.
1. A new
configuration C met in the environment (say, an unknown object) or produced by
the system (e.g., commands of a new strategy) is internally represented by the
temporal coordination of a pattern P of components; its objects can be receptor
agents, or components linked to them such as records in the memory recognizing
parts of C. The configuration will be memorized by the formation of a new
object of the memory, called the record
of C, which integrates the pattern in a higher level unit by strengthening its
links. For this, each CR will memorize the attributes of P which it may
distinguish, and the interplay among their strategies conjugates their actions
to form the record M. A later occurrence of C will reactivate the strengthened
pattern P (which we call a decomposition of
M), thus leading to the recall of M and the recognition of C.
For instance, let us analyze how an orchestra will
learn a new partition C. Each player acts as a particular CR. A pianist E sorts
out the part for piano, which he translates in his actual landscape into a
sequence of notes and tunes to be integrated as a unit in his own memory, called
the E-record of C. Independently the other players learn their part. During the
repetitions of the orchestra, the interplay among the (strategies of) the
players will synchronize and harmonize their parts, and integrate them into an
object M of the collective memory of the orchestra, which we call the record of C; the various E-records of C
become a reflection of this global record. If the orchestra takes back the
partition later on, each player will easily recall his part by reactivating his
E-record, and they'll have no problem synchronizing their parts to reactivate
the global record.
# A new
configuration C is internally represented by the formation of a pattern P of
linked components synchronously activated. A particular CR, say E, will
perceive in its actual landscape a pattern pE
of perspectives coming from a sub-pattern (possibly void) PE of P.
An objective of the strategy of E will be to memorize this pattern pE. This will be reflected to
the system into the command to bind together the pattern PE (image
of pE by the distortion
functor). Simultaneously, other CRs form their own record of C. The interplay
among the strategies will integrate the various commands, so that the
complexification process with respect to the global strategy thus obtained will
add a colimit M of P, called the record M
of C; its perspective in the landscape of E becomes a colimit of pE, called the E-record of C. A later presentation of C
reactivates P, hence also its different E-records which, by the interplay among
strategies, are binded together, thus the recall of the record M.
2. The formation of the record of C corresponds to
its "assimilation" in the sense of Piaget (1940). Afterwards it will
be consolidated and adapted to gradual enough temporal modifications of the system
and of the environment (Piaget's "accommodation"). In this way, the
record takes its own identity as a component of the system, with its successive
states becoming more and more independent from the particular decomposition
used in its formation.
For instance, the replacement of one or two players
will not prevent the orchestra to play C, with minor variations of its record;
a law can be revised to keep track of progressive changes in a society; a
scientific theory can be adapted to new facts, before a change of paradigm
(Kuhn 1972). The rate of change is measured by the stability span of the
record; during stability periods, the span is long, while it is shorter during
periods of development or of decline.
# If M has been constructed at t as the colimit of a pattern P of lower
level components, the evolutions of M and of P may remain correlated during a
certain period, but diverge at a time t'
> t, so that the state of M at t'
is no more the colimit of the new state of the pattern P (e.g., if objects of P
have been suppressed or replaced). The stability
span of M at t is defined as the
larger real dt such that there exists
a pattern Q of lower level components whose state, for each time s between t and t+dt, has for colimit
the state of M at s.
3. The development of the memory does
not only consist in the formation of records, but also in the formation and
strengthening of links between them, thus increasing the cohesion of knowledge.
Let us mention some of these links.
When a configuration C is memorized, the
links between the objects Pi
of the internal pattern P it activates are strengthened; and new links are
formed from each Pi to the
record M of C (heredity links). Later M may participate to the formation of the
record of a more complex configuration having C as a constituent, and thus
become linked to this new record.
A link can also be formed from the
record M to (the record of) a strategy in response to C; e.g., classical or
operant conditioning (Pavlov 1927, Skinner 1938) creates such links between a
stimulus C (the sound of a bell) and a conditioned response (the dog
salivates). A later presentation of C recalls M, and, if the link is strong
enough, it will lead to the automatic recall of the strategy. This is done, as
above, via the different CRs and the interplay among their strategies, with a
risk of fracture if some structural temporal constraints cannot be respected.
