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ELIS – Multimedia Lab
Semantics of Notation3 Logic:
A solution for implicit quantification
Dörthe Arndt, Ruben Verborgh, Jos De Roo, Hong Sun,
Erik Mannens, and Rik Van De Walle
Multimedia Lab, Ghent University - iMinds, Belgium
Agfa Healthcare - Ghent, Belgium
RuleML 2015, Berlin, August 04, 2015
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ELIS – Multimedia Lab
Outline
Notation3 Logic
Implicit Quantification
Formal Semantics
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ELIS – Multimedia Lab
Notation3 Logic
Notation3 Logic
What is Notation3 Logic?
Syntax
Semantics
Implicit Quantification
Formal Semantics
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ELIS – Multimedia Lab
What is Notation3 Logic?
A rule logic for the Semantic Web
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ELIS – Multimedia Lab
What is Notation3 Logic?
A rule logic for the Semantic Web
Invented by Tim Berners-Lee and Dan Connolly (∼2005)
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ELIS – Multimedia Lab
What is Notation3 Logic?
A rule logic for the Semantic Web
Invented by Tim Berners-Lee and Dan Connolly (∼2005)
Superset of RDF/Turtle
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ELIS – Multimedia Lab
Syntax
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ELIS – Multimedia Lab
Syntax
Simple Turtle triples:
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ELIS – Multimedia Lab
Syntax
Simple Turtle triples:
:Socrates a :Man.
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ELIS – Multimedia Lab
Syntax
Simple Turtle triples:
:Socrates a :Man.
Rules:
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ELIS – Multimedia Lab
Syntax
Simple Turtle triples:
:Socrates a :Man.
Rules:
{:Socrates a :Man.}
=> {:Socrates a :Mortal.}.
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ELIS – Multimedia Lab
Syntax
Statements about formulas:
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ELIS – Multimedia Lab
Syntax
Statements about formulas:
:Plato :says {:Socrates a Mortal.}.
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ELIS – Multimedia Lab
Syntax
Statements about formulas:
:Plato :says {:Socrates a Mortal.}.
Use of (quantified) variables:
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ELIS – Multimedia Lab
Syntax
Statements about formulas:
:Plato :says {:Socrates a Mortal.}.
Use of (quantified) variables:
:Plato :knows _:x. _:x a :Man.
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ELIS – Multimedia Lab
Syntax
Statements about formulas:
:Plato :says {:Socrates a Mortal.}.
Use of (quantified) variables:
:Plato :knows _:x. _:x a :Man.
{?x a :Man.}=>{?x a :Mortal.}.
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ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined
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ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
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ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
Some implementations even differ!
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ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
Some implementations even differ!
But:
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ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
Some implementations even differ!
But:
Documents like the W3C Team Submission describe the desired
semantics
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ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
Some implementations even differ!
But:
Documents like the W3C Team Submission describe the desired
semantics
Implementations such as reasoners can also be helpful.
In particular:
7 / 28
ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
Some implementations even differ!
But:
Documents like the W3C Team Submission describe the desired
semantics
Implementations such as reasoners can also be helpful.
In particular:
Cwm
7 / 28
ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
Some implementations even differ!
But:
Documents like the W3C Team Submission describe the desired
semantics
Implementations such as reasoners can also be helpful.
In particular:
Cwm
EYE
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ELIS – Multimedia Lab
What about Semantics?
The formal semantics of N3 is not defined(yet)
Some implementations even differ!
But:
Documents like the W3C Team Submission describe the desired
semantics
Implementations such as reasoners can also be helpful.
In particular:
Cwm
EYE
  We use all these as sources to define N3’s formal semantics
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ELIS – Multimedia Lab
Implicit Quantification
Notation3 Logic
Implicit Quantification
Existentials
Universals
Formal Semantics
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ELIS – Multimedia Lab
What is implicit quantification?
In N3 quantified variables can be expressed without explicitly stating
the quantifier
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ELIS – Multimedia Lab
What is implicit quantification?
In N3 quantified variables can be expressed without explicitly stating
the quantifier
Blank nodes _:x are existentially quantified variables
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ELIS – Multimedia Lab
What is implicit quantification?
In N3 quantified variables can be expressed without explicitly stating
the quantifier
Blank nodes _:x are existentially quantified variables
Variables beginning with a question mark ?x are universally
quantified
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ELIS – Multimedia Lab
What is implicit quantification?
In N3 quantified variables can be expressed without explicitly stating
the quantifier
Blank nodes _:x are existentially quantified variables
Variables beginning with a question mark ?x are universally
quantified
But: What is the scope?
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ELIS – Multimedia Lab
Simple Examples
_:x :knows :Socrates.
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ELIS – Multimedia Lab
Simple Examples
_:x :knows :Socrates. → ∃x : knows(x, Socrates)
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ELIS – Multimedia Lab
Simple Examples
_:x :knows :Socrates. → ∃x : knows(x, Socrates)
?x :knows :Socrates.
