Space-Time Viewpoints for Concurrent Processes Represented by Relational Structures

In contrast to the standard physical theories which model systems by the continuum, Petri proposed a combinatorial representation of a spacetime — concurrent structures (with occurrence nets as a special case thereof) in which notions corresponding to the relativistic concepts of world line and causal cone can be defined by means of the concurrency and causal dependence relations, repectively. As a result, the density notion of the continuum model is replaced by several properties — so called concurrency axioms (including K-density, N-density, etc.). K-density is based on the idea that at any time instant, any sequential subprocess of a concurrent structure must be in some state or changing its state. N-density can be viewed as a sort of local density. Furthermore, it has turned out that concurrency axioms allow avoiding inconsistency between syntactic and semantic representations of processes and, thereby, to exclude unreasonable processes represented by the concurrent structures. Petri’s occurrence nets model system behaviors by occurrences of local states (also called conditions) and of events which are partially ordered. The partial order is interpreted as a kind of causal dependency relation. Also, occurrence nets are endowed with a symmetric, but in general non-transitive, concurrency relation — absence of the causality. Poset models do not discriminate between conditions and events. The power and limitations of concurrency axioms in the context of occurrence nets [4, 5] and posets [6, 9, 20] have been widely studied to allow better understanding the connections of causality and concurrency relations between systems events. In contrast to these treatments, the authors of the paper [16] have dealt with causality and concurrency on cyclic processes represented by net models which do not require an underlying partial order of causality. The paper [21] has studied the interrelations between concurrency axioms in the setting of prime event structures (occurrence nets with forward hereditary (w.r.t. causality) conflicts), where the nondeterministic aspects of concurrent processes are explicitly described. In the more recent papers [1, 2], algebraic and orthomodular lattices (the elements of the lattices are the closed subsets w.r.t. a closure operator, defined starting from the concurrency relation) have been generated from occurrence nets with and