For instance, the view of a prey has no
effect for a satiated animal; but if he is hungry, he'll try to catch it. The
catching strategy requires a coordination between the visual CRs which determine
the location and size of the prey, and the motor CRs which control the motion
of the predator. If the prey runs too fast, or in an erratic manner, the visual
information on its location arrive too late to the motor CRs to adjust the
movement, and the prey flees.
# The memory is an evolutive sub-system
of the MES; its transitions correspond to complexification processes with
respect to strategies whose objectives are the formation of new records. The
construction of a complexification explicitly determines which links are formed
between the added records (cf. Section 2, a).
b)
Plasticity of knowledge
1. We have seen how a record M is consolidated to
adapt to small temporal variations of the context. But there is a more
comprehensive kind of consolidation, by extension of its domain of application.
In particular, if a strategy has been successful in
some situation and if a near enough situation arises, the different CRs will
try to use the same strategy. If it succeeds, its domain is extended; if it
fails and causes a fracture to some CRs, the fracture will be repaired by modifying
the corresponding sub-processes to adapt it to the new context. For instance, a
dog learns to bring back a particular ball sent by his master in the garden;
later on, he will use the same strategy to bring back another object sent by
someone else in an other place.
A record adapted to several contexts will be called multifold; it will be recalled by the activation
of a specific decomposition P in each context; such a decomposition can be
thought as the fixation of the values of some parameters. As we will see, the
fact that records can be(come) multifold explains the development of complex
relationships between them, leading to an interaction of all knowledge.
# In a MES, a component M which has been formed as
the colimit of a pattern P may also be, or later become, the colimit of
patterns Q non equivalent to P (Multiplicity Principle, Ehresmann &
Vanbremeersch 1996); we then speak of a multifold component, and the passage
from P to Q is called a complex switch.
2. There are two kinds of links which
are formed between records: the simple links and the complex links. For
instance, if M is a text written in French and M' its English translation, we
have a simple link 'translation' from M to M' associating to each sentence in M
its translation in M'. But the link between a sentence in M and its translation
in M' will generally not be the simple translation of each word, because of the
differing structures of the two languages (whence the difficulty of an
automatic translation!).
More generally let M and M' be two
records. If P is a decomposition of M and P' a decomposition of M', a (P,P')-simple link from M to M' binds together
a cluster of links between the objects of P and P'.
By composing a chain a simple links, we
get a link, but it is not necessarily simple. Indeed, if M' is multifold, it
can be recalled in another context through a decomposition Q' non-equivalent to
P' (corresponding to a different choice of parameters). Then if we compose a
(P,P')-simple link from M to M' with a (Q',P")-simple link from M' to a
record M", the ensuing link from M to M" may not be
(P,P")-simple; we then say that it is a complex link.
A mathematical example illustrates the
difference between simple and complex links. A topological space is the
geometric realization of several simplicial complexes. If P and P' are two
simplicial complexes associated to the topological spaces T and T', a
(P,P')-simple link from T to T' is reduced to a simplicial map from P to P';
but a complex link from T to T' is any continuous function.
# The development of the memory comes
from a sequence of complexifications. Each one introduces simple links and
complex links (cf. Section 2, a).
c) Cohesion and complexification of knowledge
The existence of complex links is at the
root of the development of more and more complex and intricate knowledge.
1. Complex links establish comprehensive
relationships between records. Indeed, a complex link from M to M" relates
not only these two records, but also the intermediate multifold records which
occur in its formation through switches between two of their decompositions
(change of parameters). Thus the cohesion it creates between M and M"
reflects more than a 'local' cohesion between lower level decompositions of M
and M"; it reflects something of the overall structure of the lower level
memory (containing the decompositions), emerging at the level of the link.
