Reverse Leibniz, and then Bend It Like Beckham Tem

Hans Akkermans

AKMC Knowledge Management BV

and

Free University Amsterdam VUA, Business Informatics Department

Amsterdam, The Netherlands

email: Hans.Akkermans@akmc.nl

ABSTRACT

I discuss and construct ontology mappings between differ-ent ontologies of time. I show how you can use them as a new method to solve significant dynamics problems, by exploiting the properties of the ontology mapping. A unique feature of a nonlinear ontology mapping I propose is that it can rigorously treat infinitesimals as strictly finite computational quantities. The approach also suggests some novel, I believe intriguing, insights into the nature of time, particularly regarding “density” and “curvature” of time. The paper provides an in-depth case study in ontology mapping, offering some evidence that ontology building, mapping, and reuse is much a substantive issue, more than a matter of generic representation language and semantic tooling.

Categories and Subject Descriptors

H.4: Information Systems Applications – Miscellaneous. General Terms

Theory, Algorithms.

Keywords

Time, ontology, mapping, signal analysis, dynamic systems INTRODUCTION

Time is a very generic upper-level ontological concept. There are many different ontologies of time [1], but the two temporal ontologies most widely used [2, 3] in science and engineering are point-based: continuous time and discrete time. In continuous-time systems, time is represented by a real-numbered parameter t∈ℜ. In discrete event-based systems, time is represented by a “step” variable S∈ℵ, i.e. an integer. Continuous and discrete approaches represent two very different ontological viewpoints on the same concept of time. They not only differ in appearance, but also come with radically different concepts and methods, witness the mathematical and computational analysis of continuous versus discrete systems, for which there exists a vast litera-ture spanning several centuries (e.g., [4] and [5]).

From the computational perspective, there is the additional problem that continuous analysis is based on the notion of derivatives and infinitesimal quantities (differential calcu-lus dating back to Leibniz’s 1684 article [4]). As the com-puter is an inherently discrete machine, computer methods for continuous systems invariably introduce approxima-tions that are in fact a kind of systematic error (known as discretization or truncation error [2, 3]). In this paper I pose – and solve – the problem: can we construct an ontol-ogy mapping between continuous and discrete ontologies of time, which is both mathematically rigorous and compu-tationally adequate, and is able to avoid systematic error in changing from one temporal perspective to the other? More simply: is it possible to reformulate any given form of continuous-time dynamics in discrete time, rigorously, without any compromise or computational approximation? The answer to this question is yes; the ontology mapping solution outlined in this paper entails a novel method that I call the T transform. Important characteristics of this new transform method are: (1) conceptually, it is a radical de-parture from the traditional view and techniques regarding the relationship between continuous and discrete time; (2) it succeeds in fundamentally avoiding computer-introduced systematic error in handling differential calculus; (3) it has informational advantages, by generating certain important systems information directly that is not so easy to obtain by conventional methods; (4) it gives rise to several new and elegant discrete algorithms for systems analysis; and (5) it has an extremely wide spectrum of applications and gener-alizations (even beyond time).

I will go through these aspects below in brief.

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Copyright 2005“NAIVE DYNAMICS”: TEMPORAL ONTOLOGIES AND THEIR MAPPING Axiomatization of Time Ontologies

Van Benthem [1] gives a tense-logical formalization of a great variety of temporal ontologies. His axiomatization for point ontologies of time is over temporal structures consist-ing of a non-empty set of time points ordered by a binary precedence relation <. It contains the following shared axioms for discrete and continuous time:

•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 has the property of “stepwise” succession.

This suffices as axiomatization of the point-based temporal ontologies I consider in this paper. Van Benthem shows that this axiomatization is syntactically complete. He also shows that it admits of several models. Thus, real-numbered time t∈ℜ (where the axiom DENS is implied by the stronger continuity axiom CONT) and discrete event-based integer time S∈ℵ (where as we shall see S indeed can be usefully read as “step”) are a specific model choice for the above ontological theories. However, these 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 rather concise and simple themselves. This turns out not to be the case for discrete-continuous ontology mappings, however. I will now proceed to show that (1) temporal on-tology mappings are important general constructs with many practical applications and implications, but (2) they are not unique, as several different useful ontology map-pings can be constructed.

