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

The authors progress in the theory of Memory Evolutive Systems (based on Category theory), already presented in former Baden-Baden Conferences (1988, 89, 90, 91). In this model for natural open self-organizing systems, such as biological, sociological or neural systems, the dynamics is modulated by the competitive interactions between the global system and a family of internal more or less specialized Centers of Regulation (CR) with a differential access to a central hierarchical Memory. Each CR operates at its own complexity level and time-scale, but their strategies are competitive, whence a 'dialectics between heterogeneous CRs which is at the root of higher order cognition, up to consciousness (cf. Baden-Baden 1991).

The problem tackled in the present paper is the emergence of Semantics through the detection of specific invariances by the CRs that leads to classify objects according to their main attributes, and form new formal units representing their invariance classes. The idea is that each CR classifies two objects B and B' as having 'the same shape' if they activate the same pattern of its actors though along different paths. This categorization is memorized as concepts in a higher sub-module of the Memory formed by the projective limits of such patterns. The concepts so defined play an essential part in the evaluation, selection and memorization of appropriate strategies, as well as in internal or external communications.

The actions of an animal, on which its fitness depends, require that it might form an adequate enough representation of its environment. In preceding papers (1987, 1989, 1991), we have shown how memorization of past experiences develops and leads to the formation of higher order units in the memory (representing Hebb (1949) assemblies of neurons, or Edelman (1989) neural groups). However, storage of items is not sufficient for action. Efficious learning relies on the recognition of constancies through changing circumstances, that is on a kind of classification of the memorized items according to their main attributes, so that the animal may react in a specific manner not only to a particular situation, but also to all similar situations. For instance, a toad will jump after a fly or after any flying object of the approximate size.

lt is necessary to distinguish a hierarchy of categorization problems:

1. Two objects being given at the same time (either external or retrieved from the memory), to compare them with respect to a particular attribute, for instance their color, or their orientation.

2. To recognize an object despite changes in some of its attributes, for instance 'grandmother with her glasses on or off'.

3. At a higher level, to characterize the class of objects having a common attribute by a unique unit, its 'concept', for instance the color 'red'.

4. To be able to communicate concepts, in particular for man thanks to the development of a language.

Remark that the difference between comparing items (1) and characterizing a class of partially similar items (3) is the same as that proposed by Leibniz between 'clear ideas' and 'distinct ideas'.

We are going to study these problems, first giving the ideas in a neural system, then transposing to a general Memory Evolutive System (§3).

The first problem to be considered is the comparison of two objects with respect to a specific attribute.

The vision of animals admits a large class of invariances: they can recognize the form of an object despite change in size, orientation, position in the visual field, movement,..., or color despite change in the spectral context of the illumination. Neurophysiological data may explain some of these perceptual constancies. For instance, Hubel and Wiesel have shown that the visual cortex of mammals contains columns of 'simple cells' which detect bars of a given orientation and of a special position in their field, but also 'complex cells' which detect bars of a given orientation whatever be their position in the field (up to some bounds), or their motion. Some mathematical theories have been developed for the generation of scale and rotation invariances in automatic pattern recognition, such as logarithmic scaling, Fourier-Mellin transforms (cf. Reitboeck 1984); but they depend on vision characteristics. Here we will propose a more general framework.

The modular theory of brain function (cf. e.g., Fodor 1981) postulates the existence of distinct 'modules' to treat specific features, such as a 'geometric' module for form, a 'color' module... The modules treat objects and discriminate two objects according to a specific attribute, without considering their resemblance or differences for other attributes.

For instance a square will be differentiated from a circle by the shape module, while if they have both the same color, the color module will not distinguish between them. But this process does not imply the module operates a real categorization of the perceptual world: the module may compare two objects and recognize if they are similar for its attribute, but this comparison does not entail a characterization of the classes of similar objects. The color module recognizes that this square and this circle have the same color, but it does not give a characterization of a 'red object'.

Before treating the case of an abstract Memory Evolutive System (MES), let us outline our classification model in the MES modeling a neural system (1991), and explain the underlying ideas in biological terms.