For instance, chains of inferences using
metaphors (Paton 1997) are powerful to reveal new overall outlooks because they
correspond to the formation of complex links; indeed, a metaphor can be
interpreted as a switch between two decompositions of a same record: the genome
with its chemical structure, or looked at as a text.
If a record has a decomposition with some complex links, it inherits not only local properties from the objects of the decomposition, but also new comprehensive properties which emerge through these complex links and rely on implicit assumptions. This cohesion of knowledge increases the risk of ambiguities in communication between people or systems having different kinds of knowledge, for they may not share the same implicit knowledge. An 'expert' may have problems to disentangle all the data necessary to construct an efficient expert system. Robots behave well only in a very simple environment, where all the conditions can be controlled.
# If a complex
link gg' is the composite of a
(P,P')-simple link g from M to M'
with a (Q',P")-simple link g' from M' to M", its properties
are deduced from the 'local' properties of the two clusters, say of level n, that g and g' bind, but also
from the fact that P' et Q' have the same colimit M'. This last condition
implicates the global structure of level n
(before the complexification), since the 'universal' property of the colimit M'
means that the two patterns impose the same constraints to any object.
2. The memory is hierarchical; when records of a
given level have been consolidated and connected by simple and complex links,
they can be assembled to form more complex records (or hyperstructures, Baas
1992), by iteration of the preceding formation process. A higher level record A
in the memory necessitates several stages to be formed (to compare with the
"stades" of Piaget 1940); the number of these stages characterizes
the complexity order of A.
For instance, in a 2 stages process, first the
strategies of the CRs and the interplay among them single out and coordinate
patterns Ri of existing records with
(simple or complex) links between them, to form new records of a higher level.
After their consolidation, the second stage similarly sorts out a pattern R
with these new records, and memorizes it by the formation of the record A; we
say that (R,(Ri)) is a ramification of A of length 2. In the
following consolidation process, A may acquire other ramifications.
Later on, A can be recalled through the unfolding of
anyone of its ramifications. This consists in the synchronous activation of the
objects of a decomposition of A, through different CRs (in their actual
landscapes); and, at a following step, the same process, applied to each of
these objects, activates one of its decompositions, through lower level CRs,
and so on if there are more stages. The choice of the ramification is done step
by step, from top to bottom: first choice of a decomposition of A, then, for
each of its objects, choice of one of its decompositions, ... Unfolding of a
ramification can be compared to the gradual filling of the different slots of a
frame in the sense of Minsky (1986). Which ramification is finally unfolded
will depend on the context, and it is selected at each step through the
interplay among the strategies of the CRs, taking into account their structural
temporal constraints. If these cannot be respected, the action will fail.
For instance, before he may walk, a child learns to
coordinate more or less innate strategies to stand up and to move forwards a
leg when he is held; the corresponding strategies are memorized in the motor
areas of his brain. And these strategies are themselves coordinated into patterns
allowing to make a step without falling. And so on up to the formation of a
complete strategy for walking, usable in the most varied situations (Josephson,
1998). But the coordination must always respect temporal constraints; if the
child tries to walk too quickly, he falls.
# A is constructed as an iterated colimit; i.e., A is the colimit
of a pattern R of objects Ri
in the memory such that each Ri
is itself the colimit of a pattern Ri,
such that each object Rik of Ri is the colimit of a pattern Rik,
and so on. We then say that A is the iterated colimit of the ramification (R,(Ri),(Rik)…). Iterated
colimits are formed through successive complexifications of the memory. It
follows from the Multiplicity Principle that A may also be(come) the iterated
colimit of other ramifications.
The complexity order of A is the smallest length of a ramification of A down to the lowest level; it is less than, or equal to, the level of A. In particular, if A is the colimit of a pattern of objects of order n, we have proved that A can also be of order n if all the distinguished links of the pattern are simple, but its order is n+1 if at least one of these links is n-complex (Ehresmann & Vanbremeersch 1996).
d)
Mental objects and higher order cognitive
processes
1. In a neural system, the record of a simple stimulus is reduced to an individual, specialized neuron; for instance, in the visual area, there exist "simple cells" representing a segment of a given direction, and "complex cells" representing a particular angle (Hubel & Wiesel 1962). But more complex stimuli, except for some exceptions (e.g., a neuron activated by a hand holding a banana for a monkey, Gazzaniga 1985), do not have their own "grand-mother neuron".