Standard Time Ontology Mapping

Let us consider first the traditional approach to continuous-discrete temporal ontology mapping, which is entrenched in today’s standard techniques for numerical analysis and simulation of system dynamics and evolution [2]. Typi-cally, one assumes that the discrete time steps or events S = 0, 1, 2, ... are embedded in continuous time t∈ℜby as-suming that the integer time point “1” (etc.) maps onto the real time point “1.000...” (etc.), as depicted in Figure 1. This looks very logical and natural indeed: formally the standard time ontology mapping between continuous and discrete time is (note: both ways) given by the simple lin-

ear function:

t / τ = S or t = Sτ, so that (1a)

X S = x t/τ . (1b)

This equation is the ontological explication of the standard operating procedure in conventional real mathematical-numerical analysis. Here, τ denotes the free (user-selectable) parameter known as the “stepsize” in continu-

ous systems simulation.

Figure 1. Traditional view on the mapping between con-

tinuous and discrete time.

How does this linear ontology mapping work in practice?

Let us take a look at the prototypical formulation of con-tinuous dynamic systems, viz., the ordinary differential equation1 (ODE):

d/dt x t = f(x t)

(2) As this is an equation in continuous time involving, more-over, infinitesimal calculus, it is not suitable for direct computer treatment. The standard approach then is to dis-cretize the ODE (2) in time. The simplest choice to do so is

1It is conceptually interesting to reread Leibniz’s original article of 1684 [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-tesimals. He does not offer any justification; in actual fact his rules are incorrect for finite quantities (they neglect the higher-order differences

that vanish in the infinitesimal case). He is sufficiently self-confident (or arrogant) to simply ignore this fundamental problem, and then saves the

day by coming up with an important useful application example (he de-

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 unlikely to survive any modern peer review. No wonder that people like

the idealist philosopher Bishop Berkeley (1734) made fun out of the be-lievers in this new calculus, deriding it as strange magic by juggling

with different kinds of zeros (he added the wider point, against non-religious rationalists such as Halley, that if you do believe in this weird calculus stuff (“ghosts of departed quantities”), you also rob yourself of

the right to criticize matters of theology). Proponents, among them Ber-noulli, Euler, Maclaurin, D’Alembert, etc. ultimately defeated him by striking back with proper theoretical foundations of the new calculus. Interestingly, this whole conceptual struggle, involving a long string of

the brightest mathematical geniuses of their time, took more than one century and a half.

by dropping the infinitesimal limit in the definition of the derivative and invoking Eq. (1b). Then

(x t+τ - x t ) / τ = f(x t ) (3a)

or

∆X S ≡ X S+1 - X S = τ f(X S ) (3b) Equation (3) is computationally a one-step forward differ-ence, generally known as the Euler algorithm . The Euler formula is essentially a direct application of the

standard continuous-discrete time ontology mapping of Eq.

(1) and Figure 1 (where I have assumed that the step size τ

= 1; this simplifies the bookkeeping and can be done with-out loss of generality). It replaces the continuous dynamic

system by a discrete-time one that is easy to compute (here,

by a one-step forward recursion). It is not really used in practice, precisely because it nicely illustrates a key prob-lem of the digital computation of continuous dynamics: it is inherently approximate (the systematic error mentioned earlier: the higher-order differences ignored by Leibniz can no longer be neglected in a finite computation). Nevertheless, Euler’s formula is rightfully seen as the

grandfather of all ODE solving algorithms. Any ODE

solver attempts to correct its shortcomings, lack of accu-racy and sometimes also of stability, by including higher

difference contributions (equivalently, orders of the Taylor

expansion) up to a prespecified order. As there are zillions

of ways to do this, each with specific advantages and

drawbacks, this has become a computing art in itself. The

famous Runge-Kutta algorithms, the universal workhorse

to simulate continuous dynamic systems, are a case in

point. However, this can only be done to a limited extent,

as explained in a very accessible and practical way in [2].

