=Paper=
{{Paper
|id=Vol-156/paper-2
|storemode=property
|title=Reverse Leibniz, and then Bend It Like Beckham: Temporal Ontology Mapping as Problem-Solving Method
|pdfUrl=https://ceur-ws.org/Vol-156/paper2.pdf
|volume=Vol-156
|dblpUrl=https://dblp.org/rec/conf/kcap/Akkermans05
}}
==Reverse Leibniz, and then Bend It Like Beckham: Temporal Ontology Mapping as Problem-Solving Method==
https://ceur-ws.org/Vol-156/paper2.pdf
Reverse Leibniz, and then Bend It Like Beckham:
Temporal Ontology Mapping as Problem-Solving Method
Hans Akkermans
AKMC Knowledge Management BV
and
Free University Amsterdam VUA, Business Informatics Department
Amsterdam, The Netherlands
email: Hans.Akkermans@akmc.nl
ABSTRACT Continuous and discrete approaches represent two very
I discuss and construct ontology mappings between differ- different ontological viewpoints on the same concept of
ent ontologies of time. I show how you can use them as a time. They not only differ in appearance, but also come
new method to solve significant dynamics problems, by with radically different concepts and methods, witness the
exploiting the properties of the ontology mapping. A mathematical and computational analysis of continuous
unique feature of a nonlinear ontology mapping I propose versus discrete systems, for which there exists a vast litera-
is that it can rigorously treat infinitesimals as strictly finite ture spanning several centuries (e.g., [4] and [5]).
computational quantities. The approach also suggests some From the computational perspective, there is the additional
novel, I believe intriguing, insights into the nature of time, problem that continuous analysis is based on the notion of
particularly regarding “density” and “curvature” of time. derivatives and infinitesimal quantities (differential calcu-
The paper provides an in-depth case study in ontology lus dating back to Leibniz’s 1684 article [4]). As the com-
mapping, offering some evidence that ontology building, puter is an inherently discrete machine, computer methods
mapping, and reuse is much a substantive issue, more than for continuous systems invariably introduce approxima-
a matter of generic representation language and semantic tions that are in fact a kind of systematic error (known as
tooling. discretization or truncation error [2, 3]). In this paper I
pose – and solve – the problem: can we construct an ontol-
Categories and Subject Descriptors ogy mapping between continuous and discrete ontologies
H.4: Information Systems Applications – Miscellaneous. of time, which is both mathematically rigorous and compu-
tationally adequate, and is able to avoid systematic error in
General Terms changing from one temporal perspective to the other?
Theory, Algorithms. More simply: is it possible to reformulate any given form of
continuous-time dynamics in discrete time, rigorously,
Keywords without any compromise or computational approximation?
Time, ontology, mapping, signal analysis, dynamic systems The answer to this question is yes; the ontology mapping
solution outlined in this paper entails a novel method that I
INTRODUCTION call the T transform. Important characteristics of this new
Time is a very generic upper-level ontological concept.
transform method are: (1) conceptually, it is a radical de-
There are many different ontologies of time [1], but the two
parture from the traditional view and techniques regarding
temporal ontologies most widely used [2, 3] in science and
the relationship between continuous and discrete time; (2)
engineering are point-based: continuous time and discrete
it succeeds in fundamentally avoiding computer-introduced
time. In continuous-time systems, time is represented by a
systematic error in handling differential calculus; (3) it has
real-numbered parameter t ∈ ℜ. In discrete event-based informational advantages, by generating certain important
systems, time is represented by a “step” variable S ∈ ℵ, i.e. systems information directly that is not so easy to obtain by
an integer. conventional methods; (4) it gives rise to several new and
Permission to make digital or hard copies of all or part of this work for elegant discrete algorithms for systems analysis; and (5) it
personal or classroom use is granted without fee provided that copies are has an extremely wide spectrum of applications and gener-
not made or distributed for profit or commercial advantage and that alizations (even beyond time).
copies bear this notice and the full citation on the first page. To copy
otherwise, or republish, to post on servers or to redistribute to lists, I will go through these aspects below in brief.
requires prior specific permission by the copyright owners.
