This project focuses on mathematical foundations and design methodologies for embedded hybrid systems (EHS). The essential feature of such systems is that software components interact not only with each other, but also with the physical world, through sensors and actuators. The discrete dynamics of computation thus adds up with the continuous dynamics of physical systems. Hybrid systems are an attempt to capture this double dynamism in a unified framework. To limit the interference of the formidable complexities of dynamical systems of both types, the continuous trajectories are usually encapsulated into states, at the static points of the discrete computational paths. The methods of continuous mathematics are then simply combined with the methods of discrete mathematics to analyze such combined systems.
The starting point of the planned research is a belief that the burgeoning field of coalgebra provides methods and techniques that will allow uniform representation and implementation methods for continuous and discrete objects and aspects. While algebraic methods allow specifying and programming of finite objects and inductive structures, such as expressions or well-founded trees, coalgebraic methods allow specifying and programming infinite objects, as coinductive structures: they include automata and various state machines on the one hand, as well as iterative function systems, analytic functions and operators, and real numbers on the other hand.
In a real sense, coinduction permeates analysis just like induction permeates arithmetic. The difference is that the latter has been recognized as a fundamental logical principle a long time ago, whereas the the former has been recognized only recently, although it has appeared implicitly for some time (e.g. in most existence-of-the-solutions theorems, although it has been recognized as backwards induction in game theory).
The task is now to make explicit and systematize the use of coinductive and coalgebraic methods, and to apply them in analysis and design of embedded hybrid systems.