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ELIS – Multimedia Lab
Simple Examples
_:x :knows :Socrates. → ∃x : knows(x, Socrates)
?x :knows :Socrates. → ∀x : knows(x, Socrates)
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ELIS – Multimedia Lab
Both types of variables
?x :loves _:y.
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ELIS – Multimedia Lab
Both types of variables
?x :loves _:y.
∀x∃y : loves(x, y)
"Everybody loves
someone."
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ELIS – Multimedia Lab
Both types of variables
?x :loves _:y.
∀x∃y : loves(x, y)
"Everybody loves
someone."
or
∃y∀x : loves(x, y)
"There is someone who
is loved by everyone."
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ELIS – Multimedia Lab
Both types of variables
?x :loves _:y.
∀x∃y : loves(x, y)
"Everybody loves
someone."
or
∃y∀x : loves(x, y)
"There is someone who
is loved by everyone."
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ELIS – Multimedia Lab
Both types of variables
“If both universal and existential quantification are specified for
the same formula, then the scope of the universal quantification
is outside the scope of the existentials”.
Source: W3C Team submission; cwm and EYE give the same result.
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ELIS – Multimedia Lab
Existentials
_:x :says {_:x a :Mortal}.
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ELIS – Multimedia Lab
Existentials
_:x :says {_:x a :Mortal}.
∃x : says(x, Mortal(x))
There is someone who
says about himself that he
is mortal.
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ELIS – Multimedia Lab
Existentials
_:x :says {_:x a :Mortal}.
∃x : says(x, Mortal(x))
There is someone who
says about himself that he
is mortal.
or ∃x1 : says(x, (∃x2 : Mortal(x2)))
There is someone who
says that someone is
mortal.
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ELIS – Multimedia Lab
Existentials
_:x :says {_:x a :Mortal}.
∃x : says(x, Mortal(x))
There is someone who
says about himself that he
is mortal.
or ∃x1 : says(x, (∃x2 : Mortal(x2)))
There is someone who
says that someone is
mortal.
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ELIS – Multimedia Lab
Existentials
“When formulae are nested, _: blank nodes syntax [is] used to only
identify blank node in the formula it occurs directly in. It is
an arbitrary temporary name for a symbol which is existentially
quantified within the current formula (not the whole file). They can
only be used within a single formula, and not within nested
formulae.”
Source: W3C Team submission; cwm and EYE give the same result.
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ELIS – Multimedia Lab
Universals
{{?x :p :a.} = {?x :q :b.}.}
=
{{?x :r :c.} = {?x :s :d.}.}.
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ELIS – Multimedia Lab
Universals
{{?x :p :a.} = {?x :q :b.}.}
=
{{?x :r :c.} = {?x :s :d.}.}.
(∀x1 : p(x1, a) → q(x1, b))
→
(∀x2 : r(x2, c) → s(x2, d))
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ELIS – Multimedia Lab
Universals
{{?x :p :a.} = {?x :q :b.}.}
=
{{?x :r :c.} = {?x :s :d.}.}.
(∀x1 : p(x1, a) → q(x1, b))
→
(∀x2 : r(x2, c) → s(x2, d))
or
∀x : ((p(x, a) → q(x, b))
→
(r(x, c) → s(x, d)))
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ELIS – Multimedia Lab
Universals
{{?x :p :a.} = {?x :q :b.}.}
=
{{?x :r :c.} = {?x :s :d.}.}.
(∀x1 : p(x1, a) → q(x1, b))
→
(∀x2 : r(x2, c) → s(x2, d))
or
∀x : ((p(x, a) → q(x, b))
→
(r(x, c) → s(x, d)))
Here the reasoning results differ!
EYECwm
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ELIS – Multimedia Lab
Who is right?
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ELIS – Multimedia Lab
Universals
The team submission states:
“Apart from the set of statements, a formula also has a set of URIs
of symbols which are universally quantified, and a set of URIs of
symbols which are existentially quantified. Variables are then in
general symbols which have been quantified. There is a also a
shorthand syntax ?x which is the same as :x except that it implies
that x is universally quantified not in the formula but in its
parent formula.”
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ELIS – Multimedia Lab
Universals
Which is the parent?
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ELIS – Multimedia Lab
Universals
Which is the parent?
:Plato :says { :Socrates a Mortal.
Formula
}.
Parent formula
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ELIS – Multimedia Lab
Universals
Which is the parent?
:Plato :says { :Socrates a Mortal.
Formula
}.
Parent formula
 The parent formula p of a formula f is the formula containing {f }
as a component.
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ELIS – Multimedia Lab
Universals
But: Universal quantification also counts for descendants
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ELIS – Multimedia Lab
Universals
But: Universal quantification also counts for descendants
{?x :p :a.}={ :s :q { ?x :r :b.
Formula
}.
Parent formula
}.
