The term ‘axiomatic richness’ is used in various places to talk about a certain property of an OWL ontology, mostly meaning ‘how much do we say about a particular concept’. Axiomatically rich ontologies are in some way considered better and more interesting than axiomatically lean ones. There is, however, no clear definition of the term. A quick google search for ‘axiomatic richness’ throws up only a few distinct sources that attempt to answer the question ‘what makes an (OWL) ontology axiomatically rich?’. In what follows, I discuss some of the main points of the papers and blog posts I have found.
‘Possibility of Deriving Inferences’
Robert Stevens and Sean Bechhofer discuss the term in their post on the OntoGenesis blog:
The axiomatic richness of an [ontology] refers to the level of axiomatisation that is present. […] A lack of axiomatic richness limits the possibility of deriving inferences from an [ontology]. […] Axiomatic richness could be measured in a number of ways. Hayes for example, in the Naive Physics Manifesto, discusses density. […]
(from http://ontogenesis.knowledgeblog.org/257, 2010)
It also states that in order to be axiomatically rich, the information in the ontology has to be “in a form amenable to machine processing”; plain text descriptions, such as in a SKOS vocabulary, are not sufficient.
This states that axiomatic richness is somehow related to the inferential potential in the ontology, but doesn’t give any further hints as to how we can measure axiomatic richness, or how we can tell whether ontology A is ‘richer’ than ontology B.
‘Large Number of Justifications’
Further down the list of search results, I happened to stumble across my own paper about the Justificatory Structure of OWL ontologies (OWLED 2010), where I state that
[…] taxonomic ontologies containing only trivial axioms of the form (A SubClassOf: B) are commonly regarded as axiomatically weak. A simple indicator for axiomatic richness could be a large average number of justications for entailments.
(from http://owl.cs.manchester.ac.uk/explanation/owled2010/JustStructure_OWLED2010.pdf, 2010)
“Could be” – nothing definitive here either. Many justifications (on average) for the entailments in the ontology simply means that there are many reasons why a certain entailment holds (entailment in the sense of asserted and inferred axioms that satisfy the entailment relationship with the ontology – blog post on this issue to follow soon, potentially including and discussing reviews from my DL workshop paper). While this might be an indicator of redundancy in the ontology (for which we haven’t got a definition either), the number of justifications alone doesn’t tell us much about how much we say about a particular concept, which is usually the focus when talking about axiomatic richness.
We could probably extend this guess to say “a concept A is axiomatically rich if there are 1) many justifications for 2) entailments of the form A SubClassOf B or EquivalentClasses(A,B)”, i.e. entailments that somehow define the concept. (Counter) examples might follow.
Using ‘Expressive’ Constructors
Mikel Egana Aranguren‘s thesis is a rich (haha) source of information about axiomatic richness. I found this quote quite interesting:
The OWL version of the Gene Ontology […] is implemented exploiting a rigorous formalism (OWL), but a limited fragment of the expressivity of OWL is used in axioms. On the other hand, the OBO version of the Sequence Ontology […] is axiomatically rich (e.g. symmetric properties and intersections of classes can be found in the ontology).
(from http://www.sindominio.net/~pik/thesis.pdf, 2009)
He also claims that “bio-ontologies represent biological knowledge in a limited, lean and not rigorous manner”.
A similar assumption is made in Martin Hepp’s description of “A Methodology for Deriving OWL Ontologies from Products and Services Categorization Standards”
[…] the semantic richness needed for most business scenarios will come from the usage of the huge collection of properties.
(from http://is2.lse.ac.uk/asp/aspecis/20050152.pdf, 2005)
Well. I see the point in this argument (similar to the one I made above, i.e. we can’t really say much if we only use atomic subsumptions in our ontology), but I disagree with the statement that expressivity=axiomatic richness. In many of our experiments, we have found that expressivity doesn’t really tell us much about how ‘complex’ the ontology is – reasoner performance, number and size of justifications, etc., do not correlate with the types of constructors found in the ontology (to a certain extent, obviously). Just using the constructors in some way to define a concept doesn’t necessarily make the ontology ‘richer’. Trust me, Son, I’ve seen some of those allegedly weak EL++ ontologies that could have made “the strongest man on earth whimper like a frightened kitten”.
Ontology Design Patterns
Robert Stevens and Mikel Egana Aranguren mention the term again in their paper “Applying Ontology Design Patterns in Bio-Ontologies”. They claim that Ontology Design Patterns (ODP)
[…] have already brought benefits in terms of axiomatic richness and maintainability […]
(from http://www.springerlink.com/content/d2lp476v0p281q73, 2008)
They refer to two more papers dealing with ODP in bio-ontologies, which I won’t cover here.
Locality Based Modules (LBM)
My (current and past) office neighbour Chiara Del Vescovo and Thomas Schneider drop a hint at defining axiomatic richness in a WoMo workshop talk:
[…] extract all (relevant) LBMs in order to […] draw conclusions on characteristics of an ontology:[…] What is the axiomatic richness of O?
(from http://www.informatik.uni-bremen.de/~ts/talks/1005_dl+womo.pdf, 2010)
Unfortunately, the slides don’t go into detail, and I don’t remember any discussions from the talk, so I can’t say much about this.
Non-Trivial Entailments
Yet another explanation from Manchester can be found in Pavel Klinov’s and Bijan Parsia’s paper on “Implementing an Efficient SAT Solver for Probabilistic DL“:
For axiomatically weak TBoxes, where almost all subsumptions can be discovered by traversing the concept hierarchy […]. More complex TBoxes may have non-trivial entailments involving concept expressions on both left-hand and right-hand sides […]
(from http://www4.in.tum.de/~schulz/PAPERS/STS-IWIL-2010.pdf, 2010)
To clarify, I assume the ‘non-trivial entailments’ means subsumptions that are inferred, not asserted, whose justifications involve GCIs. This sounds similar to my statement above about ‘many complex’ justifications for entailments.
Conclusion
Scio me nihil scire. I do however quite like the idea of relating axiomatic richness to the number and type of reasons (i.e. justifications) I have for an (some, all?) entailment of the ontology. We could certainly use some formal definition (or multiple, depending on which aspect is most relevant to the developer, the domain, the application…) which allows us to think of the same things when talking about ‘axiomatic richness’ and comparing ontologies. To be continued…
Sam Bail 
Just to add another key criterion, there’s the notion of being a “verified” ontology, i.e., that the only models of your ontology are the *intended* models. For some theories (e.g., mathematical ones), there’s the idea of having a *categorical* theory, i.e., one with exactly *one* model (well, up to isomorphism). See: http://philsci-archive.pitt.edu/3520/1/diarodin.pdf
Thus, one can characterise being “sufficiently axiomitized” as being axiomitized up to being verified or categorical, with the *degree* of richness reasonably possible being characterized by some measure of the *domain* complexity.
Measures of inconsistency seem relevant as well (as are distance to inconsistency measures).
Then there’s being a total theory,i.e., for every proposition P, the theory entails P or ~P. (Maximal consistent sets…)