Copyright 2005
2
“NAIVE DYNAMICS”: TEMPORAL standard time ontology mapping between continuous and
ONTOLOGIES AND THEIR MAPPING discrete time is (note: both ways) given by the simple lin-
ear function:
Axiomatization of Time Ontologies t / τ = S or t = Sτ, so that (1a)
Van Benthem [1] gives a tense-logical formalization of a
XS = xt/τ . (1b)
great variety of temporal ontologies. His axiomatization for
point ontologies of time is over temporal structures consist- This equation is the ontological explication of the standard
ing of a non-empty set of time points ordered by a binary operating procedure in conventional real mathematical-
precedence relation <. It contains the following shared numerical analysis. Here, τ denotes the free (user-
axioms for discrete and continuous time: selectable) parameter known as the “stepsize” in continu-
ous systems simulation.
• TRANS: time ordering is transitive.
• IRREF: the property of irreflexivity; together TRANS
and IRREF model the (asymmetric) notion of the flow
(or arrow or “river”) of time.
• LIN: linearity, expressing that time structures have a
single path (or river flow bed) without branching.
• SUCC: time has no end point (continuing succession
towards the future).
The difference between discrete and continuous time
comes with the choice of a final temporal axiom, either one
of the following two options:
• DENS: infinite divisibility of time, i.e. between any
two time points there is always another one.
• DISC: discreteness; time is not infinitely divisible, but Figure 1. Traditional view on the mapping between con-
has the property of “stepwise” succession. tinuous and discrete time.
This suffices as axiomatization of the point-based temporal How does this linear ontology mapping work in practice?
ontologies I consider in this paper. Van Benthem shows Let us take a look at the prototypical formulation of con-
that this axiomatization is syntactically complete. He also tinuous dynamic systems, viz., the ordinary differential
shows that it admits of several models. Thus, real- equation1 (ODE):
numbered time t ∈ ℜ (where the axiom DENS is implied d/dt xt = f(xt) (2)
by the stronger continuity axiom CONT) and discrete As this is an equation in continuous time involving, more-
event-based integer time S ∈ ℵ (where as we shall see S over, infinitesimal calculus, it is not suitable for direct
indeed can be usefully read as “step”) are a specific model computer treatment. The standard approach then is to dis-
choice for the above ontological theories. However, these cretize the ODE (2) in time. The simplest choice to do so is
are by far the most common and useful ones in scientific
practice, and that’s why I stick to them.
Clearly, the above two formal temporal ontologies are
1
rather concise and simple themselves. This turns out not to It is conceptually interesting to reread Leibniz’s original article of 1684
be the case for discrete-continuous ontology mappings, [4]. He clearly talks about dx and dt as finite differences, and then pro-
ceeds with stating the rules of differential calculus as if they are infini-
however. I will now proceed to show that (1) temporal on- tesimals. He does not offer any justification; in actual fact his rules are
tology mappings are important general constructs with incorrect for finite quantities (they neglect the higher-order differences
many practical applications and implications, but (2) they that vanish in the infinitesimal case). He is sufficiently self-confident (or
are not unique, as several different useful ontology map- arrogant) to simply ignore this fundamental problem, and then saves the
day by coming up with an important useful application example (he de-
pings can be constructed. rives the refraction law of Snellius directly from Fermat’s principle). So
it seems he was lucky to live in ancient times as his article would be
Standard Time Ontology Mapping unlikely to survive any modern peer review. No wonder that people like
Let us consider first the traditional approach to continuous- the idealist philosopher Bishop Berkeley (1734) made fun out of the be-
lievers in this new calculus, deriding it as strange magic by juggling
discrete temporal ontology mapping, which is entrenched with different kinds of zeros (he added the wider point, against non-
in today’s standard techniques for numerical analysis and religious rationalists such as Halley, that if you do believe in this weird
simulation of system dynamics and evolution [2]. Typi- calculus stuff (“ghosts of departed quantities”), you also rob yourself of
cally, one assumes that the discrete time steps or events S = the right to criticize matters of theology). Proponents, among them Ber-
noulli, Euler, Maclaurin, D’Alembert, etc. ultimately defeated him by
0, 1, 2, ... are embedded in continuous time t ∈ ℜ by as- striking back with proper theoretical foundations of the new calculus.
suming that the integer time point “1” (etc.) maps onto the Interestingly, this whole conceptual struggle, involving a long string of
real time point “1.000...” (etc.), as depicted in Figure 1. the brightest mathematical geniuses of their time, took more than one
This looks very logical and natural indeed: formally the century and a half.