The development of neural imaging shows that complex perceptual stimuli, or motor programs, are represented by the short-lived synchronized firing of a specific assembly of neurons. And learning would consist in the formation of such synchronous assemblies, through the strengthening of synapses between their neurons, following the rule already proposed by Hebb (1949): a synapse between two neurons is strengthened if the two neurons fire at the same time, and depressed if one fires while the other does not.
In the MES modeling the neural system, the record corresponding to such a synchronous assembly of neurons is called a category-neuron (or briefly cat-neuron); it operates as a unique 'higher order neuron' integrating the synchronous assembly. More complex records, also called cat-neurons, are constructed in several stages. Cat-neurons of order 2 correspond to a super-assembly (or 'assembly of assemblies') of neurons, which cannot be reduced to a (even large) synchronous assembly of simple neurons; higher order cat-neurons correspond to synchronous super-super-assemblies, and so on. They represent higher order mental objects and cognitive processes.
The above description of the links between records determines what are the possible interactions between synchronous (super-)assemblies of neurons, thus solving a problem raised by neuroscientists (von der Malsburg & Bienenstock 1986), which cannot be approached by classical models: they are the simple and complex links between the corresponding cat-neurons. And it becomes possible to 'compute' with cat-neurons, i.e., with (super-)assemblies of neurons, as if they were simple neurons, thus developing a real "algebra of mental objects" (following the proposition of Changeux 1983).
Let us remark that this model is very different from neo-connectionist models of neural systems which give only a description at the sub-symbolic level, and for a limited period, without taking into account the interactions between the different levels. In particular, these models can only describe the formation of simple cat-neurons (represented by attractors of the dynamics), but not their complex links, so that they cannot describe higher order cat-neurons modeling complex mental objects.
# In the MES modeling a neural system, the state-categories are obtained by successive complexifications of the category of neurons defined as follows: the objects are the neurons, the links between them are polysynaptic pathways; two polysynaptic paths from N to N' are identified if they have the same strength, i.e., if the probability that they propagate the activity of N (determined by its instantaneous frequency of spikes) to N' is the same, as well as the delay of propagation.
2. The representation of higher order processes by cat-neurons leads to a new approach of the philosophical problem of the identity between mental states and physical states of the brain. Indeed, a physical state, as it is seen through brain imagery, corresponds only to the activation of a synchronous assembly of neurons, modeled by a 'simple' cat-neuron. But a mental state is represented by a higher order cat-neuron whose activation requires a several steps unfolding through the various intermediate levels of a ramification, down to the level of physical states; and at each step, it can proceed along one or another non-equivalent decomposition of multifold records, with possibly a switch between them, the origin of which might be random (neural 'noise'), quantic (as proposed by Eccles, 1986), or controlled. Though such a process represents a well described 'physical event', we cannot identify it with a 'physical state': mental states emerge in a dynamic way (through the gradual unfolding of a ramification) from physical states but are not identical to them. This could qualify as an emergentist monism in the sense of Bunge (1979).
4.
Classification and interpretation of knowledge.
The knowledge of a system is distributed
among its agents though it remains mostly implicit for them, and they have only
the capacity to use part of it in appropriate situations. However there may
exist higher CRs which interpret and classify the automatic knowledge of lower
level CRs, and develop a semantics allowing for a more flexible and deliberate
application of knowledge. Here we just delineate the main ideas, referring for
more details to the article on consciousness in our Internet site (1999).
a)
Semantic memory
The consolidation of a record consists in its
adaptation to several circumstances by a change of parameters (unfolding of
different decompositions and ramifications). These parameters can come from the
recognition of constancy through variable circumstances, so that a same
response can be given to a whole class of similar situations; a frog will jump
after a fly or after any flying object of the approximate size. This relies on
a classification of the records contained in the memory into what we call
'concepts'.
