Analysis of complex events in Memory Evolutive Systems

1. Analysis of complex events in Memory Evolutive Systems by Andrée C. Ehresmann (Work in collaboration with Jean-Paul Vanbremeersch) Université de Picardie Jules Verne ehres@u-picardie.fr http://pagesperso-orange.fr/ehres http://pagesperso-orange.fr/vbm-ehr ServiceWave 2010, Ghent

2. A FRAME FOR COMPLEX EVENTS PROCESSING Dialectics between CRs Interplay of CRs We consider an evolutionary adaptive system with a tangled hierarchy of components, self-organized thanks to a multiplicity of mutually entailed functional subsystems, the CoRegulators CR, each operating at its own rhythm, with the help of a central Memory. Event processing punctuates the dynamics, both of the CRs, of their interactions and of the system itself.. This will be studied by modelling the system by a Memory Evolutive System (EV 1987-2009). First we give 2 generic examples. FUMEE 3

3. THE MEMORY EVOLUTIVE SYSTEM OF AN ENTERPRISE OUTSIDE MEMORY CRs TOP Direction MEMORY Managers Services Workshop Time The components are the members of the staff, the multi-level services and departments, and the various material or immaterial resources; the links model their interactions in the enterprise. The constantly revised "Memory" stores archives, past events and the knowledge necessary for functioning; part of it, the Archetypal Core AC, acts as a flexible internal model. The dynamics is directed by the cooperation/competition between the different departments acting as a net of coregulators CRs, each operating with its own temporality (daily at the lower level, up to years at higher levels). Events can propagate among the services from bottom to top, and also top-bottom, with higher CRs able to initiate drastic changes for leading to a better functioning of the enterprise.

4. MENS, MODEL OF A NEURO-COGNITIVE SYSTEM cQ=Complex mental object MENS Synchronous assembly of neurons cP=Mental Object NEUR The neural system of an animal has for components its neurons, connected by synaptic paths between them (directed from the first presynaptic neuron to the last postsynaptic one). It is modeled by the Evolutive System NEUR. A mental object corresponds to the activation of a syn-chronous assembly of neurons, with the possibility that 2 different synchronous assemblies P and P' activate the same mental object ('degeneracy' of the neural code, Edelman, 1989). This mental object is modeled by the 'binding' cP of P (and also P'), called a category-neuron,in a 'dynamic' model, MENS, including both the neural and mental systems. MENS is a MES obtained by successive 'complexifications' of NEUR; its hierarchy of components has neurons at level 0, and higher category-neurons representing more and more complex mental objects at higher levels.

5. GRAPHS AND CATEGORIES A (multi-)graph has a set of vertices (or 'objects') A, B, … and a set of oriented edges ('arrows' or 'links') between them. A path of the graph is a sequence of consecutive links (f, g, k). Example. The graph of neurons with neurons as vertices and synapses as arrows. A category is a graph on which there is given an internal composition associating to a 2-path (f, g) a composite fg, This composition is associative and each object has an identity. Because of associativity, any path has a unique composite. Two paths are 'functionally equiv-alent' if their composites are equal We have to develop a 'dynamic' theory of categories incorporating time to account for the dynamics of the system and its various events, in particular the possible loss or adjunction of components over time. Thus the system is not modeled by one category but by an Evolutive System, namely a family of categories Htndexed by Time, with partial 'transition' functors between them.

6. EVENT: FORMATION OF A COMPLEX OBJECT Ht C level n+1 s A levels ≤ n ci si si = = sj sj P Pi f f Pj A group of components with a common endeavour is modeled (in a category Ht) by a pattern P consisting of objects Pi with distinguished links between them. A collective link from P to a component A is a family of links si from Pi to A, correlated by the distinguished links of P. If the group becomes stable, it takes its own identity, and this event is modeled by the formation of a more complex component C with the same functional role as the group P. This C is modeled by the colimit (or binding) of P, meaning: there is a collective link (ci) from P vers C through which factors any other collective link (sI) from P to A. A category is hierarchical if its objects are divided into levels of 'complexity' so that an object C of level n +1 is the colimit of at least one pattern P with values in levels < n +1.

7. HIERARCHICAL EVOLUTIVE SYSTEM. COMPLEXIFICATION transition level n+1 level n level n-1 level 0 Time t' t A HierarchicalEvolutive SystemH consists of: A timescale Time and for each t in it a hierarchical category Ht (configuration at t); for t < t', a transition functor from a sub-category of Htto Ht,; these functors satisfy a transitivity condition so that a component C of H is a maximal set of objects linked by transitions. The standard events at the root of change (hence shaping the transitions) are: adding new objects, forming (or preserving, if it exists) the colimit of certain patterns, decomposing or removing some elements. This is described by the complexification process with respect to a procedure S having objectives of such types. We have explicitly constructed the corresponding transition leading to a category where these objectives are optimally satisfied. In particular the links in this category come from 2 kinds of events as follows.