3
by dropping the infinitesimal limit in the definition of the for infinitesimal calculus proper. Correcting for this is what
derivative and invoking Eq. (1b). Then makes the usual algorithms for continuous dynamics so
(xt+τ - xt) / τ = f(xt) (3a) complicated (or non-naive).
or Now, my aim is to retain in some form this idea of naive
dynamics. I will do this in a novel way, in fact the precise
∆XS ≡ XS+1 - XS = τ f(XS) (3b) opposite of the standard computational approach. Specifi-
Equation (3) is computationally a one-step forward differ- cally, I will start from the principle of the correctness of an
ence, generally known as the Euler algorithm. Euler-type formula as (3). The key reason is that, if you
The Euler formula is essentially a direct application of the succeed in doing this, computation and prediction of con-
standard continuous-discrete time ontology mapping of Eq. tinuous systems is extremely simple, since Eq. (3) is a one-
(1) and Figure 1 (where I have assumed that the step size τ step forward difference in discrete time, and the whole fu-
= 1; this simplifies the bookkeeping and can be done with- ture is predicted (without any approximation in the sense of
out loss of generality). It replaces the continuous dynamic built-in systematic bias, as in the standard approach) by
system by a discrete-time one that is easy to compute (here, repeated application of (3).
by a one-step forward recursion). It is not really used in The necessary consequence of this alternative route is that
practice, precisely because it nicely illustrates a key prob- one has to drop the correctness of the linear ontology map-
lem of the digital computation of continuous dynamics: it is ping (1). However, as I will show, there is no principal
inherently approximate (the systematic error mentioned reason why there can’t be alternative ontology mappings
earlier: the higher-order differences ignored by Leibniz can with beautiful and desirable conceptual and computational
no longer be neglected in a finite computation). properties – but consequently they must be nonlinear with
Nevertheless, Euler’s formula is rightfully seen as the respect to time. In other words, you have to “bend” time.
grandfather of all ODE solving algorithms. Any ODE
solver attempts to correct its shortcomings, lack of accu- T: THE TRANSFORMATION OF TIME
racy and sometimes also of stability, by including higher
Probabilistic Embedding of Events in Time
difference contributions (equivalently, orders of the Taylor
I now construct a new alternative class of temporal ontol-
expansion) up to a prespecified order. As there are zillions
ogy mappings by means of the following procedure that
of ways to do this, each with specific advantages and
embeds discrete events S in continuous time t. Imagine that
drawbacks, this has become a computing art in itself. The
the time difference (in continuous time) between the occur-
famous Runge-Kutta algorithms, the universal workhorse
rence of two subsequent (discrete) events is not fixed and
to simulate continuous dynamic systems, are a case in
constant, as in the traditional approach (cf. the constant τ in
point. However, this can only be done to a limited extent,
Eq. (1)), but actually is random. So, after the start event
as explained in a very accessible and practical way in [2].
S=0 that occurs at some given start time t0 (taken to be t=0
In essence, the standard linear ontology mapping of Eq. (1)
in the remainder), the discrete time events S >0 occur ran-
inherently and unavoidably introduces approximations in
domly at continuous time points tS, and the time intervals
the digital computation of what basically are infinitesimal
between two steps T1=t1-t0, ..., TS+1=tS+1-tS are all random
quantities.
variables, governed by some given probability distribution
“Naive Dynamics” (which I assume to be the same for all events).
Although never stated this way, the key problem of the Accordingly, let P(t, S) be the probability that in the inter-
standard algorithms for continuous dynamics transformed val [0,t] precisely S discrete events or steps have occurred.
to a computationally tractable discrete-time system is there- Then the time ontology mapping replacing Eq. (1) reads:
fore the underlying assumption of a linear mapping be-
tween time ontologies.