In essence, the standard linear ontology mapping of Eq. (1)

inherently and unavoidably introduces approximations in

the digital computation of what basically are infinitesimal

quantities.

“Naive Dynamics”

Although never stated this way, the key problem of the

standard algorithms for continuous dynamics transformed to a computationally tractable discrete-time system is there-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-ply understandable form of “naive dynamics” for complex (nonlinear) systems of the type (2), witness Eqs. (1) and (3). Equations (1) and (3) are both nice to have from the standpoint of naive dynamics. Unfortunately, scientific history has demonstrated that they cannot both be valid 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

for infinitesimal calculus proper. Correcting for this is what makes the usual algorithms for continuous dynamics so complicated (or non-naive). 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

opposite of the standard computational approach. Specifi-cally, I will start from the principle of the correctness of an

Euler-type formula as (3). The key reason is that, if you

succeed in doing this, computation and prediction of con-tinuous systems is extremely simple, since Eq. (3) is a one-step forward difference in discrete time, and the whole fu-ture is predicted (without any approximation in the sense of built-in systematic bias, as in the standard approach) by repeated application of (3). The necessary consequence of this alternative route is that one has to drop the correctness of the linear ontology map-ping (1). However, as I will show, there is no principal reason why there can’t be alternative ontology mappings with beautiful and desirable conceptual and computational properties – but consequently they must be nonlinear with

respect to time. In other words, you have to “bend” time. T: THE TRANSFORMATION OF TIME Probabilistic Embedding of Events in Time

I now construct a new alternative class of temporal ontol-ogy mappings by means of the following procedure that

embeds discrete events S in continuous time t . Imagine that

the time difference (in continuous time) between the occur-rence of two subsequent (discrete) events is not fixed and

constant, as in the traditional approach (cf. the constant τ in

Eq. (1)), but actually is random . So, after the start event

S =0 that occurs at some given start time t 0 (taken to be t =0

in the remainder), the discrete time events S >0 occur ran-domly at continuous time points t S , and the time intervals

between two steps T 1=t 1-t 0, ..., T S +1=t S +1-t S are all random

variables, governed by some given probability distribution (which I assume to be the same for all events). Accordingly, let P(t, S) be the probability that in the inter-val [0,t ] precisely S discrete events or steps have occurred.

Then the time ontology mapping replacing Eq. (1) reads:

Generally, this is a nonlinear ontology mapping, with re-spect to both time variables t and S . Equation (4) actually represents a whole class of ontology mappings, because there are many choices for the probability function P(t, S).

For this probability I now take a specific choice, namely:

In contrast, the nonlinear ontology mapping of Eq. (5) es-sentially represents the opposite situation, in which all events occur independently and so are totally un correlated. This situation often occurs in reality. For example the arri-val of incoming phone calls at a helpdesk is expressed by a Poisson process. Calls arrive not with fixed time intervals between them but irregularly; the probability distribution for the random time T S between two steps is a negative exponential, and the constant τ in Eq. (5) now represents the average waiting time between two subsequent events. What does this buy us? In essence, the time ontology map-ping is a transform expression – the case of Eq. (5) I call the T or T transform – that transforms a continuous func-tion x t into a discrete function X S. Transform methods are well-developed: they already stem from early 19th century mathematics, the Laplace and Fourier transforms probably the best known ones. Although mathematically demanding, their key idea is simple: if you can map the original prob-lem (say, the ODE (2)) from the original space (here, con-tinuous time) into a different problem in a new space where it is simple to solve, then you are done by simply back-transforming the found solution to the original space. This is what for example the Laplace transform does: it trans-forms differential equations from continuous time into sim-ple-to-solve algebraic equations in frequency space. But you already find this transform idea in the solution of the mutilated chessboard problem, or in that of the children’s game called Nim.

