Cellular automata and other discrete dynamical systems with memory

Abstract

In conventional discrete dynamical systems, the new configuration depends solely on the configuration at the preceding time step. This contribution considers an extension to the standard framework of dynamical systems by taking into consideration past history in a simple way: the mapping defining the transition rule of the system remains unaltered, but it is applied to a certain summary of past states. This kind of embedded memory implementation, of straightforward computer codification, allows for an easy systematic study of the effect of memory in discrete dynamical systems, and may inspire some useful ideas in using discrete systems with memory (DSM) as a tool for modeling non-markovian phenomena. Besides their potential applications, DSM have an aesthetic and mathematical interest on their own, as will be briefly over viewed. The contribution focuses on the study of systems discrete par excellence, i.e., with space, time and state variable being discrete. These discrete universes are known as cellular automata (CA) in their more structured forms, and Boolean networks (BN) in a more general way. Thus, the mappings which define the rules of CA (or BN) are not formally altered when implementing embedded memory, but they are applied to cells (or nodes) that exhibit trait states computed as a function of their own previous states. So to say, cells (or nodes) - canalize - memory to the mapping. Automata on networks and on proximity graphs, together with structurally dynamic cellular automata, will be also studied with memory. If time permits, systems that remain discrete in space and time, but not in the state variable (e.g., maps and spatial games), will be also scrutinized with memory. A list of references on DSM may be found in http://uncomp.uwe.ac.uk/alonso-sanz.