Grandparent formula
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ELIS – Multimedia Lab
Universals
But: Universal quantification also counts for descendants
{?x :p :a.}={ :s :q { ?x :r :b.
Formula
}.
Parent formula
}.
Grandparent formula
Is interpreted as:
∀x : p(x, a) → q(s, r(x, b))
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ELIS – Multimedia Lab
Universals
But: Universal quantification also counts for descendants
{?x :p :a.}={ :s :q { ?x :r :b.
Formula
}.
Parent formula
}.
Grandparent formula
Is interpreted as:
∀x : p(x, a) → q(s, r(x, b))
And not as
∀x1 : p(x1, a) → (∀x2 : q(s, r(x2, b)))
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ELIS – Multimedia Lab
Universals
But: Universal quantification also counts for descendants
{?x :p :a.}={ :s :q { ?x :r :b.
Formula
}.
Parent formula
}.
Grandparent formula
Is interpreted as:
∀x : p(x, a) → q(s, r(x, b))
And not as
∀x1 : p(x1, a) → (∀x2 : q(s, r(x2, b)))
((((((((((((((((((hhhhhhhhhhhhhhhhhh
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ELIS – Multimedia Lab
Formal Semantics
Notation3 Logic
Implicit Quantification
Formal Semantics
Handling variables
N3 context
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ELIS – Multimedia Lab
Handling variables
The scope of an existential variable is always only the
formula it occurs in, not its descendant
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ELIS – Multimedia Lab
Handling variables
The scope of an existential variable is always only the
formula it occurs in, not its descendant
The scope of a universal variable depends on its context,
scoping is also valid on the descendants of a quantified
formula
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ELIS – Multimedia Lab
Handling variables
We define two ways to apply a substitution σ:
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ELIS – Multimedia Lab
Handling variables
We define two ways to apply a substitution σ:
1. Component wise application σc: replace only direct
components of a formula
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ELIS – Multimedia Lab
Handling variables
We define two ways to apply a substitution σ:
1. Component wise application σc: replace only direct
components of a formula
2. Total application σt: replace all direct components and
nested components
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ELIS – Multimedia Lab
Handling variables
For
f = ?x :says {?x a :Mortal.}. and σ = {?x/:Socrates}
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ELIS – Multimedia Lab
Handling variables
For
f = ?x :says {?x a :Mortal.}. and σ = {?x/:Socrates}
we obtain:
f σc
= :Socrates :says {?x a :Mortal.}.
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ELIS – Multimedia Lab
Handling variables
For
f = ?x :says {?x a :Mortal.}. and σ = {?x/:Socrates}
we obtain:
f σc
= :Socrates :says {?x a :Mortal.}.
and
f σt
= :Socrates :says {:Socrates a :Mortal.}.
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o.
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
f = ?x :p1 {?x :p2 :o2}.
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1
f = ?y :p1 {?x :p2 :o2}.
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1
f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1
f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2
f = ?y :p1 {?y :p2 {?x :p3 :o3}}.
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1
f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2
f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3
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ELIS – Multimedia Lab
Handling variables
To cope with the behavior of universal quantification we define the
nesting level nf(?x) of a variable ?x in a formula f as the lowest
level, counted from above, where ?x can be found:
f = ?x :p :o. → nf (?x) = 1
f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1
f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2
f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3
We call a universal variable accessible in a formula f iff
0  nf(?x) ≤ 2
.
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ELIS – Multimedia Lab
Handling variables
Consider
{ {?x :p :a.} = {?x :q :b.}.
f11
} = {{?x :r :c.} = {?x :s :d.}.
f12
}.
f0
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ELIS – Multimedia Lab
Handling variables
Consider
{ {?x :p :a.} = {?x :q :b.}.
f11
} = {{?x :r :c.} = {?x :s :d.}.
f12
}.
f0
nf0 (?x) = 3
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ELIS – Multimedia Lab
Handling variables
Consider
{ {?x :p :a.} = {?x :q :b.}.
f11
} = {{?x :r :c.} = {?x :s :d.}.
f12
}.
f0
nf0 (?x) = 3 → ?x is not accessible in f0
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ELIS – Multimedia Lab
Handling variables
Consider
{ {?x :p :a.} = {?x :q :b.}.
f11
} = {{?x :r :c.} = {?x :s :d.}.
f12
}.
f0
nf0 (?x) = 3 → ?x is not accessible in f0
nf11 (?x) = 2 and nf12 (?x) = 2
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ELIS – Multimedia Lab
Handling variables
Consider
{ {?x :p :a.} = {?x :q :b.}.
f11
} = {{?x :r :c.} = {?x :s :d.}.
f12
}.
f0
nf0 (?x) = 3 → ?x is not accessible in f0
nf11 (?x) = 2 and nf12 (?x) = 2 → ?x is accessible in f11 and f12
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ELIS – Multimedia Lab
Handling variables
Consider
{ {?x :p :a.} = {?x :q :b.}.
f11
} = {{?x :r :c.} = {?x :s :d.}.