With an allusion to Hayes’s “Naive Physics Manifesto”
(1978/85), one might say that what makes the standard
linear ontology mapping attractive is that it leads to a sim- Generally, this is a nonlinear ontology mapping, with re-
ply understandable form of “naive dynamics” for complex spect to both time variables t and S. Equation (4) actually
(nonlinear) systems of the type (2), witness Eqs. (1) and represents a whole class of ontology mappings, because
(3). Equations (1) and (3) are both nice to have from the there are many choices for the probability function P(t, S).
standpoint of naive dynamics. Unfortunately, scientific
history has demonstrated that they cannot both be valid For this probability I now take a specific choice, namely:
simultaneously. The standard approach then makes the
choice that the linear time ontology mapping (1) is correct,
but precisely this assumption invalidates the basic discrete
Euler formula (3) and its descendants such as Runge-Kutta
4
This nonlinear ontology mapping (xt = T(XS) in short) em- Key Properties of the T Transform
beds discrete events in continuous time by means of a sto- Some key properties of my T (T) transform between dis-
chastic process known as the Poisson process. Although crete and continuous time are given in Table 1.
fundamentally different from Eq. (1), it likewise has an
Table 1. Properties of the T transform Eq. (5)
elegant conceptual interpretation. The linear ontology
mapping (1) essentially says that all discrete events occur Continuous-time Discrete-time func-
Property
totally correlated in continuous time: once we know the function tion
No.
time instant of the initial event and the (fixed) waiting time xt = T (XS) XS = T (xt)
constant τ between events, the time occurrence of all events I. 1 (constant) 1 (constant)
is wholly fixed, carved in stone with military precision as it
were (cf. Figure 1). II. t S
2
III. t S(S-1)
In contrast, the nonlinear ontology mapping of Eq. (5) es-
3
sentially represents the opposite situation, in which all IV. t S(S-1)(S-2)
events occur independently and so are totally uncorrelated. V. t n
S! / (S-n+1)!
This situation often occurs in reality. For example the arri- At
val of incoming phone calls at a helpdesk is expressed by a VI. e (1 + A)S
Poisson process. Calls arrive not with fixed time intervals VII. A xt + B yt A X S + B YS
between them but irregularly; the probability distribution VIII. d/dt xt ∆ XS ≡ XS+1 - XS
for the random time TS between two steps is a negative n n
exponential, and the constant τ in Eq. (5) now represents IX. d /dt xt ∆n XS
the average waiting time between two subsequent events. FS =
What does this buy us? In essence, the time ontology map- X. ft ≡ yt × xt ∑n=0n=S [S!/((S-n)!n!)]
ping is a transform expression – the case of Eq. (5) I call ∆S-n Y0 × Xn
the T or T transform – that transforms a continuous func-
tion xt into a discrete function XS. Transform methods are Proofs. There are several possible derivations of the prop-
well-developed: they already stem from early 19th century erties in Table 1, but they require some background in real
mathematics, the Laplace and Fourier transforms probably mathematical analysis. Property I immediately follows
the best known ones. Although mathematically demanding, from the observation that the sum of P(t, S) over all steps
their key idea is simple: if you can map the original prob- equals unity by definition, because it is a probability func-
lem (say, the ODE (2)) from the original space (here, con- tion. Property VII also follows immediately by direct alge-
tinuous time) into a different problem in a new space where braic manipulation. To prove property VIII, we differenti-
it is simple to solve, then you are done by simply back- ate both sides of Eq. (5) with respect to t and rearrange
transforming the found solution to the original space. This terms at the right-hand side, with a simple change of dis-
is what for example the Laplace transform does: it trans- crete-time variable S. Property IX then follows (for exam-
forms differential equations from continuous time into sim- ple) by repeating this procedure and complete induction
ple-to-solve algebraic equations in frequency space. But towards the order of differentiation. Properties II-V all fol-
you already find this transform idea in the solution of the low from differentiating Eq. (5) and invoking I, VIII and
mutilated chessboard problem, or in that of the children’s IX. Finally, properties VI and X are discussed in more de-
game called Nim. tail in the next section in the context of various ODE appli-
My transform idea expressed in Eq. (4) and in the T or T cations. □
transform (5) is new and special in the sense that it trans- The first property (No. I) is interesting in that it conceptu-
forms a problem formulated in continuous time into one ally implies that any constant of the motion in continuous
that is formulated in discrete time. Discrete problems are time (think of energy, momentum, angular momentum,
much more suitable for solution by a computer than con- probability, flux) is also a constant of the motion in discrete
tinuous ones; once the discrete solution is found we simply time. The second property (No. II) states that linear func-
back-transform it into the continuous solution we are actu- tions in continuous time transform to linear functions in
ally looking for by using (4) or (5).2 That this idea practi- discrete time. These are properties that the nonlinear ontol-
cally works I am going to show now. ogy mapping (5) shares with the linear one of Eq. (1).
