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

Focus Symposium on Emergence




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.