Part I: Formal Meaning

Formal meaning

Meaning is formalized with 2 functions:

Interpretation function ($I$)

From terms (syntax) to resources and properties (semantics).

Extension function ($Ext$)

From properties (semantics) to pairs of resources (semantics).

The combination of an specific interpretation function and a specific extension function is a model ($M$).

Meaning of terms

Meaning of terms: example

Meaning of sentences: Truth-conditional Semantics

Truth of a triple

$$\langle s,p,o \rangle \text{ is true} \iff \langle I(s),I(o) \rangle \in Ext(I(p))$$

$$\langle \text{the Louvre}, \text{owns}, \text{the Mona Lisa} \rangle \text{ is true} \iff$$ $$\langle I(\text{the Louvre}), I(\text{the Mona Lisa}) \rangle \in Ext(I(\text{owns})) \iff$$ $$\langle 🏰, 🖼 \rangle \in Ext(\text{concept-of-ownership}) \iff$$ $$\langle 🏰, 🖼 \rangle \in \{ \langle 🏰, 🖼 \rangle \}$$ ✓

$$\langle \text{the Mona Lisa}, \text{owns}, \text{the Louvre} \rangle \text{ is true} \iff$$ $$\langle I(\text{the Mona Lisa}), I(\text{the Louvre}) \rangle \in Ext(I(\text{owns})) \iff$$ $$\langle 🖼, 🏰 \rangle \in Ext(\text{concept-of-ownership}) \iff$$ $$\langle 🖼, 🏰 \rangle \in \{ \langle 🏰, 🖼 \rangle \}$$ ❌

Meaning of a triple

The meaning of a triple is the set of models in which the triple is true.

Meaning of a graph

The meaning of a graph is the set of models in which the triples in the graph are true.

Open World Assumption (OWA) & Closed World Assumption (CWA)

Difference between OWA and CWA

Database

“Apples are green.”

Question

“Are apples delicious?”

CWA answer

“No.”

OWA answer

“I don't known.”

Why does the SW need OWA?

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.

Example of OWA

• I(the Louvre) = 🏰
• I(the Mona Lisa) = 🖼
• Ext(I(owns)) = {〈 🏰,🖼〉}
• Ext(I(exhibits)) = {〈 🏰,🖼〉}

With CWA, owning and exhibiting would have the same meaning. With OWA, they may still be different.

• I(owns) = concept-of-ownership
• I(exhibits) = concept-of-exhibiting

Corner case example (1/2)

rdfs:Class a rdfs:Class.

• I(rdfs:Class) = 🐕
• I(a) = 🐘
• Ext(🐘) = {〈🐕,🐕〉,…}

This statement cannot be expressed with CWA, due to the axiom of regularity: ∀x (x≠∅ → ∃y∈x(x∩y=∅))

Corner case example (2/2)

owl:sameAs owl:sameAs owl:sameAs.

• I(owl:sameAs) = ~
• Ext(~) = {〈~,~〉,…}

〈owl:sameAs,owl:sameAs,owl:sameAs〉is true iff〈I(owl:sameAs),I(owl:sameAs)〉∈ Ext(I(owl:sameAs))

Notice that an RDF graph is a special kind of graph, i.e., vertices and edges may overlap.

Instances belong to classes

Domain & range

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.

SPARQL Algebra

SPARQL algebra

• Variables V
• Solution mapping μ:V → T
• Pattern instance mapping P := μ ∘ σ
• Basic Graph Pattern (BGP) x
• μ is a solution for x from G, if (∃P)(P(x) ⊆ G and μ is the restriction of P to V(x)

Example 1

Dataset

prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>
rdfs:Class a rdfs:Class.

BGP

?x a ?y.

Solution μ = {(?x,rdfs:Class),(?y,rdfs:Class)}

Example 2

Dataset

prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>
rdfs:Class a rdfs:Class.

BGP

_:1 a _:2.

• Solution μ = ∅
• σ = {(_:1,rdfs:Class), (_:2,rdfs:Class)}

Alternative semantics for owl:sameAs

Definitions of identity

Indiscernibility of identicals (Leibniz's Law)

$$a = b \rightarrow (\forall \phi \in \Psi)(\phi(a) = \phi(b))$$

Identity of indiscernibles

$$(\forall \phi \in \Psi)(\phi(a) = \phi(b)) \rightarrow a = b$$

Equivalence relation

Identity is the smallest equivalence relation

Formal Concept Analysis (FCA)

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 〈X,Y〉: X'=Y and Y'=X