My transform idea expressed in Eq. (4) and in the T or T transform (5) is new and special in the sense that it trans-forms a problem formulated in continuous time into one that is formulated in discrete time. Discrete problems are much more suitable for solution by a computer than con-tinuous ones; once the discrete solution is found we simply back-transform it into the continuous solution we are actu-ally looking for by using (4) or (5).2 That this idea practi-cally works I am going to show now.

2For 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-cial case of the probabilistic transform (4). It is the limiting case in which the waiting-time distribution between events is the Dirac delta function δ(t-τ). Reworking Eq. (4) on this basis by using its Laplace transform, one is led to a generating function method known as the z Key Properties of the T Transform

Some key properties of my T (T) transform between dis-

crete and continuous time are given in Table 1.

Table 1. Properties of the T transform Eq. (5)

Property

No.

Continuous-time

function

x t = T (X S)

Discrete-time func-

tion

X S = T (x t)

I. 1 (constant) 1 (constant)

II. t S

III. t2 S(S-1) IV. t3 S(S-1)(S-2) V. t n S! / (S-n+1)!

VI. e At(1 + A)S

VII. A

x t + B y t A

X S + B Y S

VIII. d/dt

x t∆ X S≡ X S+1 - X S

IX. d n/dt n x t∆n X S

X. f t≡ y t× x t

F S =

∑n=0n=S [S!/((S-n)!n!)]

∆S-n Y0× X n

Proofs. There are several possible derivations of the prop-

erties in Table 1, but they require some background in real mathematical analysis. Property I immediately follows

from the observation that the sum of P(t, S) over all steps equals unity by definition, because it is a probability func-

tion. Property VII also follows immediately by direct alge-

braic manipulation. To prove property VIII, we differenti-

ate both sides of Eq. (5) with respect to t and rearrange terms at the right-hand side, with a simple change of dis-

crete-time variable S. Property IX then follows (for exam-

ple) by repeating this procedure and complete induction towards the order of differentiation. Properties II-V all fol-

low from differentiating Eq. (5) and invoking I, VIII and

IX. Finally, properties VI and X are discussed in more de-

tail in the next section in the context of various ODE appli-cations. □

The first property (No. I) is interesting in that it conceptu-

ally implies that any constant of the motion in continuous

time (think of energy, momentum, angular momentum, probability, flux) is also a constant of the motion in discrete time. The second property (No. II) states that linear func-

tions in continuous time transform to linear functions in discrete time. These are properties that the nonlinear ontol-

ogy mapping (5) shares with the linear one of Eq. (1).

transform, which in digital control theory is also sometimes called the discrete Laplace transform. It directly yields the linear ontology map-

ping of Eq. (1b). Hence – although this is hardly ever explicitly recog-

nized – also the standard numerical approach using the linear ontology mapping Eq. (1) is fundamentally based on a transform idea.

Other properties are unique to my T transform (5). In particular, continuous-time functions map onto similar (e.g. same-order polynomials, cf. properties III-VI) but not iden-tical functions in discrete time. This is in stark contrast to the assumption in the standard received view that employs the same function in both continuous and discrete time (cf. Eq. (3a)). Property VII says that the T transform is a linear transform 3.

Properties VIII and IX are the crucial ones: the T transform maps the derivative d/dt onto a finite first-order discrete forward difference ∆. Hence, Euler-type formulas similar to (3) will be correct under the T transform. Moreover, this extends to the higher-order derivatives, which are simply found by repeated application of the finite forward differ-ence ∆. As a consequence, a beautiful and important prop-erty of the nonlinear time ontology mapping (5) is that it in discrete time produces the higher-order derivatives of con-tinuous time, one by one and exactly. The production of the X S values in discrete time yields a tableau (see Figure 2), by simple subtraction or addition, that contains all desired

information.

Figure 2. The T transform yields a tableau that contains

the discrete solution to derivatives of any order. If for example x t denotes the position in a space at a certain time, the tableau not only gives the solution for the location (coefficients X S ) but it simultaneously solves the question as to its velocity (∆X S ), acceleration (∆2X S ), etc. In addition it is able to reconstruct, by using only the T transform equation (5), the values of the continuous variables at any desired point in continuous time t . These are all major in-formational and computational advantages that show the power of the T transform temporal ontology mapping.

