Trace semantics for concurrent objects

The theory of traces (the theory of free partial commutative monoids) is used to describe the behavior of systems of concurrent objects. A composition operation on systems and a synchronization operation on behaviors are presented: they allow to derive the behavior of complex systems from the behavior of its components. To address fundamental issues of concurrent semantics for objects in general, we concentrate on a simple model where objects communicate via asynchronous message passing and are only capable of sending messages, creating more objects and changing the way they react to incoming messages. The semantics is built from two simple concepts: a set of events representing the reception of messages by objects, and a binary symmetric and irre exive relation on events|independence|representing permissible concurrency. Causality, the dual notion of concurrency, is expressed by the dependence relation|the complement of independence. Executions of systems are described by a traces: labeled acyclic graphs where nodes are labeled with events and the only edges are between di erent nodes labeled with dependent events. The behavior of a system is the set of traces representing all possible executions. We achieve a simple and intuitive description of the behavior of concurrent objects where both causality and concurrency are explicit. When compared to other semantics for concurrent objects, trace semantics yields a more abstract description, in the sense that more similarly behaving computations are identi ed.