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Section: Research Program

Background on non-standard analysis

Non-Standard analysis plays a central role in our research on hybrid systems modeling [1], [20], [17], [16]. 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  [25], and a recent presentation of non-standard analysis, not axiomatic in style, due to the mathematician Lindström  [47].

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  [53], [41], [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.  [43] first proposed using non-standard analysis to discuss the nature of time in hybrid systems. Bliudze and Krob  [26], [25] 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.