Patterns for modeling qualities
In the current section we review two patterns frequently proposed for modeling temporal information, i.e. reification of n-ary relations and 4d fluents. First, we present the patterns and then discuss their application in three scenarios of distinct temporal distribution of qualities.
Reification
The reification of n-ary relations is a popular strategy for modeling temporally changing information. It interprets a time-indexed quality as a 3-ary relationship linking an object, its quality and the time at which the quality is assigned to the object. Next, the relation is reified and introduced to the model as a class.
Pattern 3 depicted in Fig. 5 presents the application of this reification strategy to our use case. In contrast to patterns 1 and 2, a Quality Assignment is not modeled as owl:ObjectProperty but instead as a reified owl:Class acting as a proxy between an Object and its Quality.
A reified QA represents a specific assignment of a Quality to a particular Object and as such is dependent on both the Object and the Quality. That means that each Quality Assignment is inherent in exactly one Object and is the assignment of exactly one Quality. The former constraint is represented by the cardinality restriction on the carries link between Object and Quality Assignment and the latter - by the cardinality restriction on the of_quality link between Quality Assignment and Quality. Time-index is attributed directly to QA by means of the at_time property linking QA with the Time class.
The bottom part of Fig. 5 illustrates the application of pattern 3 to the CTO use-case. The object property carries links the Cell class with the ShapeAssignment class, which is a subclass of Quality Assignment. ShapeAssignment represents a quality assignment at a given time and has two OWL properties: of_quality and at_time. The former specifies the value of a quality, i.e. a specific shape, whereas the latter - the time index of the parameter.Footnote 3
This pattern can be applied to annotate a single cell with two distinct shapes at two different time points:
In many situations it is not the time index of quality assignments that is relevant but only their t8emporal order. This may also be true for some cell tracking experiments. In such cases pattern 3 can be simplified: in the upper part of Fig. 6 the property at_time and the class Time can be replaced with the property is_next, establishing the temporal order of quality assignments. The implementation of this pattern to our case is presented in the bottom part of Fig. 6.
Pattern 3 overcomes the limitations of patterns 1 and 2 reported above, since time-indexed quality value assignments are represented as instances only. Thus, even in situations where many time-indexed value assignments occur, the number of classes and properties in the ontology remains constant (and is relatively low).
On the other hand, as observed in [31], the model introduces additional OWL classes and OWL properties for representing time-indexed quality ascriptions, reducing its lucidity.
4d Fluents
An alternative to the n-ary relation reification is the so-called 4d fluents pattern [30]. It is inspired by four-dimensionalism [29], a philosophical theory explaining the persistence of objects through time, called perdurance, in analogy to their extension in space: similarly to an object occupying some space s having parts occupying parts of s, an object occupying some time t may have temporal parts occupying parts of t. In that understanding, time-extended objects are considered as the so-called 4d worms, which can be sliced into temporal parts, as 3d objects can be sliced into their spatial parts. The top-part of Fig. 7 presents a 4d pattern. In contrast to the reification pattern, the idea behind 4d fluents is not to reify a temporally indexed relation but instead a temporal part of an object. For instance, in order to model the fact that a cell c has a round shape at time t1 one can reify a temporal part of c and then assign a quality directly to the reified part:
Conceptually, the two patterns seem quite distinct, yet when comparing Fig. 5 and Fig. 7 one can observe that structurally they are almost the same and both are based on introducing an association class. The only structural difference is the cardinality constraint determining the number of qualities linked to the reified class. In the reification pattern it is 1, whereas in the 4d fluents pattern it is 0..n. Therefore, when modeling an object’s single quality assignment, both patterns are in fact equal.
The difference between the patterns can be well illustrated when applying the patterns to relations rather than quality assignments. Let us consider the CTO relation of cell-cell contact, denoting the fact that two cells touch each other. In order to model a temporally indexed cell-cell contact, the reification pattern requires the introduction of a single reified time-indexed relation, whereas the fluent pattern would introduce a reified and time-indexed slice for each cell participating in the contact.
The difference between the patterns can also be observed in cases where numerous quality assignments are being represented, which is in fact the real challenge of ontology engineering. Therefore, we will analyze the patterns using three different cases of temporal distribution of qualities:
-
Temporally non-overlapping quality assignments. For instance, a cell can have an oval shape at one time and an elongated shape at another, but it can never have both shapes at the same time.
-
Temporally equal quality assignments. Thit is a typical scenario in time lapse experiments where at a single time point numerous distinct qualities are observed, e.g. shape, location, etc.
-
Temporally overlapping, but not temporally equal quality assignments. This is a common situation when qualities change independently from one another, as is the case with the location and shape of a cell.
Temporally non-overlapping quality assignments of a single quality
As a starting point, we consider the simplest case, in which no two quality assignments of an object are located at the same temporal location. Such a situation is natural for many qualities when considered separately, e.g. typically a cell has a single location or a single shape at any given time. This scenario is often assumed in the works devoted to the modeling of temporal information, e.g. in [24, 30, 31]. In such a case it can be easily observed that both patterns behave the same, in fact there is no difference when applying them. Modeling n quality assignments of a single quality of a single object we need to introduce n instances of a reified class in both cases.
Both models are equally extensible, i.e. to introduce a new characteristic a new instance must be added to the model. Finally, the adequacy of both solutions seems to be merely a matter of personal taste since the choice between the patterns generates no structural differences in the models.
Temporally equal quality assignments
The above discussion is justified when considering a single quality in isolation. Yet, when considering numerous qualities of an object, it is clear that there can be two or more quality assignments which overlap temporally, e.g. a cell at a given time point can have some location and some shape. This is a typical scenario in cell tracking experiments, where at a single time point more than one quality is observed. In such cases the application of the reification pattern results in a model with redundantly time-indexed Quality Assignments: for each quality observed at a given time point a separate Quality Assignment instance has to be introduced.
