Accessibility setting

Section: Research Program

Background on non-standard analysis

Non-Standard analysis plays a central role in our research on hybrid systems modeling [3] , [6] , [15] , [14] . 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 [3] , a chapter of Simon Bliudze's PhD thesis  [21] , and a recent presentation of non-standard analysis, not axiomatic in style, due to the mathematician Lindström  [41] .

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  [48] , [35] , [10] . 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.  [37] first proposed using non-standard analysis to discuss the nature of time in hybrid systems. Bliudze and Krob  [22] , [21] 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  [21] is very pleasant and we take the liberty to borrow it. This presentation was originally due to Lindstrøm, see  [41] . 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 ${lim}_{n\to \infty }(u_n-v_n)=0$.

Motivation and intuitive introduction

We begin with an intuitive introduction to the construction of the non-standard reals. The goal is to augment $ℝ\cup \{\pm \infty \}$ by adding, to each $x$ in the set, a set of elements that are “infinitesimally close” to it. We will call the resulting set ${}^{*}ℝ$. Another requirement is that all operations and relations defined on $ℝ$ should extend to ${}^{*}ℝ$.

A first idea is to represent such additional numbers as convergent sequences of reals. For example, elements infinitesimally close to the real number zero are the sequences $u_n=1/n$, $v_n=1/\sqrt{n}$ and $w_n=1/n^2$. Observe that the above three sequences can be ordered: $v_n>u_n>w_n>0$ where 0 denotes the constant zero sequence. Of course, infinitely large elements (close to $+\infty$) can also be considered, e.g., sequences $x_u=n$, $y_n=\sqrt{n}$, and $z_n=n^2$.

Unfortunately, this way of defining ${}^{*}ℝ$ does not yield a total order since two sequences converging to zero cannot always be compared: if $u_n$ and $u_n^{\text{'}}$ are two such sequences, the three sets $\{n\mid u_n>u_n^{\text{'}}\}$, $\{n\mid u_n=u_n^{\text{'}}\}$, and $\{n\mid u_n<u_n^{\text{'}}\}$ may even all be infinitely large. The beautiful idea of Lindstrøm is to enforce that exactly one of the above sets is important and the other two can be neglected. This is achieved by fixing once and for all a finitely additive positive measure $\mu$ over the set $\mathbb{N}$ of integers with the following properties: (The existence of such a measure is non trivial and is explained later.)

1. $\mu :2^{\mathbb{N}}\to \{0,1\}$;

2. $\mu \left(X\right)=0$ whenever $X$ is finite;

3. $\mu \left(\mathbb{N}\right)=1$.

Now, once $\mu$ is fixed, one can compare any two sequences: for the above case, exactly one of the three sets must have $\mu$-measure 1 and the others must have $\mu$-measure 0. Thus, say that $u>u^{\text{'}},u=u^{\text{'}}$, or $u<u^{\text{'}}$, if $\mu (\{n\mid u_n>u_n^{\text{'}}\}=1)$, $\mu \left(\{n\mid u_n=u_n^{\text{'}}\}\right)=1$, or $\mu \left(\{n\mid u_n<u_n^{\text{'}}\}\right)=1$, respectively. Indeed, the same trick works for many other relations and operations on non-standard real numbers, as we shall see. We now proceed with a more formal presentation.

Construction of non-standard domains

For $I$ an arbitrary set, a filter $ℱ$ over $I$ is a family of subsets of $I$ such that:

1. the empty set does not belong to $ℱ$,

2. $P,Q\in ℱ$ implies $P\cap Q\in ℱ$, and

3. $P\in ℱ$ and $P\subset Q\subseteq I$ implies $Q\in ℱ$.

Consequently, $ℱ$ cannot contain both a set $P$ and its complement $P^c$. A filter that contains one of the two for any subset $P\subseteq I$ is called an ultra-filter. At this point we recall Zorn's lemma, known to be equivalent to the axiom of choice:

Lemma 1 (Zorn's lemma) Any partially ordered set $(X,\le )$ such that any chain in $X$ possesses an upper bound has a maximal element.

