Combinatory Systems and Combinatory Automata:simulating Self-Organization and Chaos in collective phenomena

I aim to present two ideas – Combinatory System and Combinatory Automaton – and to demonstrate their connection in order to describe collective behaviour. In plain words I define as (social) combinatory systems a particular class of unorganized systems made up of a collectivity of similar agents (not functionally specialized, not necessarily interconnected by evident interactions) each of which is capable of producing a micro behaviour, and a micro effect, analogous to that of the others. If, on the one hand, the macro behaviour of the System, as a whole, derives from the combination – appropriately specified (sum, product, average, min, max, etc.) – of the analogous behaviours (or effects) of its similar agents (hence the name Combinatory System), on the other hand the macro behaviour (or the macro effect) determines, or conditions, or directs, by necessity, the subsequent micro behaviours. A Combinatory Automaton is a simple tool to simulate combinatory systems. This is composed of a lattice, each of whose cells contains a variable representing the state of an agent. The value of each cell at time th depends on a synthetic global variable whose values derive from some operation carried out on the values of the cells and that represents the synthetic state of the automaton. The micro-macro feedback connects the analytical values of the cells and the synthetic state of the automaton. I will try to demonstrate, through simple examples, that combinatory systems represent a wide range of the behaviours of collectivities, that combinatory automata are a powerful tool for simulating the most relevant combinatory systems, and that combinatory systems, despite their simplicity, can show chaotic dynamics and, of course, path dependence.