2
For the real connoisseur, I mention in passing that the linear ontology
mapping of Eq. (1) can be interpreted, like my T transform (5), as a spe- transform, which in digital control theory is also sometimes called the
cial case of the probabilistic transform (4). It is the limiting case in discrete Laplace transform. It directly yields the linear ontology map-
which the waiting-time distribution between events is the Dirac delta ping of Eq. (1b). Hence – although this is hardly ever explicitly recog-
nized – also the standard numerical approach using the linear ontology
function δ(t-τ). Reworking Eq. (4) on this basis by using its Laplace
mapping Eq. (1) is fundamentally based on a transform idea.
transform, one is led to a generating function method known as the z
5
Other properties are unique to my T transform (5). In forward difference formula (3b) as the correct expression
particular, continuous-time functions map onto similar (e.g. for differentiation of a continuous variable. In a sense, we
same-order polynomials, cf. properties III-VI) but not iden- have achieved this by conceptually reversing Leibniz.
tical functions in discrete time. This is in stark contrast to Leibniz talked about infinitesimals as finite quantities, and
the assumption in the standard received view that employs subsequently invented the correct rules of differential cal-
the same function in both continuous and discrete time (cf. culus. We took differential calculus, and subsequently in-
Eq. (3a)). Property VII says that the T transform is a linear vented a temporal ontology mapping that makes it actually
transform3. correct to treat infinitesimals as finite quantities, just by
Properties VIII and IX are the crucial ones: the T transform switching from continuous to discrete time!
maps the derivative d/dt onto a finite first-order discrete
forward difference ∆. Hence, Euler-type formulas similar A FEW APPLICATIONS AND
to (3) will be correct under the T transform. Moreover, this IMPLICATIONS
extends to the higher-order derivatives, which are simply
found by repeated application of the finite forward differ-
Computer Right, Man Wrong (Save Euler)
I first show how you can solve large-scale linear differen-
ence ∆. As a consequence, a beautiful and important prop-
tial systems by temporal ontology mapping. This turns out
erty of the nonlinear time ontology mapping (5) is that it in
to have an interesting side implication on the conceptual
discrete time produces the higher-order derivatives of con-
interpretation of what algorithms do.
tinuous time, one by one and exactly. The production of the
XS values in discrete time yields a tableau (see Figure 2), A widely used special case of the ODE (2) is the linear
by simple subtraction or addition, that contains all desired system:
information. d/dt xt = A xt (6)
This equation also describes dynamic systems in many di-
mensions; then, A is not to be interpreted as a one-
dimensional constant (scalar) but as a matrix. The deriva-
tion below is then generally valid for any number of di-
mensions.
To solve this by ontology mapping, we first transform the
problem from continuous time to discrete time. Using
property VIII of Table 1, the discrete version of Eq. (6) is:
∆ XS = A XS (7)
Next, we construct the solution in discrete time starting
from the known initial condition x0 = X0, and repeatedly
applying the forward difference definition of the operator
∆. In effect, the whole discrete solution is stepwise pro-
Figure 2. The T transform yields a tableau that contains duced (also in many dimensions) by the Euler algorithm
the discrete solution to derivatives of any order. (3b). From Eq. (7) it is easy to see that the discrete solution
If for example xt denotes the position in a space at a certain is:
time, the tableau not only gives the solution for the location XS+1 = (1 + A) XS ⇒ XS = (1 + A)S X0 (8)
(coefficients XS) but it simultaneously solves the question
Finally, this solution is back-transformed to continuous
as to its velocity (∆XS), acceleration (∆2XS), etc. In addition
time by using Eq. (5). In the general case this can be done
it is able to reconstruct, by using only the T transform
computationally by various methods (e.g. by successive
equation (5), the values of the continuous variables at any
sequences of one-step recursions or, parallelized, by matrix
desired point in continuous time t. These are all major in-
methods), where as a bonus you have a free choice for the
formational and computational advantages that show the
time points t you are actually interested in. In the present
power of the T transform temporal ontology mapping.
