Combinatory Systems and Automata: Simulating Self-Organization and Chaos in Collective Phenomena

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) represents a global information that 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 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.