The 'modules' are modeled by the internal centers of Regulation, or Coregulators (CR, cf. §3), Let E be a specific CR (say the 'color' module); its units (which are neurons or 'category-neurons', cf. 1991) are called actors. The actual landscape of E at a time t is formed by all the synaptic paths from a unit of the system which activate some actors at t. Let us suppose a stimulus is presented to the animal at t. It activates first a pattern B of linked units B_i of the system (in the receptors, or eventually in the memory). The pattern is transmitted to E by the synaptic paths which link one of the B_i's to the actors A_j's. These paths are interconnected by the distinguished links of the pattern on one hand, and the operating links between the actors on the other hand. So we get a pattern in the actual landscape which is called the E-trace of B. For a color module, the trace will consist of all the paths along which the color characteristics of the stimulus (wave length, illumination,...) are transmitted to the units of the module, forgetting all other information which are treated by other modules.

In the neural system, we know that the activity of a neuron A is computed from the sum of the activities of all the neurons which activate a synaptic path arriving to A, pondered by the weight of the paths. The activity of an actor induced from the pattern B will be computed in this way taking the sole paths constituting the trace, and we so obtain a pattern A of actors in the module, which is called the pattern E-induced by B, and which represents the constraints forced on the actors by the trace of B.

The pattern A records the activities imposed on the actors, not the paths along which they are transmitted. It follows that 2 patterns B and C may induce the same pattern on E, while they induce different patterns on other CRs. In this case, we say that B and C have the same E-shape.

For instance, a red circle and a red square induce the same pattern in the color module but not in the shape module: they have the same color-shape, but different geometric-shapes. Or if we consider the two patterns of retinal cells activated by two vertical bars not in the same position in the visual field, they activate two different simple cells, but if the bars are near enough, they activate the same complex cell, that means they have the same shape for the orientation module.

If we consider a higher level associative CR (such as a 'conscious CR' in the sense of our 1991 Baden-Baden paper), two patterns will have the same shape for this CR if they have the same shape for the lower CRs it controls. If it so controls many attributes, the two patterns might be recognized as representing the 'same' object despite changes in a few other attributes (in fact, as long as the controlled attributes correspond to a sub-pattern having the same cohesive binding as the whole pattern). For instance, an object is identified whatever be its size or location in the visual field.

In preceding papers, we have introduced the notion of a MES to model complex natural systems such as bio-sociological systems or neural systems. The architecture of a MES is a compromise between a parallel distributed processing with a modular organization, and a hierarchical associative network, in which the dynamics is shaped by the dialectics between internal regulatory organs (CRs), each with its own complexity level and time-lag, which leads to the characteristic functioning of a complex system. Contrary. to the complete opacity of a computer levels between them, here the CRs have both partial direct connections and indirect interconnections through the 'fractures' they may generate in other levels.

Let us recall the definition of a MES (cf. Ehresmann-Vanbremeersch 1991, 1992).

The state of the system at a given time is modeled by a category K, formed by its components and the interactions between them (the links modeling transfers of information, energy or constraints).

The system has an organizational hierarchy, with its objects separated into various complexity levels: an object of level k+1 is the cohesive binding (or 'colimit', in the category) of a pattern formed by its own components of the lower level k and some specific links between them.

The dynamics of the system is represented by transition functors between successive state-categories, which model the change of state depending on internal modifications, on the flux of information between the levels and on exchanges or constraints originating from the environment. It is regulated by a family of sub-systems (or modules), the internal Centers of Regulation CR_n, each with its own complexity level, its time-scale and its period (or time-lag) represented by a real d_n. In the lower levels, the CRs represent specialized modules receiving direct information from the environment; in the higher levels more associative CRs with longer time-lags supervise several other CRs. These CRs operate in parallel by a trial-and-error learning process with cooperative, or eventually conflicting, strategies to modulate the general dynamics of the system.

The learning process for a CR, say CR_n , is done stepwise, according to its scale of time in which the length of a step (or 'actual present') must be greater than 2d_n. At each step, CR_n as an observational organ, constructs its own internal representation P of the global system, called its actual landscape. As a command organ, its actors coordinate their 'goals' to select a strategy on P consisting in the addition or subtraction of some elements, disassociation of some complex objects, cohesive binding of some patterns (e.g., by strengthening of their links) so that they become new (complex) units of a higher level. The anticipated landscape P' at the end of the step should be the 'complexification' of P with respect to this strategy. However, since there is a competition between the CRs and each one has only a distorted view point of the whole, the strategy may not be enforced and there will be a difference between P' and the 'real' landscape at the end of the step. As a control organ, CR_n measures this difference (by the comparison functor).

