Causal semantics for the algebra of connectors

The Algebra of Connectors is used to model structured interactions in the BIP component framework. Its terms are connectors, i.e. relations describing synchronization constraints between the ports of component-based systems. Connectors are structured combinations of two basic synchronization protocols between ports: rendezvous and broadcast. They are generated from the ports of P by using a binary fusion operator and a unary typing operator. Typing associates with terms (ports or connectors) synchronization types: trigger or synchron. In a previous paper, we studied interaction semantics for which defines the meaning of connectors as sets of interactions. This semantics reduces broadcasts into the set of their possible interactions and thus blurs the distinction between rendezvous and broadcast. It leads to exponentially complex models that cannot be a basis for efficient implementation. Furthermore, the induced semantic equivalence is not a congruence. For a subset of, we propose a new causal semantics that does not reduce broadcast into a set of rendezvous and explicitly models the causal dependency relation between triggers and synchrons. The Algebra of Causal Trees formalizes this subset. It is the set of the terms generated from interactions on the set of ports P, by using two operators: a causality operator and a parallel composition operator. Terms are sets of trees where the successor relation represents causal dependency between interactions: an interaction can participate in a global interaction only if its parent participates too. We show that causal semantics is consistent with interaction semantics. Furthermore, it defines an isomorphism between and the set of the terms of involving triggers. Finally, we define for causal trees a boolean representation in terms of causal rules.