f12
}.
f0
nf0 (?x) = 3 → ?x is not accessible in f0
nf11 (?x) = 2 and nf12 (?x) = 2 → ?x is accessible in f11 and f12
Those are exactly the formulas where the two ?x are quantified:
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ELIS – Multimedia Lab
Handling variables
Consider
{ {?x :p :a.} = {?x :q :b.}.
f11
} = {{?x :r :c.} = {?x :s :d.}.
f12
}.
f0
nf0 (?x) = 3 → ?x is not accessible in f0
nf11 (?x) = 2 and nf12 (?x) = 2 → ?x is accessible in f11 and f12
Those are exactly the formulas where the two ?x are quantified:
{ {?x :p :a.} = {?x :q :b.}. }= { {?x :q :c.} = {?x :r :d.}. }
(∀x1 : (p(x1, a) → q(x1, b))) → (∀x2 : (r(x2, c) → s(x2, d)))
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ELIS – Multimedia Lab
N3 context
Our model definition is oriented on the RDF semantics with the
following additions:
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ELIS – Multimedia Lab
N3 context
Our model definition is oriented on the RDF semantics with the
following additions:
Ground implications have the usual first order meaning
26 / 28
ELIS – Multimedia Lab
N3 context
Our model definition is oriented on the RDF semantics with the
following additions:
Ground implications have the usual first order meaning
Except if they occur in implications, graphs are handled as
simple URIs
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ELIS – Multimedia Lab
N3 context
Our model definition is oriented on the RDF semantics with the
following additions:
Ground implications have the usual first order meaning
Except if they occur in implications, graphs are handled as
simple URIs
We add grounding steps for any formula f in the following
order:
26 / 28
ELIS – Multimedia Lab
N3 context
Our model definition is oriented on the RDF semantics with the
following additions:
Ground implications have the usual first order meaning
Except if they occur in implications, graphs are handled as
simple URIs
We add grounding steps for any formula f in the following
order:
1. If f contains accessible universal variables, it is valid iff f σt
is
valid for every ground substitution σ for those variables.
26 / 28
ELIS – Multimedia Lab
N3 context
Our model definition is oriented on the RDF semantics with the
following additions:
Ground implications have the usual first order meaning
Except if they occur in implications, graphs are handled as
simple URIs
We add grounding steps for any formula f in the following
order:
1. If f contains accessible universal variables, it is valid iff f σt
is
valid for every ground substitution σ for those variables.
2. If f contains existentials on top level it is valid iff there exists a
ground substitution σ for those variables such that f σc
is valid.
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ELIS – Multimedia Lab
Notation3 Logic
Implicit Quantification
Formal Semantics
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ELIS – Multimedia Lab
Conclusion
It is possible to define the semantics as in the team submission
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ELIS – Multimedia Lab
Conclusion
It is possible to define the semantics as in the team submission
It is difficult and rather unusual to define the scope of
universals as it is proposed in the team submission
28 / 28
ELIS – Multimedia Lab
Conclusion
It is possible to define the semantics as in the team submission
It is difficult and rather unusual to define the scope of
universals as it is proposed in the team submission
Implicit universal quantification could be handled easier, for
example on formula level
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ELIS – Multimedia Lab
Conclusion
It is possible to define the semantics as in the team submission
It is difficult and rather unusual to define the scope of
universals as it is proposed in the team submission
Implicit universal quantification could be handled easier, for
example on formula level
Let’s simplify this!
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ELIS – Multimedia Lab
Example: Explicit quantification
@forAll #x. @forSome #y. #x #loves #y .
@forSome #y. @forAll #x. #x #loves #y .
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ELIS – Multimedia Lab
Example: Explicit quantification
@forAll #x. @forSome #y. #x #loves #y .
@forSome #y. @forAll #x. #x #loves #y .
Have the same meaning:
∀x∃y : loves(x, y).
29 / 28
ELIS – Multimedia Lab
Example: Variable of not?
@forSome :y. :x :p :y. @forAll :x. :x :q :o.
30 / 28
ELIS – Multimedia Lab
Example: Variable of not?
@forSome :y. :x :p :y. @forAll :x. :x :q :o.
Is the first :x quantified?
