Anticipation in MES – Memory Evolutive Systems

Abstract

The aim of this chapter is to study anticipation in autonomous adaptive systems, such as biological and social systems, which have a multilevel multi-agent organization and develop a robust though flexible long-term memory. The problem is to find the characteristics allowing the system, through some of its agent(s), (i) to enhance its comprehension of the nature and role of anticipation (what Riel Miller (Futures J Policy Plan Future Stud 39:341–362, 2007) calls “Futures Literacy”) and (ii) to use this knowledge to search for possible procedures and virtually evaluate their impact on behavior, decision-making, and/or future action. The present study concerns open systems during their ongoing evolution. Thus it is different from Rosen’s anticipatory systems in which anticipation results from the existence of an internal predictive model of the invariant structure of the system.

The study is done in the frame of the Memory Evolutive Systems (Ehresmann and Vanbremeersch, Bull Math Bio 49(1):13–50, 1987; Memory evolutive systems: hierarchy, emergence, cognition. Elsevier, 2007), a mathematical approach to “living” systems, based on a “dynamic” category theory incorporating time. The main characteristic making these systems capable of developing anticipatory processes is identified as a kind of “operational redundancy” called the Multiplicity Principle. MP allows the progressive emergence, in the memory, of multifaceted dynamical records of increasing complexity which are flexible enough to adapt to changes. In social systems, a group of interacting people can develop a shared higher-level hub of the memory of the system, its archetypal pattern, which acts as a motor in the development of anticipatory processes. An application is given to the “Futures Literacy” program of Riel Miller, with a comparison of its three phases with the three types of creativity distinguished by Boden (The creative mind; myths and mechanisms, 2nd edn. Routledge, 2004).

Appendix

B asic Notions of Category Theory

For more details on Category Theory, cf. the book of Mac Lane (1971).

In this chapter, a graph (or more precisely directed multi-graph) consists of a set of objects (its vertices) and a set of arrows (or directed edges) between them.

A category K is a graph equipped with an internal composition law which maps a path (f: A → B, g: B → C) from A to C on an arrow gf: A → C (called its composite) and which satisfies the conditions: it is associative and each vertex A has an identity id_A: A → A; a vertex of the graph is called an object of K and an arrow a morphism or a link.

A functor k from K to K’ is a map which associates to each object of K an object k(A) of K’, to each arrow from A to B in K an arrow from k(A) to k(B) in K’ and which preserves identities and composition. A partial functor from K to K’ is a functor from a sub-category of K to K’.

Yoneda Lemma. Two objects A and B of K are isomorphic if and only if the functors Hom(A, −) and Hom(B, −) are equivalent, where Hom(A, −): K → Sets associates to an object C the set Hom(A, C) of morphisms f from A to C, and to g: C → C’ the map Hom(A, g) mapping f onto the composite gf.

An Evolutive system (EV 1987) is a functor from the category associated to a total order (in this paper, the order induced by R on a part T of R) to the category ParCat of partial functors between (small) categories.

A pattern in K (also called a diagram) is defined as a homomorphism P of a graph sP to K mapping an object i of sP on P_i and an arrow x: i → j of sP on the morphism f = P(x): P_i → P_j. The P_i are called components of P and the P(x) the (given) links of P.

A cone from a pattern P to A is a family of morphisms s_i: P_i → A satisfying the equations s_i = s_j P(x) for each x: i → j of sP. The definition of a cone using explicitly the composition law: a cone cannot be defined in a graph which is not a category.

cP is the colimit (or binding) of the pattern P in K if there is a colimit-cone (c_i)_i from P to cP such that, for each cone (s_i)_i from P to A, there is one and only one morphism s binding it, meaning that s: cP → A satisfies the equation s_i = sc_i for each i. If the colimit exists, it is unique (up to an isomorphism); however, different patterns may have the same colimit without being isomorphic.

A cluster from P to a pattern P′ is a maximal set G of links satisfying the following conditions: (i) For each P_i there is at least one link in G from P_i to some P’_j; and if there are several such links, they are correlated by a zigzag of links of P′. (ii) The composite of a link in G with a link of P′ and the composite of a link of P with a morphism in G belong to G.

