Defining Definitions

Ontology can be thought of as the definition of entities in a domain of interest. It is useful, however, to talk about the word definition in a little more depth as it is a word with a range of usages within the discipline.

Definitions are applied to both classes and relationships. Open Biomedical Ontologies (OBO) has a set of definitions for standard relationships. For example:

[Typedef]
id: OBO_REL:part_of
alt_id: OBO_REL:0000002
name: part_of
def: "For continuants: C part_of C' if and only if: given any c that
instantiates C at a time t, there is some c' such that c' instantiates C' at
time t, and c *part_of* c' at t. For processes: P part_of P' if and only if:
given any p that instantiates P at a time t, there is some p' such that p'
instantiates P' at time t, and p *part_of* p' at t. (Here *part_of* is the
instance-level part-relation.)" [PMID:15892874]
comment: Parthood as a relation between instances: The primitive
instance-level relation p part_of p1 is illustrated in assertions such as:
this instance of rhodopsin mediated phototransduction part_of this instance of
visual perception. This relation satisfies at least the following standard
axioms of mereology: reflexivity (for all p, p part_of p); anti-symmetry (for
all p, p1, if p part_of p1 and p1 part_of p then p and p1 are identical); and
transitivity (for all p, p1, p2, if p part_of p1 and p1 part_of p2, then p
part_of p2). Analogous axioms hold also for parthood as a relation between
spatial regions. For parthood as a relation between continuants, these axioms
need to be modified to take account of the incorporation of a temporal
argument. Thus for example the axiom of transitivity for continuants will
assert that if c part_of c1 at t and c1 part_of c2 at t, then also c part_of
c2 at t. Parthood as a relation between classes: To define part_of as a
relation between classes we again need to distinguish the two cases of
continuants and processes, even though the explicit reference to instants of
time now falls away. For continuants, we have C part_of C1 if and only if any
instance of C at any time is an instance-level part of some instance of C1 at
that time, as for example in: cell nucleus part_ of cell.
inverse_of_on_instance_level: OBO_REL:has_part
is_reflexive: true
is_anti_symmetric: true
is_transitive: true

Is the definition from the Relations Ontology for the part_of relationship. Simply put, this definition is what this particular relationship means. Note that there is a logical definition (written as plain text, not a computational form) as well as a natural language comment on the definition.

The same goes for class definitions; it states what the class means. Such definitions are usually in termrs of characteristics of the entities being described.

Within the OBO community, a definition means a natural language or textual description of that class. The OBO Foundry principles advocate a particular style for these textual definitions of genus and differentia along Aristotelian lines. The genus can be thought of as stating what characteristics are inherited from the more general class. The differentae are those characteristics that differentiate members of that class from other members of the superclass. A very general form of such a definition would be “X is a y that z …”.

Taking as an example the class “assay” (OBI_0000070) from the ontology of biomedical investigations (OBI) (version 1.0) we have the following text definition:

"A planned process with the objective to produce information about some evaluant."

this states that an assay is a kind of planned process and that it is differentiated from other planned processes by having the objective of producing information about something being evaluated.

As a contrast we can look at a sibling of "assay": data transformation (OBI_0200000). It has the textual definition "A data transformation is a process which produces output data from input data". this states that a data transformation is a kind of planned process and is differentiated by having a data input and output.

These definitions tell a human reader what it means to be a member or instance of the assay or data transformation class.

One can also have a logical definition of a concept. Here, at least in OWL speak, definition has a narrower meaning; that of necessity and sufficiency. A necessary condition is a condition that an instance must fulfill to be a member of that class. Meeting that condition is, however, not enough to recognise that an instance is a member of a class. For example, having an x-chromosome is a necessary condition for being a member of the (cannonical) class of human male – it is not however sufficient because members of the class human female also have an x-chromosome. Having an x-chromosome is not enough to recognise a particular human as being a Man, even though a man must necessarily have an x-chromosome.

A necessary and sufficient condition is one, that when fulfilled (perhaps along with others), is enough to recognise that an instance is a member of a class. For example, being Person and having a y-chromosome is a necessary and sufficient condition for being a member of the class of (cannonical) human male. A person with a y-chromosome is a Man or male human. Having two x-chromosomes is enough to recognise a person as being a (cannonical) woman or female human.

we can take the same OBI class of assay for which we saw a text definition; its logical description is:

Class: 'assay'
EquivalentTo: 'planned process',
achieves_planned_objective some 'assay objective'

SubClassOf: 'planned process',
and has_specified_input some (material_entity and 
                             (has_role some 'evaluant role'))
and has_specified_output some 
             ('information content entity' and 
                    ('is about' some (continuant and 
                        (has_role some 'evaluant role'))))
and realizes some 'evaluant role'
and has_specified_output only 'information content entity'

This logical description in OWL has both necessity and sufficiency conditions, together with those that are just necessary. The text definition corresponds here to the "equivalence" axiom; that is, the logical definition. The conditions that are only necessary do not come in to the definition.

The textual definition in this case does correspond to the logical definition (the equivalence axiom). However, many textual definitions do seem to contain both necessary and sufficient conditions and those that are just necessary.

In the OWL world, Necessary and sufficient conditions are also called complete. Classes with only necessary conditions are called partial or primitive descriptions. Defined classes in OWL are a way of exploiting the reasoners that are available for OWL.

A definition is the criteria or conditions by which members of a class may be recognised. Labels for classes are not enough; it is possible to be misled by simple words—after all, that is one of the problems that ontologies attempt to overcome. A text definition is necessary for humans to understand what it means to be a member of that class (or other entity). A logical definition does the same, but for computers (and humans, but in a less accessible manner).

Authors

Robert Stevens
School of Computer Science,
The University of Manchester,
Oxford Road,
Manchester,
UK

Duncan Hull
EMBL Outstation – Hinxton,
European Bioinformatics Institute,
Wellcome Trust Genome Campus,
Hinxton,
Cambridge,
CB10 1SD,
UK