A duration calculus with neighborhood modalities

Abstract

To reason about continuous processes in some areas of artificial intelligence and embedded systems one has to express real-time properties. For such purpose a real-time logic has to be considered. Various such logics have been proposed. Some of these formalisms interpret formulas over intervals of time. These are called interval logics. Zhou Chaochen and Michael Hansen have introduced one such first-order interval logic called Neighborhood Logic (NL) which has two expanding modalities ◊r and ◊l. They have shown the adequacy of these modalities by deriving other unary and binary modalities from them; in particular, they show that chop can be expressed in terms of these modalities. The logic is subsequently expanded to get a Duration Calculus (called DC/NL) by expressing temporal variables as integrals of state variables. Liveness and fairness properties of a computing system can now be expressed in DC/NL by means of neighborhood (expanding) modalities. We take up an investigation of DC/NL in detail. We present a proof system for it. We further show that most of the results that hold in the original Duration Calculus continue to hold in our logic with some modifications. In this process we discuss soundness, completeness, and decidability results for DC/NL. Thus this paper establishes DC/NL as a useful formalism for specifying and verifying properties of real-time computing systems.