## Section: Research Program

### Background on non-standard analysis

Non-Standard analysis plays a central role in our research on hybrid systems modeling [1], [21], [18], [17]. The following text provides a brief summary of this theory and gives some hints on its usefulness in the context of hybrid systems modeling. This presentation is based on our paper [1], a chapter of Simon Bliudze's PhD thesis [27], and a recent presentation of non-standard analysis, not axiomatic in style, due to the mathematician Lindström [50].

Non-standard numbers allowed us to reconsider the semantics of hybrid
systems and propose a radical alternative to the *super-dense
time semantics* developed by Edward Lee and his team as part of the
Ptolemy II project, where cascades of successive instants can occur in
zero time by using ${\mathbb{R}}_{+}\times \mathbb{N}$ as a time index. In the non-standard
semantics, the time index is defined as a set
$\mathbb{T}=\{n\partial \mid n\in {}^{*}\mathbb{N}\}$, where $\partial $ is an
*infinitesimal* and ${}^{*}\mathbb{N}$ is the set of *non-standard
integers*. Remark that 1/ $\mathbb{T}$ is dense in ${\mathbb{R}}_{+}$, making it
“continuous”, and 2/ every $t\in \mathbb{T}$ has a predecessor in $\mathbb{T}$ and a
successor in $\mathbb{T}$, making it “discrete”. Although it is not effective from
a computability point of view, the *non-standard semantics*
provides a framework that is familiar to the computer
scientist and at the same time efficient as a symbolic
abstraction. This makes it an excellent candidate for the development
of provably correct compilation schemes and type systems for hybrid
systems modeling languages.

Non-standard analysis was proposed by Abraham Robinson in the 1960s to allow the explicit manipulation of “infinitesimals” in analysis [56], [42], [12]. Robinson's approach is axiomatic; he proposes adding three new axioms to the basic Zermelo-Fraenkel (ZFC) framework. There has been much debate in the mathematical community as to whether it is worth considering non-standard analysis instead of staying with the traditional one. We do not enter this debate. The important thing for us is that non-standard analysis allows the use of the non-standard discretization of continuous dynamics “as if” it was operational.

Not surprisingly, such an idea is quite ancient. Iwasaki et al. [46] first proposed using non-standard analysis to discuss the nature of time in hybrid systems. Bliudze and Krob [28], [27] have also used non-standard analysis as a mathematical support for defining a system theory for hybrid systems. They discuss in detail the notion of “system” and investigate computability issues. The formalization they propose closely follows that of Turing machines, with a memory tape and a control mechanism.

The introduction to non-standard analysis in [27] is very pleasant and we take the liberty to borrow it. This presentation was originally due to Lindstrøm, see [50]. Its interest is that it does not require any fancy axiomatic material but only makes use of the axiom of choice — actually a weaker form of it. The proposed construction bears some resemblance to the construction of $\mathbb{R}$ as the set of equivalence classes of Cauchy sequences in $\mathbb{Q}$ modulo the equivalence relation $\left({u}_{n}\right)\approx \left({v}_{n}\right)$ iff ${lim}_{n\to \infty}({u}_{n}-{v}_{n})=0$.