GFO Part I Basic Principles
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15.1 Examples

15.1.1 Example for Comparison: The Statue

The following example is discussed in (39), for the DOLCE and other approaches therein. We refer to the formalization in the framework of DOLCE only. A formalization in GFO is expounded and then compared to the DOLCE formalization. Source Material

The example is stated as follows in (39):

``A statue of clay exists for a period of time going from $t_{1}$ to $t_{2}$. Between $t_{2}$ and $t_{3}$, the statue is crashed and so ceases to exist although the clay is still there.'' Ontological Analysis

Many entities can be identified on the basis of the statement. The term ``statue'' may have different meanings; we assume that ``statue'' denotes a persistant $st$ of material objects, with a certain lifetime $c$, which we assume to be a chronoid. ``clay'' is an amount of substrate $cl$. The statue $st$ consists of the amount of clay $cl$. More precisely, at each time-boundary at which a presential instantiates the persistant $st$, there is a presential amount of substrate of which the instance of $st$ consists: \begin{equation*}
\forall x,t (\Gpersist(st, x, t) \Land \Gsubstrate(cl, y, t) \rightarrow
\Gconsist(x, y))
\end{equation*} The demolition is a process $cr$, in which many different (sub-)processes and material structures may be involved. The demolition is projected onto a framing chronoid, say $d$, with starting time-boundary $s$, and ending time-boundary $t$: \begin{equation*}
prt(cr, d) \land lb(s, d) \land rb(t,d)
\end{equation*} The original statement refers to three time-boundaries, $t_{1}$, $t_{2}$, and $t_{3}$, and the following ordering holds among them: $t_1 \le t_2 \le s \le t \le t_3$29. The statue exists from $t_1$ to $t_2$, thus one can assume that $c$ starts with $t_1$, therefore $\Glb(t_1, c)$. We may further expect that at $s$, the statue is present, but at $t$, the statue ceased to exist. Further, $st$ participates in the beginning of the demolition, $cl$ in the whole event. \begin{equation*}
\forall x,y (\Gpersist(st, x, s) \land \Gprocb(cr, s, y) \rightarrow
\Gcpart(x, y))
\end{equation*} \begin{equation*}
\Gperstpartic(st, cr) \land \Gperstpartic(cl, cr)
\end{equation*} The lifetime of $st$ and the framing chronoid $d$ overlap, more exactly there is a chronoid $f$, such that $f$ is an end-segment of $c$ and at the same time an initial segment of $d$30: \begin{equation*}
\exists f (\Gprocstarts(f,d) \land \Gprocends(f, c))
\end{equation*} The process-boundary at $t$ does not contain a constituent part that is an instance of the persistant $st$, but there is a material structure which is the ``successor'' of $st$, in the sense that its instances are ontically connected with those of $st$:

& \exists st',z ( \GMatPerst(st') \land
\Gpersist(st',z, t) \l...
...\exists r v w (\Gpersist(st, v, r) \land \Gpersist(st', w, r)) )

Finally, let us consider the point in time when the statue ceases to exist. This can be understood as an extrinsic change, such that before the change, $st$ still persists, whereas after the change, it does not: \begin{equation*}
\exists u,v (s \le u \le t \land \Gtcoinc(u,v) \land \Grb(c, u...
...ersist(st, x, u)) \land \Lnot \exists y
(\Gpersist(st, y, v)) )
\end{equation*} Figure 2 provides a graphical overview of some connections between the aforementioned entities.

Figure 2: Visualization of some aspects of the formalization
\centering {\includegraphics[width=.925\linewidth]{}}
\end{figure} Comparison with the DOLCE Formalization

We consider $t_{i}$ to refer to time-boundaries, which is not possible in DOLCE, because it does not have this notion of a boundary. However, we consider time-boundaries more adequate, based on the expressions ``going from $t_{1}$ to $t_{2}$'' and ``between $t_{2}$ and $t_{3}$''. Altogether, relating the entities to time (and space) is different in DOLCE as compared with GFO, because there is no direct projection (e.g. $\Gprt$), but DOLCE establishes the link to time and space as a relation to qualities.

