Why & How I Found This
While working through the Introduction to Knowledge Graphs, I encountered description logics and semantics for the first time. Since these concepts were entirely new to me, I sought supplementary resources to build a foundational understanding. As I explored various materials, I became particularly intrigued by the use of formal mathematical languages for proving—grammars were my favourite part of mathematics. I found the formal languages used in description logics fascinating.
Introduction
Basic Building Blocks of DL Ontologies
DL ontologies are based on three building blocks:
- concepts: set of individuals $\rightarrow$ Mother: set of all mothers
- roles: binary relationship between individuals $\rightarrow$ parentOf: relationship between parent and their children
- individual names: represent single individuals in the domain $\rightarrow$ julia: an individual Julia
Each of these terms is similar to uniary predicates, binary predicates, and constants of FOL, respectively.
DL ontology
→ fully describes situation (x)
→ consists of set of statements (o) → axioms
axiom is a set of statement such that it captures partial knowledge about the described situation. It is generally separated into three types.
Asserting Facts with ABox Axioms
ABox axiom: capture knowledge about named individual.
- where ind. belong to: concept assertion $\rightarrow$ Mother(julia)
- ind.’s mutual rel: role assertion $\rightarrow$ parentOf(julia, john)
| ABox Axiom | Natural Language Interpretation | Precise Interpretation |
|---|---|---|
| Mother(julia) | Julia is a mother | individual named julia is an instance of the concept Mother |
| parentOf(julia, john) | Julia is a parent of John | individual named julia is related by parentOf to the individual named john |
Here, we should also consider that Julia and John are not different individuals. Intuitively, they are different individuals. However, we cannot simply assume that they are different individuals based on the axiom parentOf(julia, john). This is because most DLs does not make unique name assumption (different name $\Longrightarrow$ different individual).
What we need to guarantee that unique name assumption is individual inequality assertion ($\not\approx$). This is functionally opposite of individual equality assertion ($\approx$).
For example,
- julia $\not\approx$ john: Julia and John are different individuals.
- julia $\approx$ john: Julia and John refer to the same individual.
Expressing Terminological Knowledge with TBox Axioms
TBox axiom: describe relationships between concepts (set of individuals). The two simple TBox axioms are concept inclusion and concept equivalence.
Concept inclusion axioms asserts one concept is a subconcept of the other.
Consider Mother ⊆ Parent:
- natural language: all mothers are parents.
- precise description: a concept Mother is subsumed by the concept Parent.
So, if we know Mother(julia) and Mother ⊆ Parent we can infer that Parent(julia) in other words, Julia is a parent. Notice that this ability to infer from the existing information is the characteristics of DL.
Concept equivalence axioms asserts two concepts have same instances.
Consider Person ≡ Human:
- natural language: all persons are human and vice versa.
- precise description: a concept Person is equivalent to the concept Human.
While these TBox axioms are useful, there are many complex relationships. So, we employ other complex concept expressions.