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.
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@cite_2: We present a new block diagram language for describing synchronous software. It coordinates the execution of synchronous, concurrent software modules, allowing real-time systems to be assembled from precompiled blocks specified in other languages. The semantics we present, based on fixed points, is deterministic even in the presence of instantaneous feedback. The execution policy develops a static schedule--a fixed order in which to execute the blocks that makes the system execution predictable.We present exact and heuristic algorithms for finding schedules that minimize system execution time, and show that good schedules can be found quickly. The scheduling algorithms are applicable to other problems where large systems of equations need to be solved.
@cite_8: A logic circuit is said to be combinational if the function it computes depends only on the inputs applied to the circuit, and is sequential if it depends on some past history in addition to the current inputs. Circuits that have an underlying topology that is acyclic are combinational, since feedback is a necessary condition for them to be sequential. However, it is not a sufficient condition since there exist combinational logic circuits that are cyclic. These occur often in bus structures in data paths. Traditional formal techniques in logic synthesis, logic analysis, and timing analysis of combinational circuits have restricted themselves to acyclic combinational circuits, since they have been unable to handle the analysis of circuits with cycles. Thus, in practice, these circuits are handled using clumsy work-arounds, which is obviously undesirable. This paper presents a formal analysis of these circuits and presents techniques for the logical and timing analysis of such circuits. These techniques are practically feasible on reasonably large circuits encountered in practice. >
Our idea to use relations with an unknown value is inspired from fix-point based methods such as , which are used to reason about instantaneous feedback in cyclic combinational circuits and synchronous languages @cite_8 @cite_11 @cite_2 . These methods deal with monotonic functions over the flat CPO with unknown as the least element. Because they assume total functions, they are limited to deterministic and input-receptive systems. Refinement is also absent from these frameworks.