1. A lower CR, say E, operates only an automatic
classification of the records according to the attributes it distinguishes: two
records are 'acted' as equivalent if their traces in its landscape activate the
same pattern of agents. For instance, birds recognize squares, whatever their
size, from triangles; or, at a more biological level, it is the same pattern of
neurons of a color-CR which is activated by all blue objects.
But this classification remains implicit at the
level of E, and it is interpreted as such only by a higher level CR, with a
longer period, able to overview what is common to the different records in a
same class. A pigeon does not 'know' that he classifies geometric forms; this
'knowledge' is attributed to him by the observer. This higher CR memorizes each
invariance class under the form of an object, called an E-concept, whose instances
are the different records of the class. For example, the instances of the
color-concept 'blue' are all the blue objects.
The CR-concepts, with respect to the various CRs,
form a semantic memory, which is extended
by the formation, firstly, of concepts simultaneously classifying several
features (as a blue triangle), and then
of more abstract concepts obtained by
assembling such 'concrete' concepts. A concept can be thought of as an abstract
prototype for a class of objects with a "family resemblance" (in the
sense of Wittgenstein 1953); it does not necessitate the existence of a
language.
# The automatic classification operated by a CR, say
E, is modeled using the notion of "same shape", in the sense of
Borsuk (1975) "shape theory" (cf. Cordier & Porter 1989). An
E-concept will be defined as the (projective) limit of the pattern of agents of
E activated by all the instances of a given invariance class. More general
concepts are obtained as iterated limits of patterns of E-concepts (for
different E) linked by complex links; they are formed through successive
complexifications.
2. The development of semantics increases the
plasticity, but also the fine-tuning, of knowledge. Indeed, if a record M has a
decomposition P, a new decomposition can be obtained by exchanging an object Pi of P by another instance of
the concept of Pi; the dog
who has learnt to fetch a particular ball will later bring back any ball. Thus
the number of ramifications of a record increases, making it useful in more
contexts. Conversely, there is a fine-tuning of the classification: if the
exchange of Pi by another
instance of the same concept does not function (causing a fracture), the
classification will have to be refined; a child begins to assimilate all moving
vehicles, but later learns to distinguish a car from a train. Thus extensive
consolidation of knowledge, specially of pragmatic knowledge, and development
of a fine-tune semantics are interactive with mutual benefit, thanks to the
recognition of distinctions as well as of similarities.
Moreover, the semantics leads to the formation of conceptual knowledge, based on records which are themselves concepts. In particular, the choice of a strategy by a higher CR will be done under the form of a concept, instead of a specific strategy of its invariance class. It adds a new degree of freedom in the interplay among the strategies of the CRs, since it will be possible to select among the strategies of the invariance class chosen by a CR the best adapted one, taking into account the strategies relayed by the other CRs.
b) Intentional and conscious CRs
1. A system can have, or develop, higher level
agents with particular capacities for handling knowledge in a more deliberate
manner. In MES, it is modeled by the existence and/or formation of higher CRs,
which act as associative CRs controlling lower CRs.
A CR will be called D-intentional, or intentional in the sense of Dennett (1990), if it
acts 'as if' it was able to optimize its choice of strategy in its landscape.
This is possible if some of its agents are evaluators which classify the
records of strategies available in the landscape, depending on their result for
the CR. However this classification can be 'automatic', as in the case of
reflexes. If it is deliberate, and the CR has access to the semantics, we speak
of an intentional CR. It may be
difficult to recognize if the behavior of an animal is intentional or only 'as
if'.
