Contents 1 Reactive Modules 1

block diagrams In block diagrams, we may choose to hide the internal structure of a module and view it as a black box with input and output wires. If the atom structure of a module is suppressed, we draw the module boundary as a solid line instead of a dotted line. In order to compose such abstract block diagrams, every module needs to be annotated with information about the await dependencies between variables. The amount of compatibility information that is both necessary and su cient is captured by the following de nition. Given a module P , a derived await dependency x dP y of P consists of an external variable x and an interface variable y such that x +P y. The derived await dependency x d P y indicates that the initial value of the interface variable y may depend on the initial value of the external variable x, and in every update round, the next value of y may depend on the next value of x. Therefore, P cannot be composed with a module Q with external variable y, interface variable x, and y dQ x. Exercise 1.6 fT2g [Derived await dependencies] Consider two modules P and Qwhose variables satisfy conditions (1a){(1c) for module compatibility. (a) Show that the derived await dependencies of the two modules contain exactly the information that is necessary and su cient for determining compatibility; that is, prove that P and Q are compatible i the transitive closure ( dP [ d Q) + is asymmetric. (b) Assuming P and Q are compatible, construct the derived await dependencies of the compound module PkQ from the derived await dependencies of the component modules. A block diagram for a module P must show either its internal structure or its derived await dependencies. Since it is immaterial if a derived await dependency is an actual await dependency (contained in P ) or only \derived" from other await dependencies (contained in +P ), we often omit the superscript d from the symbol .