Similarly, on the basis of time-boundaries, GFO can formalize the extrinsic change covering the particular moment when the statue is no longer considered as existent. Note that this depends on the granularity of the model, while the granularity is not yet explicitly expressible in GFO.

The remainder of the formalization appears rather similar to that of DOLCE. The relationship between the statue and the clay is covered ($DK$ in DOLCE, $\Gconsist$ in GFO), but note that this relation will be extended and revised in terms of the theory of levels, cf. sect. 4. The participation of the statue and the clay in the demolition are expressed in DOLCE (by $PC$) and GFO as well ($\Gperstpartic$ and $\Gprocb$).

15.1.2 Race Example

This example will provide a rough overview of the GFO ontology in a single, coherent, (but rather simple) modeling case. It employs many, yet not all applicable GFO categories. Source Material

Let us consider a 100-metre sprint, in which two runners take part: $runner_1$ and $runner_2$. The race starts with the signal at time $t_1$ and lasts until $t_4$, when the last runner crosses the finishing line.

$runner_1$ quickly reaches a high speed and takes the first position, while $runner_2$ does not accelerate that rapidly but manages to pass $runner_1$ at $t_2$. At $t_3$, $runner_2$ crosses the finishing line, winning the race. The victory of $runner_2$ is a big surprise for the audience, so the race is broadly discussed and is announced to be the most surprising and interesting race of the decade. Ontological Analysis

For brevity, let $m \in \{1,2\}$ and $1 \le n \le 4 $ in all formulae. Situoids:
The whole race can be interpreted as one complex entity extended in time, namely a situoid $race$, spatially delimited by a topoid $tp$: $\Gsframe(race, tp)$, and temporally framed by a chronoid $c$: $\Gtframe(race, c)$. $race$ is associated with certain universals, which select the point of view and granularity. Here we assume that these universals are $runner$, $track$ and $audience$, which delimit the context in which we analyze the race. So, we have $\Gassoc(runner, race)$, $\Gassoc(track,race)$ and $\Gassoc(audience,race)$. Chronoids and Time Boundaries:
We have identified the chronoid $c$, framing the race. It has a left boundary $t_1$ as the race starts, $\Glb(t_1, c)$, and a right boundary $t_4$, where $runner_1$ crosses the finish line, $\Grb(t_4,c)$. Moreover, we identify two inner boundaries, $\Ginnerb(t_2, c)$ and $\Ginnerb(t_3,c)$, that are of special interest: $t_2$, where $runner_2$ takes the lead and $t_3$ where he wins the race. Persistants:
The persistence of the runners throughout the entire race is provided by viewing them as two persistants, $runner_m$, which are instantiated by ontically connected presentials $runner_{m,t}$ present at each time boundary $t$ of the race. Each persistant persists through time, or more precisely, through the time boundaries on which its instances exist. Moreover, each persistant $runner_m$ participates in the process of the race. Analogous considerations apply to the persistence of the audience and the track. Space Regions and Topoids:
The location of the race is determined by the topoid $tp$ framing the situoid $race$. The topoid $tp$ is assumed to be a convex closure of the mereological sum of all space regions occupied by the material structures constituting the situations of $race$. In our case, $tp$ is the sum of space regions of the presential runners, the track and the audience, across the overall period of the race. Situations:
At each time boundary $t$ in the course of the race, one can project the race to its boundaries $race_t$, which are situations. In particular, one may consider the situations at $t_{n}$ which are referred to in the example. Each of these situations is a compound of several constituents, of which those are of particular interest. They are determined by the universals associated with the situoid race. Therefore, we focus on $runner_{m,t_n}$, $track_{ind,t_n}$, and $audience_{ind,t_n}$. Constituents / Material Structures:
All constituents of the situations of the race considered here are material structures, and as such occupy some spatial region (cf. the remarks on space regions and topoids above), and consist of some presential amount of substrate. For example, we could say that $body$ is a solid substrate of the runner: $\Gconsist(runner_{m,t_n}, body_{m,t_n})$. Properties:
Moreover, each material structure comes together with its individual properties. The runners or the track, for example, inhere qualities like speed, blood pressure or hardness (here: of the track). In the case of the property universal $speed$, for example, at each time boundary of the race we find an individual speed for each runner, as well as individual values of those property individuals: let the speeds of the runners at $t_2$ be 25$ $km/h and 30$\;$mph, respectively. We observe that the individual property values are instances of the categorial property values belonging to two different measurement systems. The first measurement system is a set of values in the form of pairs of a number and the unit ``km/h'', while the second is a set of values with unit ``mph''. Nevertheless, the individual quality values 25$\;$km/h and 30$\;$mph are comparable, since the individual qualities they refer to, say $speed\mbox{-}runner_{m,t_n}$, are instances of the same property speed.