case, the continuous solution is just a matter of simple table
look-up, see property VI in Table 1. Hence, the ontology
Leibniz Reversed
mapping method solves the dynamic problem (6) by first
Consequently, we have achieved the earlier stated aim of
transforming the problem to a new (discrete) space, next
“naive dynamics” by showing the validity of the one-step
solve it there, and then transform this solution back to the
original (continuous) space, where it reads:
3
Perhaps this sounds a bit confusing, but linearity is only a relative no-
tion. As stated earlier, the T transform is nonlinear with respect to the
xt = eAt x0 (9)
time variables t and S (see Eq. (5)). With respect to the temporal func- So, we have solved a problem involving infinitesimal cal-
tions XS and xt, however, it is linear, in accordance with property VII. culus in a strictly discrete fashion, not by directly attacking
6
the computation of derivatives in an approximate fashion Nonlinearity and the Curvature of Time:
(which is the standard way of doing it), but indirectly by Bend It Like Beckham
changing the problem space first. I mention in passing that The next important step is to show that the T transform
the above also yields a proof of property VI in Table 1: just method also handles nonlinear dynamics well. This gives it
insert Eq. (8) into Eq. (5) and carry out the summation. a major advantage over other transforms such as the
The case discussed here has several important general ap- Laplace and z ones. I will give a basic example of this, by
plications. For example, it applies to both random walks considering a special case of the ODE (2), namely:
and master equations; both have many practical applica-
tions in many different disciplines. As a bonus, the T trans- d/dt xt = A xt (1- xt ) (10)
form proves that they are their mutual discrete and con- which is generally known as the logistic equation.
tinuous-time equivalents, see also [6].
Logistic equation models. Varieties of it are widely used
A possibly even broader application is that it is applicable
in practice, for example in population models of competing
to the modern state-space approach to control systems the-
species in ecology. In one dimension, the linear term repre-
ory: adding a control signal term to Eq. (6), i.e. a function
sents exponential growth, but the nonlinear (quadratic)
explicitly dependent on t, yields the fundamental systems
term models self-limiting effects: lambs eat grass, but if
formulation underlying control engineering of multi-
there are too many in a territory (outside paradise), their
dimensional continuous systems. The methods developed
population growth is ultimately restricted due to resource
in this paper open up the opportunity to treat such systems
limitations. In more dimensions, the logistic equation can
by strictly discrete computer methods.
model interactions between species: lions eat lambs, but if
The above results give some (I believe entertaining) reha- they eat too many, first the number of available lambs will
bilitation of the Euler algorithm. Let me quote a statement drop, and ultimately their own population numbers will go
from [2], a remark that is prototypical for any modern text- down. It is easy to imagine that such models often lead to
book treatment of numerical methods: “There are several (nonlinear) oscillatory cycles in population growth, with
reasons that Euler’s method is not recommended for practi- time delays between those of interacting species.
cal use, among them, (i) the method is not very accurate
(...), and (ii) neither is it very stable” ([2], p. 704). In con- Solving the nonlinear logistic system (10) follows the same
flict with this statement, the dynamic problem (6) has been transform procedure as discussed above. Now, however,
solved here exactly by the Euler method. However, you we have to use property X of Table 1 for its time transfor-
should interpret the results of the algorithm not as rough mation. This property might seem mathematically complex,
direct estimates of the continuous-time point solution (see but it is actually a discrete convolution that is computation-
Eq. (3a), the standard interpretation). Instead, it is to be ally very simple to handle (it’s just a sequence of basic
seen as an indirect method producing the exact solution, additions and multiplications). Property X can be formally
however, in discrete time (according to Eq. (3b)). In con- proven by (rather tedious) algebraic manipulation, properly
clusion, (1) evasive maneuvers do solve problems, and (2) rearranging terms at the right-hand side (a much more ele-
the computer got it all right all these years, but man’s con- gant derivation uses symbolic operator algebra, but this is
ceptual interpretation of its outputs has always been wrong beyond the space of this article). This results in an analyti-
(except for Euler, of course). cal solution of the nonlinear ODE (10) in discrete time:
Shoham’s Extended Prediction Problem Does
Not Exist
In his book “Reasoning about Change” (1988), Shoham The first term of this solution gives the linear part (as dis-
worries that the usual differential dynamics (cf. Eqs. (2) cussed above), and the second term yields the nonlinear
and (6)) only gives a prediction of an infinitesimally small effects. Again a variant of the Euler-type algorithm is
time step forward from the considered current time point t. suited to the task of prediction: from Eq. (11) it is easy to
So how is it actually possible at all to make predictions see that the discrete solution XS obtains by successive sin-
over extended and finite periods of time on this basis? He gle-step forward recursions starting from the known initial
calls this the extended prediction problem. My ontology condition X0 and then going forward in time: S=1, next S=2
mapping gives a direct solution to this: it turns the deriva- etc. The probabilistic T map (5) then delivers the solution
tive into a strictly discrete and finite one-step forward dif- in continuous time for any desired time point t.