30 / 28

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RuleML 2015: Semantics of Notation3 Logic: A Solution for Implicit Quantification

  • 1. ELIS – Multimedia Lab Semantics of Notation3 Logic: A solution for implicit quantification Dörthe Arndt, Ruben Verborgh, Jos De Roo, Hong Sun, Erik Mannens, and Rik Van De Walle Multimedia Lab, Ghent University - iMinds, Belgium Agfa Healthcare - Ghent, Belgium RuleML 2015, Berlin, August 04, 2015 1 / 28
  • 2. ELIS – Multimedia Lab Outline Notation3 Logic Implicit Quantification Formal Semantics 2 / 28
  • 3. ELIS – Multimedia Lab Notation3 Logic Notation3 Logic What is Notation3 Logic? Syntax Semantics Implicit Quantification Formal Semantics 3 / 28
  • 4. ELIS – Multimedia Lab What is Notation3 Logic? A rule logic for the Semantic Web 4 / 28
  • 5. ELIS – Multimedia Lab What is Notation3 Logic? A rule logic for the Semantic Web Invented by Tim Berners-Lee and Dan Connolly (∼2005) 4 / 28
  • 6. ELIS – Multimedia Lab What is Notation3 Logic? A rule logic for the Semantic Web Invented by Tim Berners-Lee and Dan Connolly (∼2005) Superset of RDF/Turtle 4 / 28
  • 7. ELIS – Multimedia Lab Syntax 5 / 28
  • 8. ELIS – Multimedia Lab Syntax Simple Turtle triples: 5 / 28
  • 9. ELIS – Multimedia Lab Syntax Simple Turtle triples: :Socrates a :Man. 5 / 28
  • 10. ELIS – Multimedia Lab Syntax Simple Turtle triples: :Socrates a :Man. Rules: 5 / 28
  • 11. ELIS – Multimedia Lab Syntax Simple Turtle triples: :Socrates a :Man. Rules: {:Socrates a :Man.} => {:Socrates a :Mortal.}. 5 / 28
  • 12. ELIS – Multimedia Lab Syntax Statements about formulas: 6 / 28
  • 13. ELIS – Multimedia Lab Syntax Statements about formulas: :Plato :says {:Socrates a Mortal.}. 6 / 28
  • 14. ELIS – Multimedia Lab Syntax Statements about formulas: :Plato :says {:Socrates a Mortal.}. Use of (quantified) variables: 6 / 28
  • 15. ELIS – Multimedia Lab Syntax Statements about formulas: :Plato :says {:Socrates a Mortal.}. Use of (quantified) variables: :Plato :knows _:x. _:x a :Man. 6 / 28
  • 16. ELIS – Multimedia Lab Syntax Statements about formulas: :Plato :says {:Socrates a Mortal.}. Use of (quantified) variables: :Plato :knows _:x. _:x a :Man. {?x a :Man.}=>{?x a :Mortal.}. 6 / 28
  • 17. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined 7 / 28
  • 18. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) 7 / 28
  • 19. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) Some implementations even differ! 7 / 28
  • 20. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) Some implementations even differ! But: 7 / 28
  • 21. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) Some implementations even differ! But: Documents like the W3C Team Submission describe the desired semantics 7 / 28
  • 22. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) Some implementations even differ! But: Documents like the W3C Team Submission describe the desired semantics Implementations such as reasoners can also be helpful. In particular: 7 / 28
  • 23. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) Some implementations even differ! But: Documents like the W3C Team Submission describe the desired semantics Implementations such as reasoners can also be helpful. In particular: Cwm 7 / 28
  • 24. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) Some implementations even differ! But: Documents like the W3C Team Submission describe the desired semantics Implementations such as reasoners can also be helpful. In particular: Cwm EYE 7 / 28
  • 25. ELIS – Multimedia Lab What about Semantics? The formal semantics of N3 is not defined(yet) Some implementations even differ! But: Documents like the W3C Team Submission describe the desired semantics Implementations such as reasoners can also be helpful. In particular: Cwm EYE   We use all these as sources to define N3’s formal semantics 7 / 28
  • 26. ELIS – Multimedia Lab Implicit Quantification Notation3 Logic Implicit Quantification Existentials Universals Formal Semantics 8 / 28
  • 27. ELIS – Multimedia Lab What is implicit quantification? In N3 quantified variables can be expressed without explicitly stating the quantifier 9 / 28
  • 28. ELIS – Multimedia Lab What is implicit quantification? In N3 quantified variables can be expressed without explicitly stating the quantifier Blank nodes _:x are existentially quantified variables 9 / 28
  • 29. ELIS – Multimedia Lab What is implicit quantification? In N3 quantified variables can be expressed without explicitly stating the quantifier Blank nodes _:x are existentially quantified variables Variables beginning with a question mark ?x are universally quantified 9 / 28
