Non-standard semantics of hybrid systems modelers

Hybrid system modelers have become a corner stone of complex embedded system development. Embedded systems include not only control components and software, but also physical devices. In this area, Simulink is a de facto standard design framework, and Modelica a new player. However, such tools raise several issues related to the lack of reproducibility of simulations (sensitivity to simulation parameters and to the choice of a simulation engine). In this paper we propose using techniques from non-standard analysis to define a semantic domain for hybrid systems. Non-standard analysis is an extension of classical analysis in which infinitesimal (the ε and η in the celebrated generic sentence ∀ε∃η . . . of college maths) can be manipulated as first class citizens. This approach allows us to define both a denotational semantics, a constructive semantics, and a Kahn Process Network semantics for hybrid systems, thus establishing simulation engines on a sound but flexible mathematical foundation. These semantics offer a clear distinction between the concerns of the numerical analyst (solving differential equations) and those of the computer scientist (generating execution schemes). We also discuss a number of practical and fundamental issues in hybrid system modelers that give rise to non reproducibility of results, nondeterminism, and undesirable side effects. Of particular importance are cascaded mode changes (also called “zero-crossings” in the context of hybrid systems modelers). Foreword: in memory of Amir Pnueli When we were invited to contribute to this special issue in honor of Amir Pnueli, we first felt deeply honored. Then, we asked ourselves what we could contribute that would best fit the recollection our community has of him. Amir loved opening new avenues, rich in surprises. While he established himself as a leading figure in computer science, he had a profound mathematical background. IThis work was supported by the SYNCHRONICS large scale initiative of INRIA. ∗Corresponding author Email addresses: [email protected] (Albert Benveniste), [email protected] (Timothy Bourke), [email protected] (Benôıt Caillaud), [email protected] (Marc Pouzet) Preprint submitted to Elsevier July 8, 2011 Finally, Amir is one of the founders of the area of hybrid systems from the perspective of the computer science community. These considerations led us to contribute to this special issue with a new approach to hybrid systems modelers that builds on the heterodox — but, we think, nonetheless useful — mathematical area of non-standard analysis.