Introduction to knowledge graphs (section 4.2): Deductive knowledge – Semantics and entailment
This article is section 4.2 of part 4 of the Introduction to knowledge graphs series of articles. Recent research has identified the development of knowledge graphs as an important aspect of artificial intelligence (AI) in knowledge management (KM).
The conditions listed in the previous tables give rise to entailments; for example, the definition 
and edge 
entail 
per the SYMMETRIC condition of Table 2. Hogan and colleagues’ comprehensive tutorial article1 describes how these conditions lead to entailments.
Model-theoretic Semantics
Each axiom described by the previous tables, when added to a graph, enforces some condition(s) on the interpretations that satisfy the graph. The interpretations that satisfy a graph are called models of the graph. If we considered only the base condition of the ASSERTION feature in Table 1, for example, then the models of a graph would be any interpretation such that for every edge 
in the graph, there exists a relation 
in the model. Given that there may be other relations in the model (under the OWA), the number of models of any such graph is infinite. Furthermore, given that we can map multiple nodes in the graph to one entity in the model (under the NUNA), any interpretation with (for example) the relation 
is a model of any graph so long as for every edge 
in the graph, it holds that 
in the interpretation (in other words, the interpretation maps everything to 
). As we add axioms with their associated conditions to the graph, we restrict models for the graph; for example, considering a graph with two edges – 
and 
– the interpretation with 
is no longer a model as it breaks the condition for the irreflexive axiom.
Entailment
We say that one graph entails another if and only if any model of the former graph is also a model of the latter graph. Intuitively, this means that the latter graph says nothing new over the former graph and thus holds as a logical consequence of the former graph. For example, consider the graph 
and the graph 
. All models of the latter must have that 
, but so must all models of the former, which must have 
and further must satisfy the condition for SUBCLASS, which requires that 
also hold. Hence, we conclude that any model of the former graph must be a model of the latter graph, and thus the former graph entails the latter graph.
Next part (section 4.3): Deductive knowledge – Reasoning.
Header image source: Crow Intelligence, CC BY-NC-SA 4.0.
References:
- Hogan, A., Blomqvist, E., Cochez, M., d’Amato, C., Melo, G. D., Gutierrez, C., … & Zimmermann, A. (2021). Knowledge graphs. ACM Computing Surveys (CSUR), 54(4), 1-37. ↩