Further, one can find properties of the whole race, which seem to be indicated by the expression: ``it was the most surprising and interesting race of the decade''. Here we identify being-the-most-interesting-race-of-decade as the quality value of the individual quality, level-of-entertainment-of-the-race. It is clear, however, that this quality does not belong to the material, but rather to the social level. Here we do not say it inheres in $race$. Processes:
The race as a process is a combination of several processes, among them $run\mbox{-}runner_m$ processes. Here we can observe that either of these processes is a coherent process, the boundaries of which contain material structures, namely instances of the persistants referred to above, $runner_{m,t}$. Hence, we have $\Gprocb(runner_{m,t_n}, run\mbox{-}runner_{m})$, and all of those instances are ontically connected (for the same $m$). Changes:
Moreover, we observe certain dynamics between those processes, which can be modelled using intrinsic and extrinsic changes. First, the changes in the speed of the runners can be interpreted as intrinsic changes. Second, we may identify an extrinsic change at $t_2$, when $runner_2$ takes the lead. To represent this change we identify two parts of the process $race$, namely $leading\mbox{-}runner_1$ and $leading\mbox{-}runner_2$.

These processes meet at $t_2$ which means that $t_2$ and a coincident time-boundary are the pair of the right boundary of the projection of $leading\mbox{-}runner_1$ and the left boundary of the $leading\mbox{-}runner_2$. The extrinsic change of taking the lead - or switching from the position of losing the race to the position of winning - by $runner_2$ is represented as change( $b_1, b_2, loosing, winning,
position\mbox{-}in\mbox{-}race$), with $b_1$ and $b_2$ representing the process boundaries at the end and at the beginning of $leading\mbox{-}runner_1$ and $leading\mbox{-}runner_2$, respectively. Analogously, the crossing of the finish line by the $runner_2$ could be represented, which is a change from winning to being the actual winner. Levels:
So far we have concentrated on the material aspects of the race, where runners are identified as material objects with inherent material qualities. But we should keep in mind that the runners and the race cannot be reduced to the movement of two material objects along the line of the track. Rather we identify runners as the social roles of some individuals, just as the track is the role of some solid object of a certain shape with certain properties. We see that the situoid $race$, in part, does not belong to the material level, but to the social and conceptuals level as well. At the social level, we do not consider bodies with material qualities, but rather social objects, their roles, e.g. being runners or the audience, and their social qualities, together with their corresponding values, like those of winning or losing.

15.1.3 Staging Example

This example is taken from the domain of clinical trials, one of the major fields for application of the research group Onto-Med. The example is a first attempt to define the term ``staging'' using GFO, and illustrates the method for ontological mappings, cf. sect. 2.4. Source Material

There are various sources for defining staging, including discussions with medical experts. Therefore, we provide our own definition, based on discussions with our medical experts, respective literature, e.g. ``Pschyrembel'' (47) and ``Harrison'' (10), and several websites31.