8. BINDING EVENT: SIMPLE LINKS A C cG levels ≤ n cluster G Pj Pj = Pi = P Pi = P'i' P' P'k If P and P' are 2 patterns, a cluster G from P to P' consists of links from each Pi to at least one P'k, these links being well correlated by the distinguished links of P and P' . If P and et P' bind into C and A respectively, the cluster G binds into a unique link cG from C to A, called a (P, P')-simple link, or an n-simple link if P and P'are contained in the levels ≤ n. A composite of n-simple links binding adjacent clusters is n-simple. The n-simple links are entirely determined by links between the components of P and P'. Thus their formation iis an event just translating properties of the lower levels at level n+1. Theorem. The hierarchy of a HES deduced from level 0 by a sequence of complexifications is based on level 0, meaning the links are deduced from level 0 by iterated bindings of clusters and compositions of links already constructed.

9. MULTIPLICITY PRINCIPLE AT THE ROOT OF EMERGENCE level n+1 level n+1 complex link A C n-simple link n-simple link cluster G Q' = cluster P' = = Pi Q Pj levels ≤ n levels ≤ n The flexibility (and impredicability) of complex systems comes from the following "degeneracy property" (in the sense of Edelman 1989): there are patterns which are functionally equivalent, but not interconnected, e.g. a mental object can activate different neuronal assemblies. Formally: Multiplicity Principle (MP): There are objects C, called n-multiform, which bind 2 patterns Q and P of levels ≤ n which are not connected by a cluster. The passage from P to Q is a complex switch. This principle, which extends to a complexification, mplies the existence of complex links which are the composites of n-simple links binding non-adjacent clusters (cf. figure). Their formation is an event properly emerging at the level n+1since they cannot be recognized 'locally' through links between the extremal patterns Q' and P'; they depend on the 'global' structure of the levels ≤ n.

10. COMPLEXITY ORDER level n+1 C level n P Pi Pi P level 0 In a HES, a component C of level n+1 binds at least one pattern P of strictly lower levels (in a category Ht). Now each Pi also binds a pattern of lower levels, and so on. We thus construct a ramification of Cdown to the level 0 of the hierarchy. Its length can be < level of C. Since at each step there are several decompositions, perhaps non-connected, C may have different ramifications with possible complex switches between them. A complex event implicating C may consist in the unfolding of anyone of its ramifications down to the level 0. The ramifications of C have not always the same length. We define the complexity order of C as the smallest length of such a ramification. It measures the smallest number of events necessary for constructing C from level 0 up by successive binding processes.

11. EMERGENCE OF COMPLEX EVENTS Ht' level n+1 Ht level n level 0 Time EMERGENCE THEOREM. In a Hierarchical Evolutive System, the Multiplicity Principle is the condition characterising the existence of components of complexity order > 1, and the possibility of emergence over time of components of strictly increasing complexity order. If the MP is not satisfied, any component is the simple binding of a pattern contained in the level 0. This would characterize a 'pure' reductionism. In the SEM we consider, the MP is always satisfied, and it allows the emergence of higher complexity at the root of the processing of complex events. We can speak of an emergentist reductionism (in the sense of Mario Bunge).

12. MEMORY EVOLUTIVE SYSTEM (MES) AC An evolutionary hierarchical adaptive system will be modeled by a Memory Evolutive System MES. It is a HES whose self-organization is directed by the cooperative and/or competitive interactions between a net of specialized functional evolutive subsystems, the coregulators. with the help of an evolutive sub-system Mem modeling a central flexible memory (which may develop a robust though flexible internal model AC of the system). We suppose that the MP is satisfied and that each link has a propagation delay and a strenth (both in R+). Each CR has its own complexity level, its own discrete timescale extracted from the continuous timescale of the system, and a differential access to Mem; it participates in the formation of the transitions (via complexification processes) by selecting procedures depending on its function.

13. ONE STEP OF A CR WITH ITS VARIOUS EVENTS S S Landscape Lt S Effecteurs Temps Evaluation t' Formation of Lt t Choice of S Command of S A CR acts stepwise at its own rhythm as a hybrid system using both its own discrete timescale and the continuous time between 2 successive times t and t' of this scale. (i) The first event of the step is the formation of its landscape at t (modeled by a category Lt) with the partial information it can access. (ii) The second event is the choice, with the help of Mem) of a procedure S to respond (it should lead to the complexification AL of Lt by S). (iii) Sending commands to the effectors to realize S starts a dynamic process which unfolds during the continuous time of the step. It is directed by differential equations, implicating the propagation delays and strengths of the links, and it should move the landscape to an attractor. (iv) The result is evaluated at the beginning t' of the next step by comparing AL to the new landscape. In the event the objectives are not attained, we speak of a fracture for the CR.

