Lambda the Ultimate

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.