Wouter Beek (wouter@triply.cc)
Meaning is formalized with 2 functions:
The combination of an specific interpretation function and a specific extension function is a model ($M$).
The interpretation of a node is the resource it refers to.
The interpretation of a predicate term is the property it denotes.
The extension of a property is the collection of pairs of resources that are connected by that property.
$$\langle s,p,o \rangle \text{ is true} \iff \langle I(s),I(o) \rangle \in Ext(I(p))$$
$$\langle \text{the Louvre}, \text{owns}, \text{the Mona Lisa} \rangle \text{ is true} \iff$$ $$\langle I(\text{the Louvre}), I(\text{the Mona Lisa}) \rangle \in Ext(I(\text{owns})) \iff$$ $$\langle π°, πΌ \rangle \in Ext(\text{concept-of-ownership}) \iff$$ $$\langle π°, πΌ \rangle \in \{ \langle π°, πΌ \rangle \}$$ β
$$\langle \text{the Mona Lisa}, \text{owns}, \text{the Louvre} \rangle \text{ is true} \iff$$ $$\langle I(\text{the Mona Lisa}), I(\text{the Louvre}) \rangle \in Ext(I(\text{owns})) \iff$$ $$\langle πΌ, π° \rangle \in Ext(\text{concept-of-ownership}) \iff$$ $$\langle πΌ, π° \rangle \in \{ \langle π°, πΌ \rangle \}$$ β
The meaning of a triple is the set of models in which the triple is true.
The meaning of a graph is the set of models in which the triples in the graph are true.
In a local database, we can assume that we always have full access to all the relevant data.
On the web, we may not always have access to all the relevant data all the time.
With CWA, owning and exhibiting would have the same meaning. With OWA, they may still be different.
rdfs:Class a rdfs:Class.
rdfs:Class
) = πa
) = π
This statement cannot be expressed with CWA, due to the
axiom of regularity: βx (xβ β
β βyβx(xβ©y=β
))
owl:sameAs owl:sameAs owl:sameAs.
owl:sameAs
) = ~γowl:sameAs,owl:sameAs,owl:sameAsγis true iffγI(owl:sameAs),I(owl:sameAs)γβ Ext(I(owl:sameAs))
Notice that an RDF graph is a special kind of graph, i.e., vertices and edges may overlap.
foaf:knows rdfs:domain foaf:Person.
ex:john foaf:knows ex:mary.
=> ex:john a foaf:Person.
foaf:knows rdfs:range foaf:Person.
ex:john foaf:knows ex:mary.
=> ex:mary a foaf:Person.
prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>
rdfs:Class a rdfs:Class.
?x a ?y.
Solution ΞΌ =
{(?x
,rdfs:Class
),(?y
,rdfs:Class
)}
prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>
rdfs:Class a rdfs:Class.
_:1 a _:2.
_:1
,rdfs:Class
),
(_:2
,rdfs:Class
)}
owl:sameAs
$$a = b \rightarrow (\forall \phi \in \Psi)(\phi(a) = \phi(b))$$
$$(\forall \phi \in \Psi)(\phi(a) = \phi(b)) \rightarrow a = b$$
Identity is the smallest equivalence relation
Context: instances (X), properties (Y), mapping (m)
$$X' \,:=\, \{ y \in Y \,\vert\, (\forall x \in X)(m(x,y)) \}$$
$$Y' \,:=\, \{ x \in X \,\vert\, (\forall y \in Y)(m(x,y)) \}$$
Formal concept γX,Yγ: X'=Y and Y'=X
@WGJBeek