ference into the future. Once you have solved this finite
and discrete problem, you simply transform its solution It is instructive to compare the solution (11) of the continu-
back for any desired time t using the T transform (5). The ous ODE (10) with (i) the discrete solution (8) of the linear
extended prediction problem thus seems to satisfy the system (6), and with (ii) the nonlinear discrete dynamic
quoted Bishop Berkeley 1734 characterization concerning system that is usually seen as its discrete analog (and there-
“ghosts of departed quantities”. fore is known as the logistic map):
7
XS+1 = A XS (1 – XS) (12) • First, it changes the function f, seen as a function of x
only, into a similar but not identical function F (wit-
This logistic map is famous because it is more or less the
ness for example the properties III-VI and X). This is
simplest system that exhibits chaotic dynamic behaviour
already an important difference with the standard ap-
(in contrast to the logistic ODE). Again, there is a linear
proach depicted in Figure 1.
term and a quadratic nonlinearity, now in discrete time. But
there is an essential structural difference between the dis- • Second, it also changes the function x seen as a func-
crete solution (11) to the logistic ODE on the one hand, and tion of t (since the same properties apply again).
the linear system (8) and logistic map (12) on the other If we attempt to visualize this latter effect, we get a picture
hand. The latter are iterated maps, i.e., result from repeated radically different from Figure 1. What happens is that the
function application; to obtain the value at the next time- average density of the occurrence of (discrete) events is not
point one only needs the preceding timepoint. constant but changes over the (continuous) time axis.
In contrast, Eq. (11) shows that in the solution of the logis- You might visualize this by imagining that the discrete time
tic ODE all previous time points are involved. So, this con- axis gets curved, and in continuous time you only see its
tinuous dynamic system has a memory in discrete time, projection onto the continuous time axis (see Figure 3). It is
even though this is not at all evident from the ODE formu- actually not difficult to find examples where the discrete-
lation (10) that involves a single continuous timepoint. Al- time “curvature” becomes so strong that it creates a singu-
though they share the name, the logistic ODE and the logis- larity in (note: finite) continuous time.
tic map are totally different in their dynamic behaviour. Thus, the metaphor of the flow of time as a river [1] gets
Lorenz chaos. Property X of Table 1 also enables to solve strangely bent due to nonlinearities: it’s possible to create
in discrete time the well-known Lorenz model (1963), de- something like a black hole in the river bed of the timeline!
veloped to better understand atmospheric dynamics for
long-range weather prediction. It became prominent be-
cause it was the first demonstration of the occurrence of
chaotic behaviour in deterministic systems, with a so-called
strange attractor (the famous “butterfly” shape to which the
system tends in phase space). The Lorenz model is a simple
3D system with quadratic-type (in fact, bilinear) nonlineari-
ties. So, property X directly applies, and the discrete solu-
tion of the Lorenz model has the same structure as Eq. (11).
In general, nonlinearity in continuous dynamics has the
effect that it “bends like Beckham” the solution in discrete
time: in contrast to the linear system Eq. (8), all values at
time points before S play an explicit role in the full solution
at time S, although the solution itself can always be com-
puted by a one-step forward algorithm that also maintains
the normal causal order of events, both in continuous and
discrete time. Figure 3. The T temporal ontology mapping may lead to
flows of time that are “curved”.
