Analog and Hybrid Computation: Dynamical Systems and Programming Languages

The purpose of this article is to serve as a light-weight introduction into the mysteries of analog and hybrid computing models from a dynamical systems and programming languages perspective. Hybrid systems are the dynamical systems that combine both models of computation, i.e., have interacting discrete and continuous dynamics. They have found widespread application as models for embedded computing in embedded systems as well as in cyber-physical systems. The primary role hybrid systems have played so far is to allow us to model how a (discrete) computer controller interacts with the (continuous) physical world and to analyze by means of formal proofs or reachability analyzes whether this interaction is safe or not. Without any doubt, such analyzes are of tremendous importance for our society, because they determine whether we can bet our lives on those systems. But this article argues that hybrid systems also have computational consequences that make them an interesting subject to study from a computability theory perspective. Hybrid systems are described by hybrid programs or hybrid automata, both hybrid generalizations of corresponding discrete computational models. The phenomenon of discrete and continuous interplay, which hybrid systems provide, is fundamental and raises interesting computability questions. For example: what is computable using the analogue computation capabilities of continuous dynamical systems? How do the discrete computation capabilities of discrete dynamical systems relate to classical models of computation a la Church–Turing? What happens in hybrid computation, where discrete and continuous computation interact? Are the two facets of computation, discrete and continuous, of fundamentally different character or are they two sides of the same computational coin? This article answers some of these questions using the rich theory that a logical characterization of hybrid systems in differential dynamic logic of hybrid programs provides. But the article is meant primarily as a manifesto for the significance and inherent beauty that these questions possess in the first place.