The system has a hierarchical sub-system, called the Memory, to which each CR_n has a differential access through its actual landscapes and which it concurs to extend by the memorization of its successive strategies and of their results.

The notion of patterns of the same shape for a CR explained in §2 is conceptualized in a MÉS as follows (cf. our 1992 paper):

Consider a pattern B in the state of the MES at a certain time t. Let CR_n be one of the CRs, and P its actual landscape at t (formed by the perspectives for the actors of the objects of B which are observable by CR_n during its actual present). The CR_n-trace of B is the pattern induced on P by the category T_nB in which:

- the objects are the links b from an object B_j of the pattern to an actor A_j,

- a link from b to another b' from B_i' to A_j' consists of a commutative square (b,a;h,b'), in which h from B_i to B_i' is a distinguished link of the pattern, and a from A_j to A_j' a link in CR_n such that ba = hb',

- two adjacent squares are combined by combining their horizontal edges.

(This category is a 'comma-category', cf. Mac Lane 1971.)

Two patterns B and C have the same CR_n-shape if the categories T_nB and T_nC are isomorphic, so that their images by the functor which maps the above square on a give the same pattern of CR_n, called the pattern CR_n-induced by B. (This notion has been suggested by mathematical results on Shape Theory, cf. Cordier & Porter (1989); for patterns with a unique object, it means they are isomorphic in an appropriate 'shape-category of Holsztynski'). For the MES associated to a neural system, we recognize the notions of trace and of 'same shape' introduced in §2.

Higher animals may pursue the discrimination task further on. Locally CR_n acts as if it might recognize that two patterns B and C have the same CR_n-shape, since its actors react to both in the same way. However this comparison is only implicit; the fact that the two patterns have the same consequences for CR_n can be apprehended only externally to CR_n, in particular if there exist higher Ievel CRs which perceive the common constraints imposed by B and C on the CR_n-actors in their totality, and on a

Ionger time-scale. This agrees with: "iI n'est de sens que par rapport á autrui et pour une temporalité différente de celle d'un présent réduit à son instantanéité" (Draï 1979). Such a CR wiII attribute a 'meaning' to the pattern A of actors CR_n-induced by B and C, namely that it is what remains invariant from B to C; the invariance so displayed will be memorized by the formation of a more complex 'categorization unit', called a CR_n-concept which represents the class of aII the patterns having the same CR_n-shape. (The passage from the shape-comparison between patterns to the concepts is similar to the passage from the partition of a set to its quotient set, in which each subset becomes a unit.) Concepts act as prototypes to which a pattern may be compared to characterize its shape.

Language consists in naming the concepts, and operating on them to form still more abstract concepts.

In a MES, the formation of the CR_n-concept of B, denoted by s_nB, consists of adding a new object in higher levels of the Memory (via a complexification step of the regular stepwise learning process directed by a higher level CR). This object will be defined as the projective limit of the pattern of actors CR_n-induced by B. Let us recall what it means.

In a category, the cohesive binding (or inductive limit) of a pattern A is defined as an object A' whose links to any object N are in 1-1 correspondence with the collective links from the pattern to N. The projective limit of A is formally obtained by the same process, but after every arrow is inverted: a collective link from an object N to the pattern consists of a family of links from N to each component A_j of the pattern which commute with the distinguished links of the pattern. Precisely, the projective limit of the pattern (cf. MacLane1971) is an object limA such that the links from any N to limA are in 1-1 correspondence to the collective links from N to the pattern.

For a pattern B in a MES, the CR_n-concept of B, denoted by s_nB, will be (if it exists) the projective limit limA of the pattern A of actors which is CR_n-induced from B. There exists a collective link (l_j) from s_nB to the pattern A; and for any collective link (f_j ) from an object N to A, there exists a unique link from N to s_nB which, combined with a link l_j gives back the f_j. All the patterns which have the same CR_n-shape as B have the same CR_n-concept.

The CR_n-concepts with appropriate links form a sub-category CR_n-Sem of the Memory. We have proved (1992) that there is a collective link ß from B to s_nB and that s_nB is the object of the category CR_n-Sem which gives the best approximation of B (it means that each collective link from B to an object in CR_n-Sem may be decomposed through ß by a unique link). The formation of concepts preserves cohesive bindings: if B has a cohesive binding B' (in particular, if B' is the unit in the Memory which memorizes B), then its CR_n-concept s_nB' is the cohesive binding in CR_n-Sem of the pattern formed by the concepts s_nB_¡ of its components B_i.

