Semantics and Communication for Memory Evolutive Systems
Andrée Ehresmann & Jean-Paul Vanbremeersch
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Source: http://cogprints.org/2075/1/Baden-Baden_92.htm
The learning process for a CR, say CRn , is done stepwise, according to its scale of time in which the length of a step (or 'actual present') must be greater than 2dn. At each step, CRn 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, CRn measures this difference (by the comparison functor).
The system has a hierarchical sub-system, called the Memory, to which each CRn 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 CRn 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 CRn during its actual present). The CRn-trace of B is the pattern induced on P by the category TnB in which:
- the objects are the links b from an object Bj of the pattern to an actor Aj,
- a link from b to another b' from Bi' to Aj' consists of a commutative square (b,a;h,b'), in which h from Bi to Bi' is a distinguished link of the pattern, and a from Aj to Aj' a link in CRn 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 CRn-shape if the categories TnB and TnC are isomorphic, so that their images by the functor which maps the above square on a give the same pattern of CRn, called the pattern CRn-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.
4. Formation of concepts
Higher animals may pursue the discrimination task further on. Locally CRn acts as if it might recognize that two patterns B and C have the same CRn-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 CRn can be apprehended only externally to CRn, in particular if there exist higher Ievel CRs which perceive the common constraints imposed by B and C on the CRn-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 CRn-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 CRn-concept which represents the class of aII the patterns having the same CRn-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 CRn-concept of B, denoted by snB, 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 CRn-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 Aj 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 CRn-concept of B, denoted by snB, will be (if it exists) the projective limit limA of the pattern A of actors which is CRn-induced from B. There exists a collective link (lj) from snB to the pattern A; and for any collective link (fj ) from an object N to A, there exists a unique link from N to snB which, combined with a link lj gives back the fj. All the patterns which have the same CRn-shape as B have the same CRn-concept.
The CRn-concepts with appropriate links form a sub-category CRn-Sem of the Memory. We have proved (1992) that there is a collective link ß from B to snB and that snB is the object of the category CRn-Sem which gives the best approximation of B (it means that each collective link from B to an object in CRn-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 CRn-concept snB' is the cohesive binding in CRn-Sem of the pattern formed by the concepts snB of its components Bi.
Hence, once the more elementary CRn-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.
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