A calculus for multi-level emergent behaviours in component-based systems and simulations

A major issue in Complexity Science is the formal description of emergent properties and behaviours in terms of lower level properties and behaviours. As a consequence, there are few techniques for empirically investigating specific emergent properties. In this paper, we introduce a general compositional approach to specifying such properties, using constraints to define representative sets of compositions. More specifically, we propose a calculus of complex events, which are compositions of events generated from component-level rule executions. Complex event types can be assembled hierarchically, giving a formal means of relating behaviours at different levels of abstraction. In being able to specify and then identify complex events of different types in systems and simulations, we have a method for empirically discovering relationships between behaviours defined at different levels. The formalism offers two important practical advantages. Firstly, higher level properties can be defined with different degrees of specificity so they can be defined with limited knowledge; we can then further sub-classify properties after they have been detected to discover differences in their constituent properties. Secondly, the formalism is related directly to the rules driving component behaviour so that all higher level behaviours can ultimately be decomposed into rule executions; this is particularly important for desirable and dysfunctional properties, and in circumstances where intervention at the component rule level is possible.