The control of synchronous systems

In the synchronous composition of processes, one process may prevent another process from proceeding unless compositions without a well-defined product behavior are ruled out. They can be ruled out semantically, by insisting on the existence of certain fixed points, or syntactically, by equipping processes with types, which make the dependencies between input and output signals transparent. We classify various typing mechanisms and study their effects on the control problem. A static type enforces fixed, acyclic dependencies between input and output ports. For example, synchronous hardware without combinational loops can be typed statically. A dynamic type may vary the dependencies from state to state, while maintaining acyclicity, as in level-sensitive latches. Then, two dynamically typed processes can be syntactically compatible, if all pairs of possible dependencies are compatible, or semantically compatible, if in each state the combined dependencies remain acyclic. For a given plant process and control objective, there may be a controller of a static type, or only a controller of a syntactically compatible dynamic type, or only a controller of a semantically compatible dynamic type. We show this to be a strict hierarchy of possibilities, and we present algorithms and determine the complexity of the corresponding control problems. Furthermore, we consider versions of the control problem in which the type of the controller (static or dynamic) is given. We show that the solution of these fixed-type control problems requires the evaluation of partially ordered (Henkin) quantifiers on boolean formulas, and is therefore harder (nondeterministic exponential time) than more traditional control questions