2. An intentional CR is said to be conscious if it is able to internalize
the semantics and the notion of time and eventually to interpret itself its
knowledge. We characterize a conscious CR by the following capacities:
·
Extension
of its actual landscape by retrospection to lower levels of the near past by an
increase of awareness, in particular after a fracture which has revealed some
errors in its handling of a situation, or lack of coherence in its
knowledge;
·
Development
of an abduction process on this extended landscape to search for the possible
causes of the fracture, taking into account preceding experiences;
·
Complex
programming of a selection of strategies covering several steps ahead, thanks
to the formation of virtual landscapes in which sequences of strategies
(selected under the form of concepts) can be 'tried' without material cost for
the system; these strategies can be recalled from the memory, or obtained by
consolidation of already known strategies, or constructed by assemblage of
known strategies whose consequences are anticipated (deduction). For instance,
a chimpanzee solves the problem to seize a banana out of reach by putting a box
on top of another and climbing on it.
3. There are several degrees of consciousness (cf.
Edelman 1989, Damasio 1999), from primary consciousness up to
self-consciousness. In a neural system, they rely on the existence of functional loops between various areas of the
cortex, which form what Edelman calls a "loop of consciousness".
Higher conscious CRs can interpret their knowledge
and make a deliberate choice of long term strategies; however they cannot
control all the consequences of these, and must rely on lower CRs to realize
them; I decide to walk, but I don't know how to directly control the motion of
my legs, whence the risk of falling if I try to think how it is directed.
Language, associating a word to a concept, permits
the formation of more and more abstract conceptual knowledge, controlled by
still higher conscious CRs which are able not only to interpret their knowledge
but also to 'know that they know'. These CRs can communicate their knowledge by
teaching, and by the production of material records. Whence the development and
diffusion of culture.
5.
Conclusion
The MES give a mathematical model, based on category
theory, for complex autonomous systems. Their framework seems well adapted to
study the problems related to the acquisition, representation, cohesion and
interpretation of knowledge.
·
Innate
and acquired knowledge of the system is recorded in a collective distributed
memory, where representations of features of the environment, past experiences
and adapted behaviors are stored in a flexible way; their records are regularly
up-dated to take into account new data, and their domain of application is
gradually extended and refined.
·
The
dynamics of the system, leading to the processes of acquisition and
consolidation of knowledge, is directed by the cooperation and/or competition
between groups of agents acting as coregulators (the CRs); each CR has only
access to part of the knowledge accumulated in the memory; and, depending on
its complexity level, it more or less participates to the development of this
memory through a 'local' trial-and-error learning process;
·
Knowledge
remains pragmatic and mostly implicit for lower level agents; but higher level
CRs in sufficiently complex systems (higher animals, social groups) are able to
classify known objects into invariance classes, thus developing a semantics
which allows for a deliberate and more precise tuning and handling of both
pragmatic and conceptual knowledge.
·
Agents
endowed with 'consciousness' can perform deduction and abduction processes,
allowing for a better long term regulation of their comportment. If they
possess language, they can self-interpret their knowledge and extensively
communicate it, thus leading to the development of culture.
Acknowledgments: We are grateful to Profs. Jerry
Chandler, George Farre and Brian Josephson for stimulating exchanges on related
problems.
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Notes
1 A(n oriented
multi-)graph consists of a set of vertices N and a set of arrows between them.
A category
is a graph on which there is given a composition law associating to each pair
of successive arrows (f: N ® N', g: N' ® N") an
arrow fg: N ® N"; this
law is associative (a path has a unique composite whatever its 2-2
decomposition), and each vertex N of the graph has an 'identity' arrow idN:
N ® N whose
composite with any arrow h beginning
or ending at N is equal to h . The
vertices of the graph are called the objects
of the category, the arrows its morphisms, or more concretely here, its links.
2 A functor from a category to another is a
mapping respecting their graph structures and their composition laws.
3 A pattern (or diagram) P in a category K
consists of a family (Pi)
of objects of K and some links x
between them, called its distinguished links. A collective link from P to an object N of K is a family of links fi: Pi ® N well
correlated by the distinguished links of P, i.e. such that xfj = fi for each x: Pi ® Pj . A colimit of P is an object C of K such that the links from C to any
object N of K are in a 1-1 correspondence with the collective links from P to
N.