The Control of Synchronous Systems, Part II

A controller is an environment for a system that achieves a par ticular control objective by providing inputs to the system without constrain ing the choices of the system For synchronous systems where system and controller make simultaneous and interdependent choices the notion that a controller must not constrain the choices of the system can be formalized by type systems for composability In a previous paper we solved the control problem for static and dynamic types a static type is a dependency relation between inputs and outputs and composition is well typed if it does not intro duce cyclic dependencies a dynamic type is a set of static types one for each state Static and dynamic types however cannot capture many important digital circuits such as gated clocks bidirectional buses and random access memory We therefore introduce more general type systems so called depen dent and bidirectional types for modeling these situations and we solve the corresponding control problems In a system with a dependent type the dependencies between inputs and outputs are determined gradually through a game of the system against the controller In a system with a bidirectional type also the distinction between inputs and outputs is resolved dynamically by such a game The game proceeds in several rounds In each round the system and the controller choose to update some variables dependent on variables that have already been updated The solution of the control problem for dependent and bidirectional types is based on algorithms for solving these games