Leibniz Reversed

Consequently, we have achieved the earlier stated aim of “naive dynamics” by showing the validity of the one-step

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 time variables t and S (see Eq. (5)). With respect to the temporal func-tions X S and x t , however, it is linear, in accordance with property VII.

forward difference formula (3b) as the correct expression for differentiation of a continuous variable. In a sense, we have achieved this by conceptually reversing Leibniz. Leibniz talked about infinitesimals as finite quantities, and subsequently invented the correct rules of differential cal-culus. We took differential calculus, and subsequently in-vented a temporal ontology mapping that makes it actually correct to treat infinitesimals as finite quantities, just by switching from continuous to discrete time!

A FEW APPLICATIONS AND IMPLICATIONS

Computer Right, Man Wrong (Save Euler)

I first show how you can solve large-scale linear differen-tial systems by temporal ontology mapping. This turns out to have an interesting side implication on the conceptual interpretation of what algorithms do.

A widely used special case of the ODE (2) is the linear system:

d/dt x t = A x t

(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:

∆ X S = A X S

(7)

Next, we construct the solution in discrete time starting from the known initial condition x 0 = X 0, and repeatedly applying the forward difference definition of the operator ∆. In effect, the whole discrete solution is stepwise pro-duced (also in many dimensions) by the Euler algorithm (3b). From Eq. (7) it is easy to see that the discrete solution is:

X S+1 = (1 + A) X S ⇒ X S = (1 + A)S X 0

(8) Finally, this solution is back-transformed to continuous time by using Eq. (5). In the general case this can be done computationally by various methods (e.g. by successive sequences of one-step recursions or, parallelized, by matrix methods), where as a bonus you have a free choice for the time points t you are actually interested in. In the present case, the continuous solution is just a matter of simple table look-up, see property VI in Table 1. Hence, the ontology mapping method solves the dynamic problem (6) by first transforming the problem to a new (discrete) space, next solve it there, and then transform this solution back to the original (continuous) space, where it reads:

x t = e At x 0

(9) So, we have solved a problem involving infinitesimal cal-culus in a strictly discrete fashion, not by directly attacking

A possibly even broader application is that it is applicable to the modern state-space approach to control systems the-ory: adding a control signal term to Eq. (6), i.e. a function explicitly dependent on t, yields the fundamental systems formulation underlying control engineering of multi-dimensional continuous systems. The methods developed in this paper open up the opportunity to treat such systems by strictly discrete computer methods.

The above results give some (I believe entertaining) reha-bilitation of the Euler algorithm. Let me quote a statement from [2], a remark that is prototypical for any modern text-book treatment of numerical methods: “There are several reasons that Euler’s method is not recommended for practi-cal use, among them, (i) the method is not very accurate (...), and (ii) neither is it very stable” ([2], p. 704). In con-flict with this statement, the dynamic problem (6) has been solved here exactly by the Euler method. However, you should interpret the results of the algorithm not as rough direct estimates of the continuous-time point solution (see Eq. (3a), the standard interpretation). Instead, it is to be seen as an indirect method producing the exact solution, however, in discrete time (according to Eq. (3b)). In con-clusion, (1) evasive maneuvers do solve problems, and (2) the computer got it all right all these years, but man’s con-ceptual interpretation of its outputs has always been wrong (except for Euler, of course).

Shoham’s Extended Prediction Problem Does Not Exist

In his book “Reasoning about Change” (1988), Shoham worries that the usual differential dynamics (cf. Eqs. (2) and (6)) only gives a prediction of an infinitesimally small time step forward from the considered current time point t. So how is it actually possible at all to make predictions over extended and finite periods of time on this basis? He calls this the extended prediction problem. My ontology mapping gives a direct solution to this: it turns the deriva-tive into a strictly discrete and finite one-step forward dif-ference into the future. Once you have solved this finite and discrete problem, you simply transform its solution back for any desired time t using the T transform (5). The extended prediction problem thus seems to satisfy the quoted Bishop Berkeley 1734 characterization concerning “ghosts of departed quantities”. Nonlinearity and the Curvature of Time: Bend It Like Beckham