If P and P′ have respectively colimits C and C’, it follows from the definition of a colimit that the cluster G binds into a unique morphism cG from C to C’; it is called a (P, P′)-simple link. (Cf. Figure 4.)

In a category, two patterns with the same colimit C are structurally non-connected if they are not isomorphic and there is no cluster between them binding into the identity of C. In this case, C is said to be multifaceted.

Reduction Theorem

In a hierarchical category, let (P, (Πⁱ)) be a ramification of an object C of level 2 (EV 1996). If all the morphisms in P are simple, then the complexity order of C is 1. If one of the morphisms of P is complex, C can be of complexity order 2. This result extends to higher levels.

In the first case, C is also the (simple) colimit of a pattern of level 0 whose objects are all the objects of the different Πⁱ and the morphisms are generated by those of the Πⁱ and those of the clusters that the morphisms of P bind. Such a pattern does not exist in the second case.

T he Complexification Process

A(n inductive) semi-sketch on a category K consists in data Pr = (S, A, Π, Λ) where S is a subset of K, A a graph, Π a set of patterns in K without a colimit in K, and Λ a set of inductive cones in K.

A model of S in a category M is a functor F from the largest sub-category included in (KUA)\S to M such that (i) the image by F of the cones in Λ are colimit-cones in M and (ii) FP’ admits a colimit cP’ in M for each P′ in Π.

Complexification Theorem

A semi-sketch Pr on a category K has an “initial” model F: K° → K’; the category K’ is called the complexification of K for Pr.

Let us sketch the construction of K’. The objects of K’ are the objects of K not suppressed by Pr and, for each pattern P′ in Π, a new object cP’ which becomes the colimit of P’ in K′ (if 2 patterns P′ and P″ have the same operational role n K, we take cP’ = cP”). The morphisms are constructed in successive steps, to “force” cP’ to become the colimit of P’ and to ensure that each path of morphisms admits a composite. For instance (cf. Fig. 5), if P and Q are patterns having a colimit C, a cluster G of links from P to P’ ϵ Π must bind into a new simple link cG from C to cP’, and a cluster H form Q’ to Q binds into a new cH; then the path (cH, cG) from cQ’ to cP’ consisting of simple links binding non-adjacent clusters must have a composite c which is a complex link from cQ’ to cP’.

The interest of the complexification process is that it allows for the emergence of higher-complexity order components with complex links between them, due to the Theorem:

Iterated Complexification Theorem

Let K′ be a complexification of a hierarchical category K satisfying MP and K″ be a complexification of K′ (EV 2007). Then K″ may admit multifold objects of a higher complexity order than K and complex links between them. And there may be no semi-sketch on K for which K″ would be the (first) complexification of K.

More precisely, let us take a semi-sketch Pr’ = (S’, Π’, Λ’) on K′ such that one of the patterns R in Π’ contains a complex link c which has emerged in K′. The complexification K″ adds a colimit cR to R, and the complexity order of cR is higher than that of the objects of R because of the Reduction Theorem.

Remark

In a complexification process, we can think of the category as a material cause and the procedure as an efficient cause. Then the fact that, in a MES representing an organism, two successive complexifications might not be reduced to a unique one justifies the hypothesis of Rosen (1985a) that organisms are characterized by the separation of material and efficient Aristotle’s causes (cf. EV 2007) .

The Archetypal Core

In the model MENS for a neuro-cognitive system (cf. section “Different Kinds of Anticipation”), the Archetypal Core (EV 2007, 2009) is an evolutive subsystem AC forming a higher-level hub of the memory. It consists of often recalled multifaceted records of higher-complexity order, which blend factual or conceptual knowledge, associated past experiences with their sensations, and affects and values; these records are strongly connected by loops of strong complex links self-maintaining their activity for some time. Formally, it is constructed through successive complexifications of the structural core of the brain, a strongly connected central part of the human cortex (discovered by Hagmann et al. 2008) which gives “an important structural basis for shaping large-scale brain dynamics.”

AC acts as a motor for the development of higher cognitive processes such as consciousness, creativity, and anticipation (EV 2009).