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.
@cite_22: The author further delineates and improves the evidence that nondeducibility on strategies is a respectable candidate for a definition of security against information compromise, at least for the class of systems that can be modeled as synchronized state machines. First, the author confirms the thesis of J.T. Wittbold and D.M. Johnson (1990) that nondeducibility on strategies is stronger than the notion of nondeducibility on inputs, defined by D. Sutherland (1986), which is generally viewed as a minimum requirement for security. Second, it is shown that nondeducibility on strategies is preserved when two machines that are secure by this definition are hooked up arbitrarily, even when loops are created by the interconnection. In order to make these more general hookups possible, it is necessary to generalize the definition of a synchronized state machine. >
In @cite_22 a model for synchronous nondeterministic systems is used for studying preservation of security properties by composition of systems, including delayed feedback. However this model does not treat non-input receptive systems.