  • 30. ELIS – Multimedia Lab What is implicit quantification? In N3 quantified variables can be expressed without explicitly stating the quantifier Blank nodes _:x are existentially quantified variables Variables beginning with a question mark ?x are universally quantified But: What is the scope? 9 / 28
  • 31. ELIS – Multimedia Lab Simple Examples _:x :knows :Socrates. 10 / 28
  • 32. ELIS – Multimedia Lab Simple Examples _:x :knows :Socrates. → ∃x : knows(x, Socrates) 10 / 28
  • 33. ELIS – Multimedia Lab Simple Examples _:x :knows :Socrates. → ∃x : knows(x, Socrates) ?x :knows :Socrates. 10 / 28
  • 34. ELIS – Multimedia Lab Simple Examples _:x :knows :Socrates. → ∃x : knows(x, Socrates) ?x :knows :Socrates. → ∀x : knows(x, Socrates) 10 / 28
  • 35. ELIS – Multimedia Lab Both types of variables ?x :loves _:y. 11 / 28
  • 36. ELIS – Multimedia Lab Both types of variables ?x :loves _:y. ∀x∃y : loves(x, y) "Everybody loves someone." 11 / 28
  • 37. ELIS – Multimedia Lab Both types of variables ?x :loves _:y. ∀x∃y : loves(x, y) "Everybody loves someone." or ∃y∀x : loves(x, y) "There is someone who is loved by everyone." 11 / 28
  • 38. ELIS – Multimedia Lab Both types of variables ?x :loves _:y. ∀x∃y : loves(x, y) "Everybody loves someone." or ∃y∀x : loves(x, y) "There is someone who is loved by everyone." 11 / 28
  • 39. ELIS – Multimedia Lab Both types of variables “If both universal and existential quantification are specified for the same formula, then the scope of the universal quantification is outside the scope of the existentials”. Source: W3C Team submission; cwm and EYE give the same result. 12 / 28
  • 40. ELIS – Multimedia Lab Existentials _:x :says {_:x a :Mortal}. 13 / 28
  • 41. ELIS – Multimedia Lab Existentials _:x :says {_:x a :Mortal}. ∃x : says(x, Mortal(x)) There is someone who says about himself that he is mortal. 13 / 28
  • 42. ELIS – Multimedia Lab Existentials _:x :says {_:x a :Mortal}. ∃x : says(x, Mortal(x)) There is someone who says about himself that he is mortal. or ∃x1 : says(x, (∃x2 : Mortal(x2))) There is someone who says that someone is mortal. 13 / 28
  • 43. ELIS – Multimedia Lab Existentials _:x :says {_:x a :Mortal}. ∃x : says(x, Mortal(x)) There is someone who says about himself that he is mortal. or ∃x1 : says(x, (∃x2 : Mortal(x2))) There is someone who says that someone is mortal. 13 / 28
  • 44. ELIS – Multimedia Lab Existentials “When formulae are nested, _: blank nodes syntax [is] used to only identify blank node in the formula it occurs directly in. It is an arbitrary temporary name for a symbol which is existentially quantified within the current formula (not the whole file). They can only be used within a single formula, and not within nested formulae.” Source: W3C Team submission; cwm and EYE give the same result. 14 / 28
  • 45. ELIS – Multimedia Lab Universals {{?x :p :a.} = {?x :q :b.}.} = {{?x :r :c.} = {?x :s :d.}.}. 15 / 28
  • 46. ELIS – Multimedia Lab Universals {{?x :p :a.} = {?x :q :b.}.} = {{?x :r :c.} = {?x :s :d.}.}. (∀x1 : p(x1, a) → q(x1, b)) → (∀x2 : r(x2, c) → s(x2, d)) 15 / 28
  • 47. ELIS – Multimedia Lab Universals {{?x :p :a.} = {?x :q :b.}.} = {{?x :r :c.} = {?x :s :d.}.}. (∀x1 : p(x1, a) → q(x1, b)) → (∀x2 : r(x2, c) → s(x2, d)) or ∀x : ((p(x, a) → q(x, b)) → (r(x, c) → s(x, d))) 15 / 28
  • 48. ELIS – Multimedia Lab Universals {{?x :p :a.} = {?x :q :b.}.} = {{?x :r :c.} = {?x :s :d.}.}. (∀x1 : p(x1, a) → q(x1, b)) → (∀x2 : r(x2, c) → s(x2, d)) or ∀x : ((p(x, a) → q(x, b)) → (r(x, c) → s(x, d))) Here the reasoning results differ! EYECwm 15 / 28
  • 49. ELIS – Multimedia Lab Who is right? 16 / 28
  • 50. ELIS – Multimedia Lab Universals The team submission states: “Apart from the set of statements, a formula also has a set of URIs of symbols which are universally quantified, and a set of URIs of symbols which are existentially quantified. Variables are then in general symbols which have been quantified. There is a also a shorthand syntax ?x which is the same as :x except that it implies that x is universally quantified not in the formula but in its parent formula.” 17 / 28
  • 51. ELIS – Multimedia Lab Universals Which is the parent? 18 / 28
  • 52. ELIS – Multimedia Lab Universals Which is the parent? :Plato :says { :Socrates a Mortal. Formula }. Parent formula 18 / 28
  • 53. ELIS – Multimedia Lab Universals Which is the parent? :Plato :says { :Socrates a Mortal. Formula }. Parent formula  The parent formula p of a formula f is the formula containing {f } as a component. 18 / 28
  • 54. ELIS – Multimedia Lab Universals But: Universal quantification also counts for descendants 19 / 28
