International Conference on Systems
Research, Informatics and Cybernetics (Baden-Baden 1994)

Focus Symposium on Emergence

EMERGENCE AND CAUSALITY IN EVOLUTIONARY
SYSTEMS

by A.C. EHRESMANN and J.-P.
VANBREMEERSCH

Faculté de Mathématique et Informatique,
33 rue Saint-Leu, 80039 Amiens. France

Abstract. Memory Evolutive Systems (cf.
6 preceding Baden-Baden Conferences, denoted by BB) give a mathematical
model, based on Category Theory, for complex self-organized systems, such as
bio-sociological or neural systems. In their frame, we have represented the
emergence (by association, classification or organization) of a new object, or
its sudden manifestation for some observers (epistemic emergence) by a 3 stages
process: a pattern of objects with some special links between them is
strengthened, it is glued together to become a new object (the 'colimit' of the
pattern), and takes an identity of its own. It is modelled through the
construction of the 'complexification', which also describes complex links corresponding
to emergent properties. We prove that iterated complexifications lead to the
emergence of higher order structures whose order cannot be reduced; this result
relies on the 'degeneracy property' which asserts that the same object admits
several decompositions in patterns of which it is the colimit and between which
it may 'switch'.

The
evolution of a MES is regulated by the competitive interactions of a net of
internal Centers of Regulation. At the level of a particular CR and on its
short term, emergence is partially caused by the actions of the CR (or its
'intentions' for higher CRs). However causality attributions become blurred for
the system as a whole, making it unpredictable, though not entirely
intractable, on the long term (several complexifications), because of the
interplay between CRs and the organisational role played afterwards by
emergent structures. In the case of neural systems, the model leads to an emergentist
monism, characterizing how higher cognitive or 'mental' levels may emerge from
brain physical states.

Key words. Emergence, complexity,
hyperstructure, category, Evolution, neural system.

1.
Emergence by association or classification.

In
a system modelled by a category, an object (or component) will be considered to
play a double role: 1. It acts as a causative agent or as an emittor, through
its links to other objects which represent its actions, or messages it sends.
2. It becomes a receptor, or an observer, through the links arriving at it,
which correspond to aspects it observes, or messages it receives, or
constraints imposed on it

This
dual situation extends to a pattern in the system, consisting of a family of
objects with some specific links between them: 1. The collective actions of the
pattern model the actions which can be performed only if all its objects
cooperate through their specific links to act together; so they consist of
those families of individual links from the components of the pattern to
another object correlated by the specific links of the pattern. For instance in
a neural system, the collective firing of an assembly of neurons may activate
another neuron. 2. The common triggers of the pattern are defined dually, just
by inverting all the arrows; they model the common messages sent by an object and
which can be globally apprehended by the pattern only if the partial messages
received by each of its components are united and coordinated through its
specific links. For instance a film sequence needs several cameras adequately
coordinated to be registered.

We have explained in (BB 1993) how a
pattern P may lead to the emergence by association of a new object, by a 3-step
process: 1. The pattern as a whole takes a functional significance for some
objects by collectively acting on them. 2. Its specific links which mediate the
collective actions are strengthened so that the pattern becomes a coherent
assembly. 3. The assembly is 'institutionalized' by the emergence of a new
object (in a complexification process), called the colimit of the pattern, denoted
by colimP. An example is given by the speciation process in the Theory of
Evolution. The colimit internalizes the operation field of the pattern, defined
as the sub-system on which the pattern may exercise a collective action not
performable by its components acting separately; its objects are all the
actions of colimP on other objects, interconnected with links compatible with
these actions.

Dually,
we have defined the emergence by classification as the transformation of a
pattern P, now considered as a receptor, into a new object, called its limit,
limP. Semantics in a neural system is obtained by such a process (cf. BB 1992).
The limit internalizes the classification field of the pattern, consisting of
the aspects or messages whose reception necessitates the cooperation of all
the components of the pattern.

In the frame of Memory Evolutive
Systems (EV, 1991), we have explicitly described the complexification process
that allows to realize a strategy calling for the emergence and/or disparition
of complex objects, and introduces both simple and complex links between them
as defined below. The evolution of a natural system modelled by a MES is shaped
by the reiteration of such complexifications. The emergence of higher order
structures that is the essence of complexity will depend on the following:

2. Degeneracy Principle and Switch
between decompositions.

The colimit of a pattern is considered
as a complex object admitting the pattern as a decomposition, or (depending on
the context) as an internal organization. The Degeneracy Principle asserts that
two non-equivalent patterns may have the same colimit A (or, dually, the same
limit). As an important consequence A may switch between its several
decompositions.

For example, a Journal is an entity by
itself, which, in the system of social groups, is modelled both as the colimit
of the editorial board and as the colimit of the publishing staff, since each
of them gives a valid representation of the Journal. If a letter is simply
addressed to the Journal, it will be dispatched to one or the other: a paper
submitted for publication will be sent to an editor, an order for subscription
to the accounting agent. And the Journal switches between both to mediate the
communication of the authors with the subscribers.

The
Degeneracy Principle takes its name from the fact that the passage of a pattern
to its colimit 'forgets' the fine-grained details of the organization of the
pattern to preserve only its overall functional activity; for instance the same
aminoacid is coded by several triplets; or in a neural system, a stimulus is
recognized in spite of noise. Conversely the passage of a complex object A to
one of its decompositions acts as the filling of a slot (as in a frame in the
sense of Minsky, 1986) to single out that one of its internal organizations
most appropriate in the context (e.g. the editorial board of the Journal or its
publishing staff), thus leading to more adapted responses through the
redundancy of the possible choices. Moreover the switch process between two
different decompositions (as the two faces of an ambiguous figure) allows for
them both to intervene simultaneously or successively to mediate the relations
with other objects. Such a switch may also exist between two patterns admitting
A as a limit, or between a pattern whose colimit is A and a pattern whose limit
is A.