The next important step is to show that the T transform method also handles nonlinear dynamics well. This gives it a major advantage over other transforms such as the Laplace and z ones. I will give a basic example of this, by considering a special case of the ODE (2), namely:

d/dt x t = A x t (1- x t ) (10) which is generally known as the logistic equation. Logistic equation models. Varieties of it are widely used in practice, for example in population models of competing species in ecology. In one dimension, the linear term repre-sents exponential growth, but the nonlinear (quadratic) term models self-limiting effects: lambs eat grass, but if there are too many in a territory (outside paradise), their population growth is ultimately restricted due to resource limitations. In more dimensions, the logistic equation can model interactions between species: lions eat lambs, but if they eat too many, first the number of available lambs will drop, and ultimately their own population numbers will go down. It is easy to imagine that such models often lead to (nonlinear) oscillatory cycles in population growth, with time delays between those of interacting species.

Solving the nonlinear logistic system (10) follows the same transform procedure as discussed above. Now, however, we have to use property X of Table 1 for its time transfor-mation. This property might seem mathematically complex, but it is actually a discrete convolution that is computation-ally very simple to handle (it’s just a sequence of basic additions and multiplications). Property X can be formally proven by (rather tedious) algebraic manipulation, properly rearranging terms at the right-hand side (a much more ele-gant derivation uses symbolic operator algebra, but this is beyond the space of this article). This results in an analyti-cal solution of the nonlinear ODE (10) in discrete time

: The first term of this solution gives the linear part (as dis-cussed above), and the second term yields the nonlinear effects. Again a variant of the Euler-type algorithm is suited to the task of prediction: from Eq. (11) it is easy to see that the discrete solution X S obtains by successive sin-gle-step forward recursions starting from the known initial condition X0 and then going forward in time: S=1, next S=2 etc. The probabilistic T map (5) then delivers the solution in continuous time for any desired time point t.

It is instructive to compare the solution (11) of the continu-ous ODE (10) with (i) the discrete solution (8) of the linear system (6), and with (ii) the nonlinear discrete dynamic system that is usually seen as its discrete analog (and there-fore is known as the logistic map):

X S+1 = A X S (1 – X S ) (12) This logistic map is famous because it is more or less the

simplest system that exhibits chaotic dynamic behaviour (in contrast to the logistic ODE). Again, there is a linear term and a quadratic nonlinearity, now in discrete time. But there is an essential structural difference between the dis-crete solution (11) to the logistic ODE on the one hand, and the linear system (8) and logistic map (12) on the other hand. The latter are iterated maps, i.e., result from repeated function application; to obtain the value at the next time-point one only needs the preceding timepoint.

In contrast, Eq. (11) shows that in the solution of the logis-tic ODE all previous time points are involved. So, this con-tinuous dynamic system has a memory in discrete time, even though this is not at all evident from the ODE formu-lation (10) that involves a single continuous timepoint. Al-though they share the name, the logistic ODE and the logis-tic map are totally different in their dynamic behaviour. Lorenz chaos. Property X of Table 1 also enables to solve in discrete time the well-known Lorenz model (1963), de-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.

Preview of Coming Attractions

It is probably most interesting here to briefly investigate the impact on the structure of time resulting from nonlinear dynamics. Namely, the above methods and results suggest some intriguing conceptual (or if you wish, philosophical) insights into the nature of time, particularly regarding “density” and “curvature” of time.

Consider the nonlinear differential equation (2) in general and how it changes under the T temporal ontology mapping (5). The left-hand side T (lhs) is easy: according to property VIII it always maps onto a simple one-step forward differ-ence. The right-hand side involves a composite function f(x(t)) that is generally nonlinear. Taking the transform T (rhs) changes our ontological view on the dynamic world in two stages:

•

First, it changes the function f, seen as a function of x only, into a similar but not identical function F (wit-ness for example the properties III-VI and X). This is already an important difference with the standard ap-proach depicted in Figure 1.