Preview of Coming Attractions
It is probably most interesting here to briefly investigate UPPER-LEVEL GENERALIZATIONS
the impact on the structure of time resulting from nonlinear This paper only outlines a small fraction of the results I
dynamics. Namely, the above methods and results suggest have developed concerning nonlinear ontology mappings
some intriguing conceptual (or if you wish, philosophical) between time, and could only hint at the underlying
insights into the nature of time, particularly regarding mathematical proofs and algorithms. A few final general
“density” and “curvature” of time. remarks are in order.
Consider the nonlinear differential equation (2) in general The uses of “old” science. First, the whole theory of tem-
and how it changes under the T temporal ontology mapping poral ontology mapping can be founded upon various treas-
(5). The left-hand side T(lhs) is easy: according to property ures stemming from rather ancient mathematics. Much of it
VIII it always maps onto a simple one-step forward differ- has more or less become extinct and superseded by modern
ence. The right-hand side involves a composite function computer (in fact, number crunching) approaches, and as a
f(x(t)) that is generally nonlinear. Taking the transform result is not treated anymore in modern textbooks on
T(rhs) changes our ontological view on the dynamic world numerical methods. Specifically, this theory can be set up
in a very elegant and concise way by means of symbolic
in two stages:
operator algebra [3] as you find it in the textbook by Boole
8
[5] (the first edition was from 1860, by the way). What I that spacetime has characteristics of randomness, and still
actually find gratifying is that these methods are not just is causally ordered, according to Eq. (5).
ancient, but if you develop them further as I tried to do in Above I discussed things from the perspective of time.
this paper, they can be actually made ready for today’s in- Surely this is a key top-level ontology concept. However,
telligent system-style computing, and made valuable be- my approach and methods also work for types of independ-
yond popular number crunching styles of computing. A ent variables other than time. Other continuous variables
few examples have been given in this paper. can be formally discretized in this way as well. For exam-
ple, one can in this manner also treat the concept of space.
Computational complexity. An issue not discussed in this
paper is the computational complexity of the algorithms
Ontology: content vs. representation. Finally, the paper
related to the T transform. Generally, they are of low-
has provided an in-depth case study in ontology mapping. I
degree polynomial complexity. The calculation of the T
submit that this provides some evidence that ontology
transform (5) itself is of order O(N2), where N is the num-
building, mapping, and reuse is much a substantive issue,
ber of timepoints considered (same order as matrix multi-
more than a matter of generic representation language and
plication; this also applies to the inverse transform, as T
semantic tooling. I note that this is already the case for such
turns out to be a so-called orthogonal transformation). The
a high-level, generic, common, and commonsensical con-
computational complexity of the solution of ODEs is com-
cept as time that does not depend on a specific domain.
monly measured in terms of the number of function evalua-
Substantive or content issues will be even more strongly
tions (the right-hand side of Eq. (2)). Then, the T transform
present in task and domain specific ontologies. But in the
method of solution is of linear complexity in time, cf. Eq.
end this is where the real semantic and web intelligence
(11); I note that this is also true in the general case.
applications will be. This is perhaps a sign that the seman-
AI temporal reasoning and android epistemology. The tic research community at some point cannot avoid signifi-
present work has important differences with respect to cant substantive issues in Web ontology, and has to be
much of the work in AI temporal reasoning (see e.g. [7]) in careful about (over)emphasis of generic representation and
terms of focus and assumptions. The present work uses tooling issues without adequate domain grounding. Or, be
standard point algebra from mathematics, and therefore sufficiently moderate in its expectations of the size of its
interval algebras and axiomatizations (such as Allen’s, see own role in building the Semantic Web.
also [1] and [7]) are not really relevant here. Important in Acknowledgment. This work has been supported in part by a
my approach is that time is a metric space, i.e. a distance travel grant from the KnowledgeWeb Network of Excellence
measure can be defined (in AI temporal reasoning usually (EU-IST-2004-507482).
called duration information). Another important difference
is the type of tasks considered. AI temporal reasoning has
spent much effort on constraint-based algorithms for estab- REFERENCES
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9