Computing with Continuous Change

A central challenge in computer science and knowledge representation is the integration of conceptual frameworks for continuous and discrete change, as exemplified by the theory of differential equations and real analysis on the one hand, and the theory of programming languages on the other. We take the first steps towards such an integrated theory by presenting a recipe for the construction of continuous programming languages — languages in which state dynamics can be described by differential equations. The basic idea is to start with an untimed language and extend it uniformly over dense (real) time. We present a concrete mathematical model and language (the Hybrid concurrent constraint programming model, Hybrid cc) instantiating these ideas. The language is intended to be used for modeling and programming hybrid systems. The language is declarative — programs can be understood as formulas that place constraints on the (temporal) evolution of the system, with parallel composition regarded as conjunction. It is expressive — it allows the definition of continuous versions of the preemption control constructs. The language is obtained by extending the general-purpose computational formalism of (default) concurrent constraint programming (Default cc) with a single temporal construct, called hence — hence A is read as asserting that A holds continuously beyond the current instant. Various patterns of temporal activity can be generated from this single construct by use of the other combinators in Default cc. We provide a precise operational semantics according to which execution alternates between (i) points at which discontinuous change can occur, and (ii) open intervals in which the state of the system changes continuously. Transitions from a state of continuous evolution are triggered when some condition starts or stops holding. We show that the denotational semantics is correct for reasoning about the operational semantics, through an adequacy theorem.