  • 55. ELIS – Multimedia Lab Universals But: Universal quantification also counts for descendants {?x :p :a.}={ :s :q { ?x :r :b. Formula }. Parent formula }. Grandparent formula 19 / 28
  • 56. ELIS – Multimedia Lab Universals But: Universal quantification also counts for descendants {?x :p :a.}={ :s :q { ?x :r :b. Formula }. Parent formula }. Grandparent formula Is interpreted as: ∀x : p(x, a) → q(s, r(x, b)) 19 / 28
  • 57. ELIS – Multimedia Lab Universals But: Universal quantification also counts for descendants {?x :p :a.}={ :s :q { ?x :r :b. Formula }. Parent formula }. Grandparent formula Is interpreted as: ∀x : p(x, a) → q(s, r(x, b)) And not as ∀x1 : p(x1, a) → (∀x2 : q(s, r(x2, b))) 19 / 28
  • 58. ELIS – Multimedia Lab Universals But: Universal quantification also counts for descendants {?x :p :a.}={ :s :q { ?x :r :b. Formula }. Parent formula }. Grandparent formula Is interpreted as: ∀x : p(x, a) → q(s, r(x, b)) And not as ∀x1 : p(x1, a) → (∀x2 : q(s, r(x2, b))) ((((((((((((((((((hhhhhhhhhhhhhhhhhh 19 / 28
  • 59. ELIS – Multimedia Lab Formal Semantics Notation3 Logic Implicit Quantification Formal Semantics Handling variables N3 context 20 / 28
  • 60. ELIS – Multimedia Lab Handling variables The scope of an existential variable is always only the formula it occurs in, not its descendant 21 / 28
  • 61. ELIS – Multimedia Lab Handling variables The scope of an existential variable is always only the formula it occurs in, not its descendant The scope of a universal variable depends on its context, scoping is also valid on the descendants of a quantified formula 21 / 28
  • 62. ELIS – Multimedia Lab Handling variables We define two ways to apply a substitution σ: 22 / 28
  • 63. ELIS – Multimedia Lab Handling variables We define two ways to apply a substitution σ: 1. Component wise application σc: replace only direct components of a formula 22 / 28
  • 64. ELIS – Multimedia Lab Handling variables We define two ways to apply a substitution σ: 1. Component wise application σc: replace only direct components of a formula 2. Total application σt: replace all direct components and nested components 22 / 28
  • 65. ELIS – Multimedia Lab Handling variables For f = ?x :says {?x a :Mortal.}. and σ = {?x/:Socrates} 23 / 28
  • 66. ELIS – Multimedia Lab Handling variables For f = ?x :says {?x a :Mortal.}. and σ = {?x/:Socrates} we obtain: f σc = :Socrates :says {?x a :Mortal.}. 23 / 28
  • 67. ELIS – Multimedia Lab Handling variables For f = ?x :says {?x a :Mortal.}. and σ = {?x/:Socrates} we obtain: f σc = :Socrates :says {?x a :Mortal.}. and f σt = :Socrates :says {:Socrates a :Mortal.}. 23 / 28
  • 68. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: 24 / 28
  • 69. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. 24 / 28
  • 70. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 24 / 28
  • 71. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 f = ?x :p1 {?x :p2 :o2}. 24 / 28
  • 72. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1 24 / 28
  • 73. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1 f = ?y :p1 {?x :p2 :o2}. 24 / 28
  • 74. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1 f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2 24 / 28
  • 75. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1 f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2 f = ?y :p1 {?y :p2 {?x :p3 :o3}}. 24 / 28
  • 76. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1 f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2 f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3 24 / 28
  • 77. ELIS – Multimedia Lab Handling variables To cope with the behavior of universal quantification we define the nesting level nf(?x) of a variable ?x in a formula f as the lowest level, counted from above, where ?x can be found: f = ?x :p :o. → nf (?x) = 1 f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1 f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2 f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3 We call a universal variable accessible in a formula f iff 0 nf(?x) ≤ 2 . 24 / 28
  • 78. ELIS – Multimedia Lab Handling variables Consider { {?x :p :a.} = {?x :q :b.}. f11 } = {{?x :r :c.} = {?x :s :d.}. f12 }. f0 25 / 28
  • 79. ELIS – Multimedia Lab Handling variables Consider { {?x :p :a.} = {?x :q :b.}. f11 } = {{?x :r :c.} = {?x :s :d.}. f12 }. f0 nf0 (?x) = 3 25 / 28
  • 80. ELIS – Multimedia Lab Handling variables Consider { {?x :p :a.} = {?x :q :b.}. f11 } = {{?x :r :c.} = {?x :s :d.}. f12 }. f0 nf0 (?x) = 3 → ?x is not accessible in f0 25 / 28
  • 81. ELIS – Multimedia Lab Handling variables Consider { {?x :p :a.} = {?x :q :b.}. f11 } = {{?x :r :c.} = {?x :s :d.}. f12 }. f0 nf0 (?x) = 3 → ?x is not accessible in f0 nf11 (?x) = 2 and nf12 (?x) = 2 25 / 28