Hence, once the more elementary CR_n-concepts are constructed (and in a neural system, there are such neurons already specialized at birth), more general concepts (with respect to several attributes) are obtained by forming cohesive bindings of patterns constructed on them.

The sub-module of Memory formed by all the concepts so constructed and their natural links (e.g. those from a concept to a more general one: from "blue triangle" to "blue") is called the semantic Memory, denoted Sem.

The development of the Memory relies on two distinct processes and their interplay : the usual memorization process (defined in 1989, 1991) leading to the empirical memory, simply called Memory, and the categorization process leading to Sem. To have more intuitive ideas, we take a neural network; the results are easily generalized to any MES.

- A pattern B is stored in the Memory through its cohesive binding B', so that each time B is activated, it is recognized through the activation of B'. Biologically, the memorization of the pattern B consists of the strengthening of its distinguished links, which transforms the pattern in a synchronous assembly of interconnected neurons in the concept of Hebb (1949). B' denotes this assembly, considered as a higher order unit in the Memory. For B' to be activated, it is necessary that the assembly as such be activated, so that all its components B_j are activated, and act synergistically through their distinguished links.

- The categorization process operates in parallel in the various CRs. For each CR_n, it decomposes in 2 steps:

1. a local step consisting in the comparison of the constraints two patterns impose on the actors of CR_n, to recognize if they have the same CR_n-shape;

2. a more complex process directed by higher level CRs in which the whole class of patterns with the same CR_n-shape is memorized by a unique concept in CR_n-Sem, the concept being activated as soon as one of the patterns of the class is.

What are the consequences and the interactions of these processes on the communications between CRs which modulate the dynamics of the system? It will recognize constancies through changing circumstances and react in a specific manner not to a particular situation but to all similar situations; for instance an object is identified whatever be its size or location in the visual field. In particular the procedural Memory Strat (cf. our paper in Baden-Baden 1991) relies on invariances: what counts in a strategy is not which pattern of the landscape is activated but what command the actors transmit to the effectors. So it is the induced pattern of actors which is important, and strategies will be chosen and memorized in terms of CR_n-concepts (we have developed this point in our 1992 paper). It follows that the choice of a strategy does not force the explicit patterns which are activated in the Memory or the effectors, as long as they have the same shape. This latitude confers a great flexibility to the model. If a strategy requires the activation of a particular concept in Sem, it activates the induced pattern A of actors; which pattern B representing the concept will be activated depends on the context.

As the different CRs all cooperate (eventually with conflicting strategies) in the dynamics of the system, the choice by one of them, say CR_n , of a strategy will have a different result according to the choices of the other CRs. It explains that a strategy in the procedural memory acts as a 'frame' in the sense of Minsky (1986), where 'slots' may be differentially filled depending on which choice of B the situation requires. For instance, the order to hold an object will not activate the same muscles depending on the location of the object, on its size, its shape,... .

Physiological data explain how the activation of a CR_n-concept G is reflected into that of a specific pattern B in its class, though there is no direct link from G = s_nB to B (there is only a link in the other way). G activates the pattern of actors A_i of which it is the limit, and the process of selecting strategies increases the attention in a diffuse way, so that the activity of all the patterns B (either in the receptors or the Memory)

having G as their concept will be increased. Then, following Hebb (1949), the strength of the synaptic links from B to A_i also increases, its two ends increasing. This increases the activity of B through a feedback from the post-synaptic neuron A_i to the pre-synaptic neurons B_j (synapses transmit prediction and information via NO fluxes, cf. Schuman & Madison 1991). In this way, the B which receives the more inputs from other sources will be activated by G.

As will be shown elsewhere, this process explains how behavior will become more and more adapted, since the occurrence of fractures may force the actors to distinguish which specific pattern must be activated in a particular context. For instance, in the development of language by a child, terms take a more precise meaning as the experiences accumulate. It also explains how several types of neural degeneracy, such as aphasies or apraxies, depend on the severing of communications between specific CRs, so disrupting the transmission of activities between a pattern and its concept or reciprocally.

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