Ontology Examples

Consider again two people modeling the relation in over the four entities of Route 2, Orono,

Bangor, and Maine. Instead of explicitly providing all the tables of data via tuples, they now also specify axioms that they think in should satisfy. These axioms enhance the explicit data

tuples for in by adding other conditions that in must satisfy. Such a combination of data and axioms is an example of an axiomatic ontology. Assume further that the two people use the

same language (e.g., the language of first-order logic) to specify the axioms for in.

Schematically, these assumptions are represented in Figure 1.7. Note the three major differences between this figure and Figure 1.6. First, Figure 1.6 deals with databases; second,

it makes no mention of a logical language; and third, it shows just one model (i.e., set of tuples) for each database. By contrast, Figure 1.7 deals with ontologies, it mentions the logical language (used for specifying axioms and for drawing inferences), and it shows more than one

model (set of tuples) for each ontology.

same nonlogical vocabulary

and same logical language

A's idea of in

multiple sets of tuples Ontology 1

same external reality

B's idea of in

different notions of in

different

models of in multiple sets_{of tuples} Ontology 2

Figure 1.7: Same symbols and logical language, different meanings

Consider the following points.

• The two people are modeling the same external reality.

• They use the same symbol—in—to represent this spatial relation.

• This same symbol and the same logical language are used by both ontologies.

• Each ontology has multiple sets of tuples, each of which is a model of the spatial relation in, reflecting the individual’s notion of in.

• Once the focus is on the ontologies (as machine-processable written artifacts), one is no

longer dealing with exactly which notions of in the two human modelers have had in mind. Instead, the ontology is taken to be the best available approximation to their no- tions, and the focus is on the meaning of in as that meaning is specified via the semantics

of the ontologies themselves.

In the case of the example databases (Figure 1.6), the model of a given database consists of just the data elements along with a single set of tuples for the relation in. Under the viewpoint

taken this thesis, this unique model defines what in means in that database.

In the case of an ontology, a model also consists of the data elements along with a single set of tuples for the relation in. The significant difference between an ontology and a simple

database is that an ontology usually has multiple models, expressed implicitly by the combi- nation of data and axioms, whereas a database has just a single model, specified exactly by the data tuples.

Each of the ontologies in Figures 1.8 and 1.9, for instance, has more than one model, because the axioms and the data of the ontology do not uniquely constrain the set of tuples that specify the semantics of the relation in. The fact that ontologies in general typically

have more than one model is central to any definition of semantic interoperability between ontologies. This is so in spite of the fact there may be other, human-significant aspects of the

relation in that are not captured by the ontology. But because ontologies do not capture any aspects of meaning outside the data and the axioms (and the formal framework of reasoning

in which they are embedded), such meanings are not amenable to the formal analysis used in this thesis.

Axioms forin Data Tuples forin

(Route 2, Orono) ∀x,in(x, x) (Orono, Maine)

Figure 1.8: OntologyO1for in

Axioms forin Data Tuples forin

(Route 2, Orono) ∀xyz,in(x, y)∧in(y, z)→in(x, z) (Bangor, Maine) (Orono, Maine)

Figure 1.9: OntologyO2for in

Consider the ontologiesO1 andO2 specified in Figures 1.8 and 1.9 above. Each contains

a single axiom and some tuples of data. The axiom of O1 states that every entity is in itself,

and the axiom of O2 states that the relation in is transitive. More significantly, the particular

axioms and data tuples of the ontologies combine to specify (implicitly) the models of each

ontology.

A model for a given ontology is a depiction of how the world could be configured that conforms to the data and the axioms. (Chapter 3 gives a more precise definition.) For instance,

in any model ofO1 the following relationships must hold: “Route 2 is in Route 2,” “Orono is

in Orono,” “Bangor is in Bangor,” and “Maine is in Maine”, because these relationships are dictated by the axiom. Similarly, in any model ofO1 the relationships “Route 2 is in Orono”

A convenient way to picture the models ofO1 is to use graphs, since the entities of Route

2, Orono, Bangor, and Maine can be represented by vertices in a graph, and the single binary

relation in can be depicted by the directed arrows in the graphs. In the top half of Figure 1.10, two of the models of O1 are shown as graphs, where the vertices labeled ‘R2,’ ‘O,’ ‘B’, and

‘M’ stand for Route 2, Orono, Bangor, and Maine, respectively. Similarly, the bottom half of

Figure 1.10 shows two models ofO2.

Figure 1.10 suggests thatO1 andO2 do not have the same sets of models (because model

1 of O1 is different from model 1 of O2). This turns out to be the case, since model 1 ofO2

could never be a model ofO1 (since the in relation in model 1 ofO2 is not reflexive), and so

the sets of models of the two ontologies cannot be the same. BecauseO1andO2have different

sets of models, they have different semantics, in particular, different semantics for the spatial relation in.

Even though bothO1 andO2use the same relation symbol, and the same logical language

to express axioms, the meanings that the two ontologies give to the symbol in are different. This is true in spite of the fact that the ontologies have at least one model in common — the the right-hand model in Figure 1.10.

Figure 1.10 exemplifies another important difference between the semantics of databases and the semantics of ontologies. Although in both cases the meaning of symbols is formalized via models, if two databases have a model in common, then they have the same semantics

(based on the discussion in the previous section), because the model for a given database is unique. Two ontologies, however, may have one or more models in common and yet have different semantics, since an ontology generally has more than one model.

One might suppose that the difference in the semantics of in betweenO1 and O2 is due

to the fact that O1 has one axiom and O2 has a different axiom. But that fact provides only

a partial explanation for the different semantics of O1 and O2. A fuller explanation is that

the different meanings are due to each ontology’s particular combinations of axioms and data that result in different sets of models for the two ontologies. There are three ways that two

ontologies might have different sets of models: (1) they have the same data and different axioms; (2) they have the same axioms but different data; or (3) they have different axioms and different data, which is the case ofO1 andO2. What is significant about the semantics of

an ontology is not the data tuples or axioms considered individually, but rather the way that the data and axioms collectively determine the models of the ontology (see Section 3.1).

R2 B M O R2 B M O model 1 of O1 model 2 of O1 R2 B M O model 1 of O2 R2 B M O model 2 of O2

Figure 1.10: Two models of Ontology 1 and of Ontology 2

Since the semantics of the ontologies is determined by their models, and sinceO1 andO2

have different sets of models, one can say that they are semantically heterogeneous. Given this particular view of semantic heterogeneity, to what extent might the ontologies nonetheless be semantically interoperable?

To answer this question, consider the following. The semantic interoperability of ontolo- gies has to do with implicit logical consequences. These consequences can be probed by

determining whether a given query (statement in the same logical language used to specify the ontology) evaluates to True in all models of the ontology. So, it makes sense that an analysis of semantic interoperability takes into account both models and queries.

Thus, among the plausible answers to the above question are that

• the sets of models ofO1andO2 overlap;

• queries to each ontology target one or more of the models they have in common;

• a given queryQposed toO1 gives the same answer when posed toO2.

Chapter 3 explores in-depth an answer to this question based on the sets of models of the two ontologies and the queries put to the ontologies.