Synchronous Structures

Synchronous languages have been designed to ease the de velopment of reactive systems by providing a methodological frame work for assisting system designers from the early stages of requirement speci cations to the nal stages of code generation or circuit produc tion Synchronous languages enable a very high level speci cation and an extremely modular design of complex reactive systems We de ne an order theoretical model that gives a uni ed mathematical formalization of all the above aspects of the synchronous methodology from relations to circuits The model has been speci ed and validated using a theo rem prover as part of the certi ed reference compiler of a synchronous programming language Introduction Synchronous languages such as Signal Lustre and Esterel have been designed to ease the development of reactive systems The syn chronous hypothesis provides a deterministic notion of concurrency where operations and communications are instantaneous In a synchronous lan guage concurrency is meant as a logical way to decompose the descrip tion of a system into a set of elementary communicating processes In teraction between concurrent components is conceptually performed by broadcasting events Synchronous languages enable a very high level spec i cation and an extremely modular design of complex reactive systems by structurally decomposing them into elementary processes The use of synchronous languages provides a methodological framework for assist ing the users from the early stages of requirement speci cations to the nal stages of code generation or circuit production while obeying com pliance to expressed and implied safety requirements In that context the synchronous language Signal is particularly interesting in that it allows the speci cation of early relational properties of systems which can then be progressively re ned in order to obtain an executable speci cation All the stages of this design process can easily be modeled and understood in isolation The purpose of our presentation is to de ne a mathematical model which gives a uni ed formalization of all the aspects of a synchronous methodology and which contains each of them in iso lation The model uses basic notions of set theory and order theory It has been speci ed and validated using the Coq proof assistant This implementation is part of a certi ed reference compiler of the Signal language It completes and extends the results of on the de nition of a co inductive trace semantics of Signal in Coq In uential Analogy In the great Italian mathematician Gero lamo Cardano wrote an important and in uential treatise on Algebra Ars Magna in which the rst complete expression for the solution of a general cubic equation was put forward Cardano noticed that in the case of some equation with three real solutions he was forced to take at a certain stage the square root of a negative number The imaginary numbers were borned Analogically we generalize the classical notion of signal with imaginary signals This extension has no material counterpart It is used to compute intermediate results For instance the temporal abstractions of signals called clocks have necessary a greatest lower bound but do not always have a real least upper bound In that case we need to de ne an imaginary least upper bound This axiomatiza tion allows to extend the notion of classical clocks a clock is a temporal abstraction of a signal with imaginary clocks and de ne a boolean lat tice of clocks In this lattice theoretical model temporal relations between signals always have a solution If the solution contains imaginary signals this means that the system has no real solution in the classical model and that it does not thus form an executable speci cation Plan We rst introduce the synchronous language Signal in the sec tion In the section we abstract the notion of control dependence in a mathematical structure that we call a synchronous structure Within this structure we formalize the notions of signals clocks and instants and their relations We de ne some internal operations on signals and clocks prove their algebraic properties prove that the set of clocks forms a boolean lattice and de ne a Cartesian closed category of signals with product and coproduct In the section we add a valuation function and a data dependency relation to synchronous structure In the section we brie y expose the outcome of our model for the compilation of programs written in the synchronous language Signal