A filter $ℱ$ over $I$ is an ultra-filter if and only if it is maximal with respect to set inclusion. By Zorn's lemma, any filter $ℱ$ over $I$ can be extended to an ultra-filter over $I$. Now, if $I$ is infinite, the family of sets $ℱ=$ $\{P\subseteq I\mid P^c\text{is}\text{finite}\}$ is a free filter, meaning it contains no finite set. It can thus be extended to a free ultra-filter over $I$:

Lemma 2 Any infinite set has a free ultra-filter.

Every free ultra-filter $ℱ$ over $I$ uniquely defines, by setting $\mu \left(P\right)=1$ if $P\in ℱ$ and otherwise 0, a finitely additive measure (Observe that, as a consequence, $\mu$ cannot be sigma-additive (in contrast to probability measures or Radon measures) in that it is not true that $\mu \left({\bigcup }_n{A}_n\right)={\sum }_n\mu \left(A_n\right)$ holds for an infinite denumerable sequence $A_n$ of pairwise disjoint subsets of $\mathbb{N}$.) $\mu :2^I\mapsto \{0,1\}$, which satisfies

Now, fix an infinite set $I$ and a finitely additive measure $\mu$ over $I$ as above. Let $𝕏$ be a set and consider the Cartesian product ${𝕏}^I={\left(x_i\right)}_{i\in I}$. Define $\left(x_i\right)\approx \left(x_i^{\text{'}}\right)$ iff $\mu \{i\in I\mid x_i\ne x_i^{\text{'}}\}=0$. Relation $\approx$ is an equivalence relation whose equivalence classes are denoted by $\left[x_i\right]$ and we define:

$𝕏$ is naturally embedded into ${}^{*}𝕏$ by mapping every $x\in 𝕏$ to the constant tuple such that $x_i=x$ for every $i\in I$. Any algebraic structure over $𝕏$ (group, ring, field) carries over to ${}^{*}𝕏$ by almost point-wise extension. In particular, if $\left[x_i\right]\ne 0$, meaning that $\mu \{i\mid x_i=0\}=0$ we can define its inverse ${\left[x_i\right]}^{-1}$ by taking $y_i=x_i^{-1}$ if $x_i\ne 0$ and $y_i=0$ otherwise. This construction yields $\mu \{i\mid y_i{x}_i=1\}=1$, whence $\left[y_i\right]\left[x_i\right]=1$ in ${}^{*}𝕏$. The existence of an inverse for any non-zero element of a ring is indeed stated by the formula: $\forall x(x=0\vee \exists y(xy=1\left)\right)$. More generally:

Lemma 3 (Transfer Principle) Every first order formula is true over ${}^{*}𝕏$ iff it is true over $𝕏$.

The above general construction can simply be applied to $𝕏=ℝ$ and $I=\mathbb{N}$. The result is denoted ${}^{*}ℝ$; it is a field according to the transfer principle. By the same principle, ${}^{*}ℝ$ is totally ordered by $\left[u_n\right]\le \left[v_n\right]$ iff $\mu \{n\mid u_n>v_n\}=0$. We claim that, for any finite $\left[x_n\right]\in {}^{*}ℝ$, there exists a unique $\mathrm{𝑠𝑡}\left(\left[x_n\right]\right)$, call it the standard part of $\left[x_n\right]$, such that

To prove this, let $x=sup\{u\in ℝ\mid \left[u\right]\le \left[x_n\right]\}$, where $\left[u\right]$ denotes the constant sequence equal to $u$. Since $\left[x_n\right]$ is finite, $x$ exists and we only need to show that $\left[x_n\right]-x$ is infinitesimal. If not, then there exists $y\in ℝ,y>0$ such that $y<|x-[x_n\left]\right|$, that is, either $x<\left[x_n\right]-\left[y\right]$ or $x>\left[x_n\right]+\left[y\right]$, which both contradict the definition of $x$. The uniqueness of $x$ is clear, thus we can define $\mathrm{𝑠𝑡}\left(\left[x_n\right]\right)=x$. Infinite non-standard reals have no standard part in $ℝ$.

It is also of interest to apply the general construction (1 ) to $𝕏=I=\mathbb{N}$, which results in the set ${}^{*}\mathbb{N}$ of non-standard natural numbers. The non-standard set ${}^{*}\mathbb{N}$ differs from $\mathbb{N}$ by the addition of infinite natural numbers, which are equivalence classes of sequences of integers whose essential limit is $+\infty$.