  • 82. ELIS – Multimedia Lab Handling variables Consider { {?x :p :a.} = {?x :q :b.}. f11 } = {{?x :r :c.} = {?x :s :d.}. f12 }. f0 nf0 (?x) = 3 → ?x is not accessible in f0 nf11 (?x) = 2 and nf12 (?x) = 2 → ?x is accessible in f11 and f12 25 / 28
  • 83. ELIS – Multimedia Lab Handling variables Consider { {?x :p :a.} = {?x :q :b.}. f11 } = {{?x :r :c.} = {?x :s :d.}. f12 }. f0 nf0 (?x) = 3 → ?x is not accessible in f0 nf11 (?x) = 2 and nf12 (?x) = 2 → ?x is accessible in f11 and f12 Those are exactly the formulas where the two ?x are quantified: 25 / 28
  • 84. ELIS – Multimedia Lab Handling variables Consider { {?x :p :a.} = {?x :q :b.}. f11 } = {{?x :r :c.} = {?x :s :d.}. f12 }. f0 nf0 (?x) = 3 → ?x is not accessible in f0 nf11 (?x) = 2 and nf12 (?x) = 2 → ?x is accessible in f11 and f12 Those are exactly the formulas where the two ?x are quantified: { {?x :p :a.} = {?x :q :b.}. }= { {?x :q :c.} = {?x :r :d.}. } (∀x1 : (p(x1, a) → q(x1, b))) → (∀x2 : (r(x2, c) → s(x2, d))) 25 / 28
  • 85. ELIS – Multimedia Lab N3 context Our model definition is oriented on the RDF semantics with the following additions: 26 / 28
  • 86. ELIS – Multimedia Lab N3 context Our model definition is oriented on the RDF semantics with the following additions: Ground implications have the usual first order meaning 26 / 28
  • 87. ELIS – Multimedia Lab N3 context Our model definition is oriented on the RDF semantics with the following additions: Ground implications have the usual first order meaning Except if they occur in implications, graphs are handled as simple URIs 26 / 28
  • 88. ELIS – Multimedia Lab N3 context Our model definition is oriented on the RDF semantics with the following additions: Ground implications have the usual first order meaning Except if they occur in implications, graphs are handled as simple URIs We add grounding steps for any formula f in the following order: 26 / 28
  • 89. ELIS – Multimedia Lab N3 context Our model definition is oriented on the RDF semantics with the following additions: Ground implications have the usual first order meaning Except if they occur in implications, graphs are handled as simple URIs We add grounding steps for any formula f in the following order: 1. If f contains accessible universal variables, it is valid iff f σt is valid for every ground substitution σ for those variables. 26 / 28
  • 90. ELIS – Multimedia Lab N3 context Our model definition is oriented on the RDF semantics with the following additions: Ground implications have the usual first order meaning Except if they occur in implications, graphs are handled as simple URIs We add grounding steps for any formula f in the following order: 1. If f contains accessible universal variables, it is valid iff f σt is valid for every ground substitution σ for those variables. 2. If f contains existentials on top level it is valid iff there exists a ground substitution σ for those variables such that f σc is valid. 26 / 28
  • 91. ELIS – Multimedia Lab Notation3 Logic Implicit Quantification Formal Semantics 27 / 28
  • 92. ELIS – Multimedia Lab Conclusion It is possible to define the semantics as in the team submission 28 / 28
  • 93. ELIS – Multimedia Lab Conclusion It is possible to define the semantics as in the team submission It is difficult and rather unusual to define the scope of universals as it is proposed in the team submission 28 / 28
  • 94. ELIS – Multimedia Lab Conclusion It is possible to define the semantics as in the team submission It is difficult and rather unusual to define the scope of universals as it is proposed in the team submission Implicit universal quantification could be handled easier, for example on formula level 28 / 28
  • 95. ELIS – Multimedia Lab Conclusion It is possible to define the semantics as in the team submission It is difficult and rather unusual to define the scope of universals as it is proposed in the team submission Implicit universal quantification could be handled easier, for example on formula level Let’s simplify this! 28 / 28
  • 96. ELIS – Multimedia Lab Example: Explicit quantification @forAll #x. @forSome #y. #x #loves #y . @forSome #y. @forAll #x. #x #loves #y . 29 / 28
  • 97. ELIS – Multimedia Lab Example: Explicit quantification @forAll #x. @forSome #y. #x #loves #y . @forSome #y. @forAll #x. #x #loves #y . Have the same meaning: ∀x∃y : loves(x, y). 29 / 28
  • 98. ELIS – Multimedia Lab Example: Variable of not? @forSome :y. :x :p :y. @forAll :x. :x :q :o. 30 / 28
  • 99. ELIS – Multimedia Lab Example: Variable of not? @forSome :y. :x :p :y. @forAll :x. :x :q :o. Is the first :x quantified? 30 / 28