The purpose of this study is to show how simple “collectivities” of non-interconnected similar agents, which the author has termed “combinatory systems” and which produce analogous micro behaviors, reveal very interesting forms of micro and macro behaviors and effects attributable to a cybernetic mechanism the author shall call “micro-macro feedback”. On the one hand, the macro behavior of the system as a whole derives from the “combination” of the analogous micro behaviors or effects of the agents, and on the other hand, the macro behavior determines, conditions or directs the subsequent micro behavior, thereby creating observable effects and patterns in the collectivity.
This paper proposes a new combinatory system theory (CSysT) by constructing a formal model that explains a vast group of phenomena produced by the cybernetic behavior of the collectivity as if an internal organizer were regulating the micro dynamics of agents, producing self-organization, synchronization, path dependence and chaos.
In addition to illustrating the CSysT, this study also proposes a new and powerful tool to simulate combinatory systems: the “combinatory automaton”. 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 each time depends on a synthetic global variable whose values derive from some operations 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.
The CSysT suggests how to control combinatory systems through external actions aimed at making the macro and micro behaviors conform to the desired behaviors. The control is carried out through suitable strengthening or weakening actions, which operate by acting directly on the macro behavior – the author will define this as macro or external control – or by influencing the micro behaviors; in this case, the control will be called micro or internal control. The macro-level control is achieved through strengthening or weakening actions aimed at modifying some recombining factor. Instead, the micro-level control acts on the necessitating factors.
The CSysT is original and represents an effective tool for observing collective behavior. Combinatory systems are not easily recognizable; nevertheless, they are widely diffused and produce most of the social and economic collective phenomena involving the accumulation of objects, the spread of features or information, the pursuit of a limit and the achievement of general progress as the consequence of the individual pursuit of particular interests.
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