There
are two essential applications of this Principle. 1. It explains how, in time,
an emerging object may take its own identity, allowing for changes in its
organization and replacement of some of its components; this relies on a
gradual transformation of the pattern from which it has emerged, by a sequence
of switches between decompositions each of which remains a decomposition during
a stability span (EV, 1987 and 1994) before being replaced by another one. This
process is examplified by the notion of a temporal species which evolves though
maintaining its overall identity. 2. It is at the root of the definition of
complex links modelling emergent properties, as we are going to explain.

3.
Emergence of complex links.

A pattern P may exert a collective
action not only on a particular object but also on a pattern Q. It means that
each object of P sends a message at least to one object of Q, and if it sends
several, those transmit the same information to Q as a whole, in the sense that
they are interconnected by a zig-zag of specific links of Q. Thus we get the
notion of a cluster from P to Q (cf. EV, 1987), that is a family of such
messages correlated by the links between the emittors (in P) as well as by the
links between the receptors (in Q). If colimP and colimQ have emerged, this
cluster glues into a unique link from colimP to colimQ; this link is said to be
(P,Q)-simple because it only 'institutionalizes' the cluster, without adding
informations not already known at the patterns level. In Embryology, the
induction of a population by another one corresponds to the formation of a
simple link.

Taking
the situation from 'above', and considering two complex objects A and A', we
define the simple links from A to A', each formed by glueing a cluster linking
particular decompositions P of A and Q of A'. Remark that, with respect to
another decomposition Q' of A', a link which is (P,Q)-simple is not always
(P,Q')-simple, because it might not exist a cluster from P to Q' that it glues,
except if the switch from Q to Q' is simple in the sense that the identity of
A' be (Q,Q')-simple (we also say that Q and Q' are equivalent). There exist
complex (i.e. non-simple) switches, e.g., between the non-equivalent genotypes
(due to the presence of alleles) of a species; these complex switches may
introduce difficulties to combine simple links.

A
simple link from A to A' and a simple link from A' to A" must combine in a
link from A to A" (by definition of a category). If these links decompose
respectively into a cluster from P to Q, and a cluster from the same Q to
another pattern R, then the combined link is also simple; indeed, by combining
the individual links of the two clusters we get a cluster from P to R, and the
combined link is the (P,R)-simple link glueing this cluster. But the two simple
links may decompose only into non-adjacent clusters, namely if the first glues
a cluster from P to Q and the second a cluster from another decomposition Q' of
A' to R. Then if the switch from Q to Q' is complex, the combined link
generally does not glue a cluster, and we call it a complex link from A to
A". For instance, the communication between authors to subscribers of a
Journal is a complex link mediated by the complex switch between Editors and
Publishers.

More
generally, there will emerge complex links between two complex objects, defined
as a link obtained by combination of a sequence of simple links but which is
not simple. Such a complex link from N to N' may be presented in several ways
as the combination of a path from N to N' formed by simple links in which two
successive links decompose into non-adjacent clusters, requiring complex
switches between the two decompositions of the middle objects; the sequence of
these underlying clusters will be called a decomposition of the complex link.
Complex links also occur to combine simple links glueing non-adjacent clusters
between limits or colimits, with a complex switch between limits, or between a
limit and a colimit. They represent emergent properties and are at the root of
the formation of higher order structures, e.g. of higher cognitive processes in
neural systems (cf. BB 1991).

4.
Ramifications of a complex object.

Natural systems have components of
increasing complexity levels (e.g., atoms, molecules, cells, tissues,...)
formed through reiterated emergences; let us study their structure. First let A
be an object admitting as one of its decompositions a pattern P such that all
the components of P are themselves complex objects, and so have at least one
decomposition into a pattern Pi , and the links of P are simple or complex
links with respect to these decompositions. We'll say that A is a 2-iterated
colimit of (P,(Pi)); or still that it admits (P,(Pi)) as a 2-ramification. The
ramification determines univocally the complex object by means of an internal
organization with two levels. But conversely, the multiplicity of
decompositions of a complex object implies that A has many 2-ramifications,
their number depending on the number of decompositions of A and of each
component of these. This number will be called the 2-entropy of A (cp. with the
definition of entropy of a gaz as the logarithm of the number of its
microcanonical states). We also define the 2-ramifications of a complex link by
considering all its decompositions and the decompositions of these.

By iteration, we define a k-iterated
colimit, and a k-ramification, whence the k-entropy of a complex object. (The
1-entropy is just the number of decompositions or 1-ramifications.) If we think
of a decomposition as a particular means to fill slots of the complex objects,
we see that a k-ramification multiplies the possibilities to fill these slots,
since each slot once filled has itself its own slots to fill, and so on to the
bottom. In other words, the complex object acquires much more degrees of
liberty, with extended possibilities of switches between ramifications. This
leads to the emergence of k-complex links where complex switches are introduced
at the various decreasing levels (cf. EV, 1994). Iterated colimits are
hyperstructures in the sense of (Baas, 1994); compare also to Eigen
hypercycles. Ramifications may also be defined for limits, or with both limits
and colimits, as in the definition of an abstract concept (cf. BB 1993).

5. Hierarchies and emergence of higher
order structures.

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