"Defining Emergent Descriptions by Information Preservation"
The task of defining emergence in a suitable way is an important issue in the research of complex systems and its difficulty is apparent from the larger number of various of formal definitions brought forward to this purpose. Recently, e.g. in (Rasmussen et al. 2001), a promising formalization has been made based on category theory. One of the problems remaining with definitions based on meta-models like category theory is that, while precise, in practice they are often computationally inaccessible and they make it difficult to allow for a natural concept of emergence arising from the intrinsic structure of a system. Guided by the need to have a more generic, computationally accessible and useable mathematical model for emer-gence, here, we therefore propose a formal concept of emergent description by a decomposition of a stochastic dynamic system into approximately independent subsystems. Our approach is inspired by the natural decompo-sition of the collective dynamics in crystals into individual oscillatory modes, phonons (Born and Huang 1954). This case is limited by the fact that it requires the systems to be linear. In the field of synergetics, an approach to decompose also nonlinear systems in a natural fashion is undertaken (Haken 1983). The natural decomposition of dynamical systems near fixed points into stable, central and unstable manifolds is reinterpreted in a heuristic way, separating fast foliations and slow manifolds in the system (Mees 1981). The slow parameters (master modes) of the system dynamics are said to enslave the fast degrees of freedom (slave modes). The master modes can be construed as emerging from the system dynamics. This system decomposition into individual subsystems arises naturally from the system dynamics and is therefore not limited to the eye of the beholder (Harvey 2000) usually assumed to be needed with emergent phenomena. Also, decomposition is not limited to systems exhibiting different time scales. In fact, under certain conditions it is possible to decompose nonlinear dynamic systems canonically into weakly coupled subsystems even if they have no separate time scales (Winter 1997). Above decompositions can be further generalized using information theory. Haken (2000) takes a step towards an information-theoretical view of synergetics; it is well-known that dynamical systems can be well described using information theory (Wolf 1999; Deco and Schuermann 2001); the dynamics is translated into the information-dynamical language via an a priori choice of the space partition. However, it would be desirable to have an approach where the decomposition emerges naturally from the structure of the dynamical system and is not imposed upon it. Our approach to define emergent descriptions is based on taking up the decomposition principles derived from a close analysis of the aforementioned simplest linear dynamical systems (see second paragraph), generalizing them to nonlinear systems using information theory. We define emergent descriptions as a complete decomposition of the system into independent subsystems which are individually predictable (Polani 2002). These three aspects can be directly formulated in an information-theoretic fashion (and thus making them, at least in principle, com-putationally accessible): if 1. a general dynamical system (discrete or continuous, deterministic or stochastic) is decomposed in such a way that the totality of subsystems always provide complete information about the state of the total system, 2. these subsystems are informationally independent from each other s dynamics and 3. the indi-vidual subsystems maximally preserve information, we consider this decomposition a natural emergent description of the system. The plausibility, impact and usefulness of the concept is explored in several examples. The present concept is related, but goes beyond the related concepts of Independent Component Analysis and clustering by deterministic annealing by introducing a temporal dimension. The individual subdynamics can be interpreted as closed subsystems (or approximations thereof). In larger systems, we expect that one would obtain a hierarchy of subsystems with different degrees of independence. This would provide a useful perspective to model biological super-organisms, organisms, and organelles; it would allow to cover both the aspect of information exchange between different units and the structural organization of the system. Evolution is then just a specific form of system dynamics. Evolutionary transitions or important events will be reflected by structural changes in the identified hierarchy. A detailed study of the ability of the model to incorporate these effects is the topic of current research. References Born, M., and Huang, K., (1954). Dynamical Theory of Crystal Lattices. Oxford, England: Clarendon Press. Deco, G., and Schuermann, B., (2001). Information Dynamics: Foundations and Applications. Springer. Haken, H., (1983). Advanced synergetics. Berlin: Springer-Verlag. Haken, H., (2000). Information and Self-Organization. Springer Series in Synergetics. Springer. Harvey, I., (2000). The 3 Es of Artificial Life: Emergence, Embodiment and Evolution. Invited talk at Artificial Life VII, 1.-6. August, Portland. Mees, A. I., (1981). Dynamics of feedback systems. John Wiley & sons, Ltd. Polani, D., (2002). On Individuality, Emergence and Information Preservation. In Nehaniv, C. L., and te Boekhorst, R., editors, Proceedings of the Symposium on Evolvability and Individuality, 18-20 September 2002, St. Albans. University of Hertfordshire. Rasmussen, S., Baas, N., Mayer, B., Nilsson, M., and Olesen, M. W., (2001). Ansatz for Dynamical Hierarchies. Artificial Life, 7:329 353. Winter, S., (1997). Zerlegung von gekoppelten Dynamischen Systemen. Diploma thesis, Johannes Gutenberg-Universitaet Mainz. (In German). Wolf, F., (1999). Berechnung von Information und Komplexitaet in Zeitreihen - Analyse des Wasserhaushaltes von bewaldeten Einzugsgebieten. Dissertation, Universitaet Bayreuth. 2