Generalising Traces

In the classical Mazurkiewicz trace approach the behaviour of a concurrent system is described in terms of sequential observations that differ only with respect to their ordering of independent actions. This paper aims to determine a full generalisation of the trace model to the case that actions can be observed as occurring simultaneously. Thus observations are sequences of steps, i.e., sets of actions. This leads to an extended trace model based on three relations between events: simultaneity, serialisability, and interleaving. Whereas the underlying causal structures of traces are based on dependencies between actions leading to a partial order interpretation, more general causal orders are needed to describe the invariant relations between the action occurrences in a generalised trace. We present a complete picture including extended dependence graphs and a characterisation in terms of the (most) general order structures.