## Section: New Results

### An algebraic theory of data-flow processing

Participants : Kenneth Johnson, Paul Le Guernic, Jean-Pierre Talpin.

The objective of this work is to define a uniform theoretical framework in the context of the algebraic theory of data for studying polychronous equational systems and their transformations from abstract specifications to real-time implementations on specific architecture. The Signal language is used as a concrete support for these studies.

In the algebraic theory of data, data types are modelled by an algebra consisting of an interface and an implementation. The interface is given by a signature consisting of symbols naming both data and operations on the data. The meaning, or implementation of the signature is given by a many sorted algebra consisting of carrier sets and functions on the sets implementing the data and operations named in the signature.

Our research will study the Signal framework as a data type of streams. A stream is a collection of of data distributed through time. Let the data come from a set V and time be points (or tags) in a set T. We model streams by partial functions assigning the data in V to points in T. Functions on streams are derived from functions on both data and time. Various sets T of tags are being defined depending upon the step in the design process. Morphisms on these sets will be used to define transformations (such as refinement, code distribution,...).

One major investigation is of the expressive power of the Signal framework. We aim to give an analysis of the type of system behaviour that may be expressed in terms of Signal equations. The choice of the Signal data and operations determine the expressiveness of the equations. Our research emphasises the process of choosing new data and new Signal operations. Our choices need not be limited to the discrete domain. We consider cases where data and tags are continuous sets such as the real numbers. Thus, we consider Signal processes operating in continuous time over continuous data.