It seems that the 4d fluents pattern solves that problem. Time slices in their simplest form are temporal parts of objects having an arbitrary temporal extension (usually considered an interval). An alternative approach, present e.g. in the General Formal Ontology (GFO), introduces temporal particles located at discrete time points (the so-called presentials) which are distinct from time extended slices [45]. In GFO, a presential is an entity that is wholly present at a single time point. For instance, a cell observed at a single time point would be considered a presential cell. A presential may have multiple assigned qualities, all present at the same time point as the presential which carries them. Thus, a presential is a snapshot of a time extended entity, i.e. a cell observed at a single time point can be considered a snapshot of a time extended cell.
Figure 8 presents the pattern for modeling time slices and presentials where both are considered temporal particles of objects. Based on that pattern a modeler can utilize both time interval slices and/or presentials, depending on the actual needs.
The annotation of an individual cell using the presential pattern would look as presented below:
In contrast to the reification pattern, the presential pattern reduces the number of instances introduced to the model. Instead of reifying each quality assignment at a given time point, all coinciding quality assignments are modeled with the help of a single presential instance.
4d fluents also scales better: each new quality assignment added to the reification-based model requires a new instance, whereas in the case of the 4d-based model no additional instance is needed when a time slice with the same time index already exists in the model.
The frugality of the 4d pattern seems to make it more intuitive than the reification pattern, especially in the case of cell tracking experiments where the presential cells are the entities physically represented in the acquired images. Thus, they can be easily identified and it seems quite natural to reify them.
Temporally overlapping characteristics
In cell tracking experiments the qualities of enduring cells are deduced on the basis of a sequence of observed presential cells and their qualities. For instance, if a cell is observed to have a round shape over the sequence taken at time points t1, t2,.., tn, then typically one can deduce that the cell has a round shape during the whole time interval (t1, tn). Clearly, in the case of numerous mutually independent quality assignments it may turn out that the temporal extensions of many of them may overlap. For instance, during a time interval (t1, t5) a cell may remain in location l1 but its shape may change from round in (t1, t3) to elongated in (t3, t5). That results in two shape quality assignments overlapping with the location quality assignment. This situation is clearly visible when the reification pattern is used, as for each observed quality assignment a reified instance is introduced.
However, when turning to 4d fluents, the first observation we make is that the adaptation of the pattern to this case is not as straightforward as in previous cases, when it was relatively easy to say what the cell’s time slice is, namely, a presential cell observable in an image and thus having its own identity. However, in the current case we want to reify not the presential (observable) cells but instead the time extended, temporal parts of cells. This raises the following question: What is a temporal part of a cell and what rules drive the slicing of an object (a cell) into its temporal parts?
It seems that at least two strategies for introducing temporal parts could come in handy. The first is based on the principal idea of 4-dimensionalism, namely that a time extended entity can be sliced into temporal slices in such a way that each slice fully represents the sliced object at a given time. This means that all qualities assigned to an object within the time span of a slice are attributed to the slice directly. We call this type of slicing vertical.
Unfortunately, modeling temporally overlapping quality assignments with vertical slices easily leads to serious redundancy and is hardly maintainable. This is due to the fact that slices are overlapping and each one represents an object in full at a given time and as such it carries all qualities attributed to the object during the slice’s lifetime.
An intuitive solution to fix this problem would be to prohibit the overlapping of slices. This results in a model in which a time extended entity is sliced into non-overlapping slices so that the sum of all the parts constitutes the full lifespan of the object.
Let us illustrate this strategy with an example, starting with the model of a cell remaining in location l1 during the interval (t1, t5):
Now, let us assume that we add to our model a new observation (fact) that the cell changes its shape from round in (t1, t3) to elongated in (t3, t5). If we add that observation to our model, we end up with two additional time particles depicting the location of the cell: one ending at t3 and the other starting at t3, thus both new particles are overlapping with:my_cell_slice. In order to fix this, one could reorganize the time slices into two non-overlapping slices, the first representing the state of the cell being round and located in l1 and the second - the state of the cell being elongated and located in l2.
Unfortunately, this strategy has its problems. As it can be seen from the above example, the change of any of the qualities may entail the reorganization of the object’s slices. That leads to a proliferation of slices, but more importantly, it makes the knowledge base strongly coupled and thus harder to extend, i.e. the addition of new information entails the reorganization of previous knowledge.
In addition, despite the fact that the strategy solves the problem of the model’s redundancy, it still results in heavily overloaded models. As presented on the listing above, each slice carries a full specification of the cell at a given time, even for those qualities which remain constant across many slices.
In order to overcome those limitations an alternative interpretation of 4d fluents could be considered: an entity could be sliced not only vertically, i.e. along the time dimension, but also horizontally, i.e. along its quality assignments. That way a slice does not fully represent an object at a given time but only some of its aspects, e.g. that a cell is located at l1 at t1 - t2. Thus, a slice is a kind of temporally indexed reified attribute of an entity.
That interpretation fixes the problem of model redundancy, but it also blurs the difference between the 4d fluents and the reification pattern, since a time slice now represents some quality assignment, i.e. some temporally indexed attribute of an entity. The actual difference between such an interpretation of time slices and reified quality assignments is hidden in the cardinality constraint on the quality role (presented in Fig. 5 and Fig. 7). While the reified quality assignment links an object with a single quality, a slice can link an object with multiple qualities when quality assignments overlap temporally. Thus, if we add the fact of a cell’s size, which is temporally equal to that of its shape, we are not forced to introduce a new temporal particle but it is sufficient to add that fact to: my_cell_slice_2: