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Summary

An exploration of the relationship between subproperties and property characteristics, in particular transitivity.

Author

Sean Bechhofer
Information Management Groups
School of Computer Science
University of Manchester
Oxford Road
Manchester
United Kingdom
M13 9PL
sean.bechhofer@manchester.ac.uk

Property Characteristics and Subproperties

Transitive properties can be very useful in ontologies. Recall that a property P is transitive if and only if the following is true:

* For all x, y, and z: P(x,y) and P(y,z) => P(x,z)

An example of a transitive property is “ancestor”. Any ancestor of an ancestor of mine is also an ancestor of mine. OWL provides us with an axiom for stating that a particular property is transitive.

ObjectProperty: ancestor
  Characteristics: Transitive

The notion of subproperties are also useful. For a property Q, R is a subproperty if and only if

* For all x, y: R(x,y) => Q(x,y)

An example of a subproperty relationship is “hasParent” and “hasFather”. Any two individuals that are related via the father relationship must be related via the parent relationship.

ObjectProperty: hasParent

ObjectProperty: hasFather
  SubPropertyOf: hasParent

Sometimes there is confusion over the way in which characteristics like transitivity interact with the sub/super property hierarchy. As far as transitivity is concerned, the characteristic is not “inherited” by sub-properties — we cannot infer that the sub property of a transitive property is transitive. The same holds for super properties.

To illustrate this, consider the following example. We have three (object) properties: knows, hasFriend, marriedTo. One of these (hasFriend) is transitive (now you might question this as a piece of modelling, but please just go with my rosy world-view that all the friends of my friends are also friends), and the properties are arranged in a hierarchy. In Manchester syntax we would have:

ObjectProperty: knows

ObjectProperty: hasFriend
  Characteristics: Transitive
  SubPropertyOf: knows

ObjectProperty: isMarriedTo
  SubPropertyOf: hasFriend

And yes I know that expecting marriage to imply friendship is again hopelessly optimistic, but I’m a hopeless optimist.

Now, consider a domain with four elements, Arthur, Betty, Charlie and Daphne. They are related as follows:

* Arthur knows Betty.

* Betty knows Charlie and Daphne.
* Betty hasFriend Charlie and Daphne.
* Betty isMarriedTo Charlie.

* Charlie knows Daphne.
* Charlie hasFriend Daphne.
* Charlie isMarriedTo Daphne.

The situation is as pictured below.

If we look at the ontology presented above, we can see that all the axioms hold — the subproperty axioms are being respected, as is the transitivity of hasFriend. Thus this situation is a model of the ontology.

Now if we consider isMarriedTo, we can see that our conditions for transitivity do not hold. There are three elements with isMarriedTo(Betty,Charlie) and isMarriedTo(Charlie,Daphne), but we do not have isMarriedTo(Betty,Daphne). So we cannot infer that isMarriedTo is transitive from the axioms. Similarly, there are three elements where knows(Arthur,Betty) and knows(Betty,Charlie) but we don’t have knows(Arthur,Charlie).

Recall that the inferences we can make from an ontology or collection of axioms are those things that necessarily hold in all models of the ontology. This little sample model provides us a “witness” for the fact that we cannot infer that knows is transitive from the axioms. Similarly, we cannot infer that isMarriedTo is transitive.

Of course, this is just saying that we can’t in general make such an inference. We are not saying that superproperties cannot (sometimes) be transitive. If we add to our interpretation the fact that Arthur knows Charlie and Daphne, then in this interpretation, knows is indeed transitive. And if we allow Betty to marry Daphne — hey, it’s 2013! — then we have a transitive subproperty (in this interpretation).

On the topic of transitivity and sub properties, the thesaurus representation SKOS uses a common modelling pattern, where a non-transitive property (skos:broader) has a transitive superproperty (skos:broaderTransitive) defined. The superproperty is not intended to be used for asserting relationships, but can be used to query for transitive chains of skos:broader relationships (assuming our query engine is performing inference). As we now know, this doesn’t mean that skos:broader is necessarily transitive.

This pattern is also often used for representing partonomy. Here we would use a (non-transitive) hasDirectPart for asserting parts of a whole, with a transitive superproperty hasPart allowing us to query the transitive closure. We can use counting with hasDirectPart — for example min or max cardinality restrictions — which we would not be able to do in OWL DL if hasDirectPart was transitive, due to restrictions relating to simple properties (see the OWL2 Structural Specification).

For other property characteristics the situation is different. For example, a subproperty of a functional property must be functional. Why? We’ll leave that as an exercise for the interested reader……

This is a review for Uli Sattler’s article on OWL the Web Ontology language. The title “OWL” looks a little strange in the table of contents, so I’d be inclined to change it to “The Web Ontology Language (OWL)”. Most of my comments are at this level, so indicates that the article is fine. I have only a few other comments.

As a background to how OWL works this K-Blog works for me. This article takes the reader on an introductory tour of OWL. It gives a genral background on OWL. It concentrates on the reasoning side of OWL – what a reasoner does. A few lines of introduction to say what the purpose the article is and what it wil not do would be useful. for instance, a guide to OWL exspressivity and modelling tips for OWL is not the purpose of this article. A link to Phil Lord’s KBlog on components and Matt Horridge’s on OWL syntax would be good.

Whilst OWL is probably the most prominent ontology language of the moment, within the biology context that is the topic of the Ontogenesis Kblog, the OBOF format should be mentioned (along with a link to the OBOF K-Blog). In this community, OBOF is very widely used and this should be acknowledged.

It would be useful to give a simple statement of what an axiom is. The word “model” and its meaning within the DL world may need more attention as well (or a warning to just overlook it). Model is used in this community in its much more general sense.

“where OWL and reasoning is used purely to make sure that the things said about classes and the resulting entailed class hierarchy are correct” – Is OWL natively used in SnoMED? I thought it used a DL, but not OWL as such. Also, “correct” needs care as it is checked for logical correctnesss rather than domain of interest correctnerss.

Minor:

• “are important: OWL comes in” → sentence change instead of colon.
• “, properties, and inviduals” → individuals
• “chose whatever relation you like” → choose
• “help us unterstand ” → understand
• “Coming back to Protege” → new paragraph at onset of this sentence.
• “currently recides in” → resides
• “Both have clearly advantages ” clearly have

reviewed by robert Stevens

School of Computer Science, University of Manchester.

The word ontology is highly overloaded and can have a number of interpretations. In addition, there a variety of representations, languages and approaches that can be used to represent knowledge. Each has characteristics that impact on their appropriate usage or the functionalities that they offer. The following attempts to tease out some of the differences between the various representation languages and solutions available. Although some of this may be “well known” or obvious, it is useful to set out clearly the vocabulary that we are using — for example, the term formal is often used to refer to languages which are what we have called here semantically strong.

Content by Sean Bechhofer and Robert Stevens.

Representation Languages and Artefacts

In the following discussion, we distinguish between knowledge artefacts and representation languages. A knowledge artefact (or more simply artefact) captures some particular knowledge about a domain. Artefacts are represented using a representation language which provides syntax (and some degree of semantics) for expressing such knowledge. An example of a representation language is OWL [link], while OBI [link], a particular OWL ontology, is an artefact.

This distinction between the languages and the artefacts is important – for example there is a relationship between expressive languages and rich representations, but one does not necessarily always produce rich ontologies when using an expressive language.

Establishing a vocabulary allows us to articulate the characteristics of our representations and artefacts, and assists in placing them within space. This will help in tasks such as choosing appropriate artefacts for particular purposes or evaluation. For example, FMA [link] and MeSH [link] are both artefacts that are referred to as ontologies, but have radically different representation languages, modelling styles and application usages.

Below we set out a number of characteristics that can apply to representation languages and artefacts.

Syntax and Semantics

Representation languages have a syntax — a way of writing statements down. Syntax sets out rules that tell us how to form expressions using the symbols available, and defines what constitutes well formed statements in the language. For example, to use Manchester OWL Syntax:

(X and Y)

is a well-formed class expression, whereas

and X Y()

is not.

Syntax on its own carries no meaning. The syntax rules simply tell us which combinations of symbols are valid. The semantics of the representation language allow us to interpret those combinations of symbols.

Semantic Strength

A strong representation language is one for which explicit semantics are given for the operators or vocabulary of the representation. The semantics tells us what well-formed statements mean, and are often defined as the set of concrete situations (models) that are consistent with a sentence or set of sentences (this is often called a model theory). A strong representation can reduce the ambiguity or vagueness that might be present in determining whether concrete situations are consistent with the statements being made in the vocabulary.

A strong representation language facilitates machine processing via inference (see reasoning).

A weak representation language is one for which explicit (machine-processable) semantics are given for the operators in the language.

For example, to compare OWL and SKOS, OWL is a stronger representation, while SKOS is weaker — the semantics of the OWL subclass relationship are given in terms of class extensions, while in SKOS intended interpretation of the hierarchical and related relationships are given in natural language, viz: A hierarchical link between two concepts indicates that one is in some way more general (“broader”) than the other (“narrower”). An associative link between two concepts indicates that the two are inherently “related”, but that one is not in any way more general than the other. Note that for SKOS, the situation is slightly complicated, as there are some conditions for which a semantics is provided (for example, the broaderTransitive relation is declared to be transitive, which allows some inferences to be drawn).

Note that here the use of the terms “strong” and “weak” are not intended to be used in a pejorative sense. A weaker representation language may be more appropriate than a stronger one in some particular application scenario.

Expressivity

The expressivity of a representation language refers to the ability of the language to distinguish between different kinds of concrete situation. For example, OWL provides different quantifiers that allow us to distinguish between situations where all of the related objects must have some characteristic (universal quantification) and situations where some related object must have a characteristic (existential quantification). Expressivity can be seen as a “sliding scale” — for example, OWL Lite is a less expressive language than OWL DL, as it lacks some of the operators provided in OWL DL (e.g. cardinality restrictions involving numbers other than 1 or 0). Expressivity is related to semantic strength.

There are also differences between languages which are “naturally” and “un-naturally” expressive. For example, negation is not explicitly present in OWL-Lite (there is no complementOf operator, but can be encoded.

Similarly, representation languages may include “syntactic sugar”, for example allowing equivalence axioms between class expressions rather than requiring explicit pairs of subclass axioms.

Cognitive complexity. Do I need to write down a lot in order to express something?

Reasoning

Reasoning refers to the process of answering some semantic based query, such as determining if one statement follows from another. Reasoning is generally performed in the context of a particular artefact, using the characteristics or properties of the representation language (for example the semantics). Reasoning thus relies on a strong representation (e.g. the representation has a semantics that can be used as the basis for that reasoning). Issues of tractability and scalability (related to expressiveness and richness). [Link to elsewhere]

Formality (artefact)

A formal ontology is an artefact in which certain philosophical principles have been applied in order to determine that the underlying content is an “accurate” or faithful representation of the world. Formality is a property of an artefact. There are a number of mechanisms and methodologies for ensuring formality (e.g. OntoClean). Upper ontologies can be used in order to ensure that the organisation of the ontology is consistent with other ontologies (rather than ad hoc). The use of an upper ontology is likely to lead to increased formality.

Formality requires a subjective, qualitative judgement. This may also depends on the ontological modelling style.

Axiomatic Richness (artefact)

The axiomatic richness of an artefact refers to the level of axiomatisation that is present. The term axiomatisation is taken to refer to the expression, in some representation language, of the explicit assumptions being made about the domain described by the artefact. A lack of axiomatic richness limits the possibility of deriving inferences from an artefact. There are no guarantees, however, that an artefact that is axiomatically rich will lead to many new inferences.

Axiomatic richness could be measured in a number of ways. Hayes [ref] for example, in the Naive Physics Manifesto, discusses density. One could also consider coverage, i.e. how much information is being captured about the domain.

Knowledge artefacts may include rich descriptions of their content in terms of natural language descriptions, scope notes etc. For example, a SKOS vocabulary with very detailed descriptions of the Concepts described in it. Such an artefact would not necessarily be considered axiomatically rich: although the artefact contains comprehensive information, this is not being presented in a form amenable to machine processing.

Application/Purpose (artefact)

What is the intended purpose of the artefact? How is it actually used (this could differ from the intended purpose)? Who are the intended consumers? Humans? Machines?

Community/Consensus (artefact)

Does the artefact capture the view of an individual or a community?

Cost

Cost applies to both representation languages and artefacts. Costs are incurred in a number of ways.

• Computational costs (time, space, memory). Cost of reasoning. Related to expressivity, axiomatic richness.
• Cognitive costs. Understanding a language can be challenging. The articulation of facts using a representation can be difficult.
• Management costs. Construction, maintenance, change management, all of which can be hard in collaborative situations.

Examples

This section requires further work.

Artefact examples?

• GALEN
• MeSH
• OBI
• FMA
• GO

Non-rich OWL, something that respects the semantics — there are things around that have been “bent” into OWL, and which thus doesn’t respect the semantics. There is a difference between things that shouldn’t be represented as OWL, because they don’t respect the semantics and things which could be represented as OWL, but there’s no point because they’re not rich enough.

Acknowledgements

This paper is an open access work distributed under the terms of the Creative Commons Attribution License 3.0 (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided that the original author and source are attributed.

The paper and its publication environment form part of the work of the Ontogenesis Network, supported by EPSRC grant EP/E021352/1.