The definition is divided into three parts of overall validity, some background facts of frequent validity and general background knowledge. Definition:
Staging is a process composed of the detection of the anatomic extent of tumor$_{1}$32 and the classification of the result with respect to a staging system. Background knowledge:
Anatomic extent refers to the size of the tumor$_{1}$, in both its primary location and in metastatic sites. The most common staging system is the TNM classification, but there are others, e.g. those used for cancers of children and those used for cancers of female reproductive organs. Staging is applied to malignant tumors$_{2}$. The result of staging, i.e., the classification in a staging system, is used for treatment planning, prognosis evaluation and the comparison of treatments.

There are four types of staging. Clinical-diagnostic staging involves what a doctor can see, feel and determine through x-rays and other tests. Surgical-evaluative staging involves exploratory surgery, biopsy or both. After surgery, the tumor$_{1}$ can be directly examined and its cells microscopically analysed, which is called post-surgical-treatment pathologic staging. If additional or new treatments are applied to the same disease, re-treatment staging uncovers the extent of the tumor$_{1}$. General background knowledge:
A tumor$_{2}$ is a disease that causes the growth of tumor$_{1}$ (often tumor tissue). Ontological Embedding into GFO

Following the axiomatic method, we begin by collecting important terms from their definitions. There are: staging, process, detection, anatomic extent, tumor$_{1}$, classification, staging system, disease, tumor$_{2}$, primary location, metastatic site, size and malignant.

Now these terms can be analyzed and ontologically embedded into GFO. Each term is subsumed by a GFO category or linked to GFO categories by means of basic relations, as specifically as possible. We analyze and group terms with respect to the basic category to which they refer. Processes:
Staging is a process that is composed of two steps, a process of detection and a process of classification. Thus, staging is a discrete process. Detection and classification are processes as well, but they are not analyzed in detail here, since they will be used as domain primitives below. Further, each disease is a process, and thus a tumor$_{2}$, as well. Topoids:
Topoids are only indirectly involved, through the notions of ``location'' and ``site''. These refer to topoids determined relative to the body of the patient and the tumor$_{1}$, respectively. A tumor$_{1}$ may spread throughout the body. The topoid occupied by that part of the tumor$_{1}$ first occurring or discovered is called the primary location. Topoids of other tumor$_{1}$ parts (metastases) are called metastatic sites. Configurations:
Consider the anatomic extent of the tumor$_{1}$, which is determined and classified during staging. This should be understood as a situation rather than a single quality, although the latter may appear appropriate at first glance. This situation refers to (a) the size of the parts of the tumor$_{1}$ at the primary location and metastatic sites, (b) the relationship between the tumor$_{1}$, the involved anatomic entity and adjacent anatomic entities, and (c) possibly more relations between the tumor$_{1,2}$ and the body (cf. the TNM staging system). Properties:
First, the sizes of connected parts of the tumor$_{1}$ are qualities that are measured in centimeters or inches. Second, there is an evaluative quality of a tumor$_{2}$, which is the degree of malignity. The simplest measurement system contains just the values ``malignant'' and ``benign'', which are mutually exclusive. Usually, malignant tumors$_{2}$ are staged. Material structures:
A tumor$_{1}$ (often tumor tissue) is a material structure, which is created and (usually) growing throughout the course of the disease, i.e., tumor$_{2}$. Symbolic structures:
In order to fully describe the notion of a staging system, the category of symbolic structures is required. A staging system is, in the simplest case, a set of symbolic structures that denote universals of anatomic extents (viewing the extent of a tumor$_{1}$ as a multi-dimensional or -faceted configuration, as introduced above). However, this cannot be further analyzed without a deeper understanding of symbolic structures and the denotation relation. Domain-specific Extension

The above descriptions provide an ontological embedding of several domain-specific terms into GFO. However, this is obviously rather weak, e.g. for staging only, a structural decomposition into two processes could be stated. In order to add domain-specific dependencies, a domain extension is necessary. That means, new primitives must be added and ontologically embedded, which can then be used to express more domain-specific interdependences.

Robert Hoehndorf 2006-10-18


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