14. interplay between CRs Fracture CR' CR Operative procedure interplay CR procedures Coregulators The procedures of the various CRs at a given time may not fit together. since their rhythms and perspectives are different. The operative procedure actually carried out on the system comes from an equilibration process between them, the interplay among the CRs, in which the complex switches play an important role. It may by-pass the procedures of some CRs, thus causing more or less severe events to them (fracture or, if the fracture persists, dyschrony). In particular each CR has structural temporal constraints to be respected so that its step beginning at t be achieved in time. They areexpressed by the inequalities ("laws of synch”): p(t) << d(t) << z(t) where p(t) is the time lag (= mean propagation delay of the links in the landscape), d(t) is the period of the CR (= mean length of its preceding steps) and z(t) is the smallest stability span of the necessary components C (= period during which C has a stable lower order decomposition).

15. DIFFERENT KINDS OF EVENTS FOR A CR The non-respect of the temporal conditions of a CR may cause various events, and in particular fractures in the following cases: (i) an increase of the time lags, so that information and commands are not sent in time: the landscape is not well constructed or the procedure is not realized in time; (ii) no admissible procedure S is found, or the commands of S cannot be effected; (iii) a decrease of the stability spans: the information is no more valid or the strategy cannot be realized. A fracture not repaired at the next step causes a more severe event, namely a dyschrony; if it persists, it might necessitate a change of period of the CR, which we call a re-synchronization of the CR. This may backfire to CRs of increasing levels, leading to systemic complex events such as a cascade of re-synchronizations, or even a systemic "dynamic disease".

16. DIALECTICS BETWEEN HETEROGENEOUS CRs fracture fracture → new procedure fracture fracture CR is a lower coregulator, with small steps, and CR' is a much higher one with much longer steps. A sequence of events at CR during successive steps, and the corresponding changes at the lower level are not transmitted in real time to CR' (propagation delays,…). However their accumulation may cause a noticeable event for CR' up to causing a fracture. The response of CR' and its change of procedure may backfire to CR by causing a fracture at its level, and the process can repeat. FUMEE 3

17. COMPLEX EVENTS PROCESSING AMONG CRs How different kinds of events for a CR backfire to CRs of other levels, with possibly severe consequences, such as a cascade of fractures, itself leading to a cascade of re-synchronizations at various levels to avoid a systemic disease. This CEP is at the root of our physiologically inspired Theory of Aging for an organism by a cascade of re-synchronizations (EV 1993), whichwould also apply to a social system

18. THE ARCHETYPAL CORE AT THE BASIS OF U-CEP Memory CR1 CR2 AC A P Q Cognitive systems develop over time a sub-system of the memory, the Archetypal Core AC, which integrates and intertwines recurring memories and notable events. and acts as a flexible internal model of the system, memorizing its identity.ACconsists of higher order components linked by strong and swift links forming self-maintained loops. Activation of part of AC is first propagated through these loops to a large domain of AC, then to ramifications of each A in it, with possible complex switches between them, thus forming an increased activated domain. This sequence of events is transmitted back to higher CRs which can unite their landscapes into a longer term global landscape on which ubiquitous complex events can be processed.

19. U-CEP. CONSCIOUS PROCESSES Attention ↑ AC is activated S Activated domain ↑ Activated intentional CRs cooperate Prospection Global Landscape CR < Retrospection In cognitive systems, a non-expected event S increases the attention, that leads to an activation of a large part of AC, hence to the formation of a global landscape GL, in which conscious processes, characterized by an integration of the time dimension, can develop: First a retrospection process(in several steps) allows recollecting the recent past for analyzing the event S and its possible causes (via an 'abduction' process). Then a prospection process can be developed in the long term GL, still using the motor role of AC, to iteratively construct virtual landscapes in which sequences of procedures (or scenarios)are tried with no-cost evaluation of the possible events, and an adequate one is selected.

20. FOR MORE INFORMATION Memory Evolutive Systems: Hierarchy, Emergence, Cognition, Elsevier, 2007. MENS, a mathematical model for cognitive systems, JMT 0-2, 2009. The following internet sites contain a number of papers, in particular more recent ones: http://pagesperso-orange.fr/ehres http://pagesperso-orange.fr/vbm-ehr THANKS