Discrete Dynamical Systems with Memory: A New Tool for Modelling Complexity

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This project will investigate a novel approach to the simulation and modelling of complex natural and artificial systems. Conventionally, a system is called complex if it consists of a large number of simpler elements and, typically, the system's behaviour cannot be predicted by any means but computer simulation. This is why computational models are so important in modern biology, sociology, advanced engineering, ecology, agriculture and urban studies. A number of computational models have been presented with which to study and/or exploit aggregate behaviour and self-organisation from a number of simple interacting components. Typically, this work does not include memory, i.e., previous state information, at the component level but memory is an essential feature of all living systems and a significant part of physical, chemical, and engineering systems. We propose to undertake systematic studies in increasing both the amount and type of memory in the components of discrete dynamical systems with the aim of identifying new underlying principles of complex systems, using random Boolean networks and cellular automata in particular as such examples are well-studied in their basic forms.It is expected that the dynamics of such systems will better capture those of a wide variety of both natural and artificial phenomena. However the inverse problem for modelling even with memory-less discrete dynamical systems, such as the identification of the update rules for cellular automata, is a non-trivial task. The following problem must be tackled: given a highly non-linear system, design a computational model that will reconstruct the local rules of the system's behaviour and then simulate the global system. In the proposed research this will be achieved by developing a universal simulator framework based on discrete dynamical systems with memory, capable of recognizing local events from a series of global descriptions/snapshots of a given system, the parallel extraction of local rules, which govern the behaviour of the system's elements, and the, mostly unsupervised, design of a minimal complete model of the given system. Our approach to this is to cast the problem as a data mining task and to exploit machine learning techniques to perform the identification, which has recently been shown by us to be effective for traditional memory-less CAs.