• Second, it also changes the function x seen as a func-tion of t (since the same properties apply again).

If we attempt to visualize this latter effect, we get a picture radically different from Figure 1. What happens is that the average density of the occurrence of (discrete) events is not constant but changes over the (continuous) time axis.

You might visualize this by imagining that the discrete time axis gets curved, and in continuous time you only see its projection onto the continuous time axis (see Figure 3). It is actually not difficult to find examples where the discrete-time “curvature” becomes so strong that it creates a singu-larity in (note: finite) continuous time.

Thus, the metaphor of the flow of time as a river [1] gets strangely bent due to nonlinearities: it’s possible to create

something like a black hole in the river bed of the timeline!

Figure 3. The T temporal ontology mapping may lead to

flows of time that are “curved”.

UPPER-LEVEL GENERALIZATIONS

This paper only outlines a small fraction of the results I have developed concerning nonlinear ontology mappings between time, and could only hint at the underlying mathematical proofs and algorithms. A few final general remarks are in order.

The uses of “old” science . First, the whole theory of tem-poral ontology mapping can be founded upon various treas-ures stemming from rather ancient mathematics. Much of it has more or less become extinct and superseded by modern computer (in fact, number crunching) approaches, and as a result is not treated anymore in modern textbooks on numerical methods. Specifically, this theory can be set up in a very elegant and concise way by means of symbolic operator algebra [3] as you find it in the textbook by Boole

(11); I note that this is also true in the general case.

AI temporal reasoning and android epistemology. The present work has important differences with respect to much of the work in AI temporal reasoning (see e.g. [7]) in terms of focus and assumptions. The present work uses standard point algebra from mathematics, and therefore interval algebras and axiomatizations (such as Allen’s, see also [1] and [7]) are not really relevant here. Important in my approach is that time is a metric space, i.e. a distance measure can be defined (in AI temporal reasoning usually 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-lishing (partial) temporal ordering, possibly under incom-plete or uncertain information. In contrast, this paper as-sumes full linear ordering in time (this is precisely what the variable S expresses), and focuses on tasks of prediction and control (in line with physics and mathematics). This paper shows that also in the point approach to temporal reasoning a lot of interesting progress can still be made.

If one refrains from delving into the mathematics, it yields some conceptual consequences for “android epistemology”. Androids (computers, robots and other discrete machines such as presumably StarTrek's Mr. Data) live in a discrete spacetime. Humanoids seem to live in a continuous space-time. So they inhabit ontologically speaking fundamentally different worlds. One might think that continuous beings can do all kinds of things in their spacetime that discrete beings cannot do in theirs – since continuous spacetime has many more points one can do something in or at than dis-crete spacetime. This paper shows this is not true: if they are sufficiently intelligent, discrete beings can do anything continuous beings can. Being “intelligent” can even be mathematically expressed here: the reasoning assumption that spacetime has characteristics of randomness, and still is causally ordered, according to Eq. (5).

Above I discussed things from the perspective of time. Surely this is a key top-level ontology concept. However, my approach and methods also work for types of independ-ent variables other than time. Other continuous variables can be formally discretized in this way as well. For exam-ple, one can in this manner also treat the concept of space. Ontology: content vs. representation. Finally, the paper has provided an in-depth case study in ontology mapping. I submit that this provides some evidence that ontology building, mapping, and reuse is much a substantive issue, more than a matter of generic representation language and semantic tooling. I note that this is already the case for such a high-level, generic, common, and commonsensical con-cept as time that does not depend on a specific domain. Substantive or content issues will be even more strongly present in task and domain specific ontologies. But in the end this is where the real semantic and web intelligence applications will be. This is perhaps a sign that the seman-tic research community at some point cannot avoid signifi-cant substantive issues in Web ontology, and has to be careful about (over)emphasis of generic representation and tooling issues without adequate domain grounding. Or, be sufficiently moderate in its expectations of the size of its own role in building the Semantic Web. Acknowledgment. This work has been supported in part by a travel grant from the KnowledgeWeb Network of Excellence (EU-IST-2004-507482).

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