Towards Compositional Feedback in Non-Deterministic and Non-Input-Receptive Systems

Feedback is an essential composition operator in many classes of reactive and other systems. This paper studies feedback in the context of compositional theories with refinement. Such theories allow to reason about systems on a component-by-component basis, and to characterize substitutability as a refinement relation. Although compositional theories of feedback do exist, they are limited either to deterministic systems (functions) or input-receptive systems (total relations). In this work we propose a compositional theory of feedback which applies to non-deterministic and non-input-receptive systems (e.g., partial relations). To achieve this, we use the semantic frameworks of predicate and property transformers, and relations with fail and unknown values. We show how to define instantaneous feedback for stateless systems and feedback with unit delay for stateful systems. Both operations preserve the refinement relation, and both can be applied to non-deterministic and non-input-receptive systems.