Reasoning about synchronous systems

Abstract

In this paper we describe a simple semantic model for synchronous systems of processes, suitable for high level functional descriptions of VLSI designs, and use it to justify formal reasoning about the behaviour of systems. Treating a system as a directed graph in which the nodes represent computational units and the arcs indicate the communication links and t ime dependencies, we define the outputs of each node in a system as a function of its inputs. The inputs and outputs are regarded as data-valued functions of t ime. Basically the idea is to specify the semantics of a system as a set of (mutually recursive) function definitions. This set of function definitions amounts to a fixed point equation whose solution is the desired semantics of the system. A solution to these equations is guaranteed to exist under reasonable assumptions about the type of computational unit used in the system. For particularly regular or simple systems, the solutions will be explicitly determinable by standard methods such as substitution. However, even in cases when this is not possible, the solutions can be found by elementary fixed point techniques. This model of systems allows extremely easy and elegant proofs of some interesting results on retimings and other system transformations such as pipelining. A retiming is a transformation of the communicat ion graph of a system which preserves the underlying graph but alters the internode delays in a uniform manner; retiming has a simple effect on the semantics of a system. These results were first obtained by Leiserson and Saxe [12], but only under certain assumptions on the underlying communication graphs of systems. We show that these assumptions are unnecessary. Moreover, these authors were able only to give a long and somewhat complicated proof, because of their choice of semantic model. The fixed point method also serves as mathematical basis for an algebraic approach to VLSI design, such as the one described by Kung and Lin [9]. Again, however, their results were obtained under certain assumptions (well-definedness) which we show to be unnecessary. We demonstrate the use of our semantic model to develop and justify a design for a palindrome recogniser, beginning from a mathematical description of the problem.