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 ${ℝ}_{+}\times \mathbb{N}$ as a time index. In the non-standard semantics, the time index is defined as a set $𝕋=\{n\partial \mid n\in {}^{*}\mathbb{N}\}$, where $\partial$ is an infinitesimal and ${}^{*}\mathbb{N}$ is the set of non-standard integers. Remark that 1/ $𝕋$ is dense in ${ℝ}_{+}$, making it “continuous”, and 2/ every $t\in 𝕋$ has a predecessor in $𝕋$ and a successor in $𝕋$, 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 $ℝ$ 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 ${lim}_{n\to \infty }(u_n-v_n)=0$.