Section:
Research Program
Background on non-standard analysis
Non-Standard analysis plays a central role in our research on hybrid
systems modeling [3] ,
[9] , [10] , [6] . 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 [40] .
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
, where is an
infinitesimal and is the set of non-standard
integers is such that 1/ is dense in , making it
“continuous”, and 2/ every has a predecessor in and a
successor in , 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 [46] , [34] , [11] .
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.
[36] 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 [40] . 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 iff
.
Motivation and intuitive introduction
We begin with an intuitive introduction to the construction of the
non-standard reals.
The goal is to augment by adding, to each in
the set, a set of elements that are “infinitesimally close” to
it. We will call the resulting set .
Another requirement is that all
operations and relations defined on should extend to .
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 ,
and . Observe that the above three sequences can be
ordered: where 0 denotes the constant zero
sequence. Of course, infinitely large elements (close to )
can also be considered, e.g., sequences , , and
.
Unfortunately, this way of defining does not yield a total order
since two sequences converging to zero cannot always be
compared: if and are two such sequences, the three sets
, , and
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 over the set of
integers with the following properties: (The existence of such
a measure is non trivial and is explained later.)
-
;
-
whenever is finite;
-
.
Now, once is fixed, one can compare any two sequences: for the
above case, exactly one of the three sets must have
-measure 1 and the others must have -measure 0. Thus, say
that , or , if ,
, or ,
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 an arbitrary set, a filter over is a family of subsets of such that:
-
the empty set does not belong to ,
-
implies , and
-
and implies .
Consequently, cannot contain both a set and its complement
. A filter that contains one of the two for any subset
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 such that any chain in
possesses an upper bound has a maximal element.
A filter over is an ultra-filter if and only if it is maximal with
respect to set inclusion.
By Zorn's lemma, any filter over can be extended to an
ultra-filter over .
Now, if is infinite, the family of sets
is a free
filter, meaning it contains no finite set. It can thus be extended to
a free ultra-filter over :
Lemma 2
Any infinite set has a free ultra-filter.
Every free ultra-filter over uniquely defines, by setting
if and otherwise 0, a finitely additive
measure (Observe that, as a consequence, cannot be
sigma-additive (in contrast to probability measures or Radon
measures) in that it is not true that holds for an infinite denumerable sequence
of pairwise disjoint subsets of .) , which satisfies
Now, fix an infinite set and a finitely additive measure
over as above. Let be a set and consider the Cartesian
product . Define iff
. Relation is an
equivalence relation whose equivalence classes are denoted by
and we define
is naturally embedded into by
mapping every to the constant tuple such that for
every . Any algebraic structure over (group, ring,
field) carries over to by almost point-wise extension. In
particular, if , meaning that we
can define its inverse by taking if
and otherwise. This construction yields
, whence in .
The existence of an inverse for any non-zero element of a
ring is indeed stated by the formula: . More generally:
Lemma 3 (Transfer Principle)
Every first order formula is true over iff it is true over .
Non-standard reals and integers
The above general construction can simply be applied to and
.
The result is denoted ; it is a field according to the transfer
principle.
By the same principle, is totally ordered by
iff .
We claim that, for any finite , there exists a unique
, call it the standard part of , such that
To prove this, let
, where denotes the constant
sequence equal to . Since is finite, exists and we only
need to show that is infinitesimal. If not, then there
exists such that , that is, either
or , which both contradict
the definition of . The uniqueness of is clear, thus we can
define . Infinite non-standard reals have no standard
part in .
It is also of interest to apply the general construction
(1 ) to , which results in the set of
non-standard natural numbers.
The non-standard set differs from
by the addition of infinite natural numbers, which are
equivalence classes of sequences of integers whose essential limit is
.
Integrals and differential equations: the standardization principle
Any sequence of functions point-wise defines a function by setting
A function so obtained is called internal.
Properties of and operations on ordinary
functions extend point-wise to internal functions of
. The non-standard version of
is the internal function
. The same notions apply to sets. An internal
set is called hyperfinite if ; the cardinal of is defined as
.
Now, consider an infinite number and the set
By definition, if , then with
hence .
Now, consider an internal function and a hyperfinite set
.
The sum of over can be defined:
If is as above, and is a standard function, we obtain
Now, continuous implies , so,
Under the same assumptions, for any ,
Now, consider the following ODE:
Assume (8 ) possesses a solution such that the function is continuous.
Rewriting (8 ) in its equivalent integral form
and using (7 )
yields
The substitution in (9 ) of , which is positive
and infinitesimal, yields . The expression in parentheses on the right hand
side of (9 ) is the piecewise-constant right-continuous
function such that, for :
By (9 ), the solutions , of ODE (8 ), and
, as defined by recurrence
(10 ), are related by .
Formula (10 ) can be seen as a non-standard
operational semantics for ODE (8 ); one which depends on
the choice of infinitesimal step parameter .
Property (9 ), though, expresses the idea
that all these non-standard semantics are equivalent from the standard
viewpoint regardless of the choice made for .
This fact is referred to as the standardization principle.