Zélus : A Synchronous Language with ODEs

Timothy Bourke, Marc Pouzet

2013

Zélus is a new programming
language for modeling systems that mix discrete logical time and
continuous time behaviors. From a user's perspective, its main
originality is to extend an existing Lustre-like
synchronous language with Ordinary Differential Equations (ODEs). The
extension is conservative: any synchronous program expressed as
data-flow equations and hierarchical automata can be composed
arbitrarily with ODEs in the same source code.

A dedicated type system and causality analysis ensure that all
discrete changes are aligned with zero-crossing events so that no side
effects or discontinuities occur during integration. Programs are
statically scheduled and translated into sequential code that, by
construction, runs in bounded time and space. Compilation is effected
by source-to-source translation into a small synchronous subset which
is processed by a standard synchronous compiler architecture. The
resultant code is paired with an off-the-shelf numeric solver.

We show that it is possible to build a modeler for explicit hybrid
systems à la Simulink/Stateflow on top of
an existing synchronous language, using it both as a semantic basis
and as a target for code generation.

Synchronous programming languages (à la Lucid
Synchrone) are language designs for reactive systems with discrete
time. Zélus extends them gracefully to hybrid discrete/continuous
systems, to interact with the physical world, or simulate it -- while
preserving their strong semantic qualities.

The paper is short (6 pages) and centered around examples rather than
the theory -- I enjoyed it. Not being familiar with the domain, I was
unsure what the "zero-crossings" mentioned in the introductions are,
but there is a good explanation further down in the paper:

The standard way to detect events in a numeric solver is
via zero-crossings where a solver monitors expressions for changes in
sign and then, if they are detected, searches for a more precise
instant of crossing.

The Zélus website has a 'publications' page with
more advanced material, and an 'examples' page with
case studies.