On Fixpoint Logics and Equivalences for Processes with Restricted Nondeterminism

Abstract
In concurrency, processes can be studied using a partial order or an interleaving semantics. In partial order semantics, at least four different kinds of behaviour can be recognised: concurrency, causality, conflict, and confusion. In interleaving semantics, only conflicts can be observed. All these features can be characterised in logical terms, and various logics have been defined for this purpose. For instance, Hennessy-Milner logic is a modal language that captures strong bisimilarity, the standard bisimulation equivalence for processes with interleaving semantics. However, when considering processes with partial order semantics, stronger equivalences are used and more discriminating logics are needed.

In the present paper, we study conditions to ease the definition of such logics and equivalences for processes with partial order semantics. More specifically, we study the impact that nondeterminism can have on some fixpoint modal logics and bisimulation equivalences for concurrent and multi-agent systems, when it is systematically restricted within the four kinds of behaviour mentioned above. Our results show that when the concurrency and confusion relations are taken to be deterministic, then the main equivalence for causal behaviour can be completely captured (even in logical and game-theoretic terms) by a simpler, weaker, more local bisimulation relation. We also provide key examples of the kinds of processes that can be modelled using deterministic confusion in order to illustrate the expressive power of the general framework defined here.