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 +× as a time index. In the non-standard semantics, the time index is defined as a set 𝕋={nn*}, where is an infinitesimal and * is the set of non-standard integers. Remark that 1/ 𝕋 is dense in +, making it “continuous”, and 2/ every t𝕋 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 as the set of equivalence classes of Cauchy sequences in modulo the equivalence relation (un)(vn) iff limn(un-vn)=0.