RuleML2015: Semantics of Notation3 Logic: A Solution for Implicit Quantification

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, BelgiumAgfa Healthcare - Ghent, Belgium

RuleML 2015, Berlin, August 05, 2015

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Outline

Notation3 Logic

Implicit Quantification

Formal Semantics

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Notation3 Logic

Notation3 LogicWhat is Notation3 Logic?SyntaxSemantics

Formal Semantics

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What is Notation3 Logic?

I A rule logic for Semantic Web

I Invented by Tim Berners-Lee and Dan Connolly (∼2005)

I Superset of RDF/Turtle

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Syntax

I Simple Turtle triples:

:Socrates a :Man.

I Rules:

{:Socrates a :Man.}

=> {:Socrates a :Mortal.}.

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Syntax

:Socrates a :Man.

I Rules:

{:Socrates a :Man.}

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Syntax

:Socrates a :Man.

I Rules:

{:Socrates a :Man.}

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Syntax

:Socrates a :Man.

I Rules:

{:Socrates a :Man.}

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Syntax

:Socrates a :Man.

I Rules:

{:Socrates a :Man.}

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Syntax

I Statements about formulas:

:Plato :says {:Socrates a Mortal.}.

I Use of (quantified) variables:

:Plato :knows _:x. _:x a :Man.

{?x a :Man.}=>{?x a :Mortal.}.

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Syntax

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Syntax

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Syntax

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Syntax

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What about Semantics?

I The formal semantics of N3 is not formally defined

(yet)I Some implementations even differ!I But:

I Documents like the W3C Team Submission describe the desiredsemantics

I Implementations such as reasoners can also be helpful.In particular:

I CwmI EYE

Ô We use all these as sources to define N3’s formal semantics

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I The formal semantics of N3 is not formally defined(yet)

I Some implementations even differ!I But:

I CwmI EYE

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I The formal semantics of N3 is not formally defined(yet)I Some implementations even differ!

I But:I Documents like the W3C Team Submission describe the desired

semanticsI Implementations such as reasoners can also be helpful.

In particular:I CwmI EYE

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I The formal semantics of N3 is not formally defined(yet)I Some implementations even differ!I But:

I CwmI EYE

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I CwmI EYE

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I CwmI EYE

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I CwmI EYE

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I CwmI EYE

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Notation3 Logic

Implicit QuantificationExistentialsUniversals

Formal Semantics

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What is implicit quantification?

In N3 quantified variables can be expressed without explicitly statingthe quantifier

I Blank nodes _:x are existentially quantified variablesI Variables beginning with a question mark ?x are universallyquantified

But: What is the scope?

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I Blank nodes _:x are existentially quantified variables

I Variables beginning with a question mark ?x are universallyquantified

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Simple Examples

_:x :knows :Socrates.

→ ∃x : knows(x , Socrates)?x :knows :Socrates. → ∀x : knows(x , Socrates)

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Simple Examples

_:x :knows :Socrates. → ∃x : knows(x , Socrates)

?x :knows :Socrates. → ∀x : knows(x , Socrates)

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Simple Examples

_:x :knows :Socrates. → ∃x : knows(x , Socrates)?x :knows :Socrates.

→ ∀x : knows(x , Socrates)

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Simple Examples

_:x :knows :Socrates. → ∃x : knows(x , Socrates)?x :knows :Socrates. → ∀x : knows(x , Socrates)

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Both types of variables

?x :loves _:y.

∀x∃y : loves(x , y)

"Everybody lovessomeone."

or ∃y∀x : loves(x , y)

"There is someone whois loved by everyone."3 7

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?x :loves _:y.

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?x :loves _:y.

"There is someone whois loved by everyone."

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?x :loves _:y.

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“If both universal and existential quantification are specified forthe 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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Existentials

_:x :says {_:x a :Mortal}.

∃x : says(x , Mortal(x))

"There is someone whosays about himself that he

is mortal."

or ∃x1 : says(x , (∃x2 : Mortal(x2)))

"There is someone whosays that someone is

mortal."7 3

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Existentials

is mortal."

mortal."7 3

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Existentials

is mortal."

mortal."

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Existentials

is mortal."

mortal."7 313 / 26

Existentials

“When formulae are nested, _: blank nodes syntax [is] used to onlyidentify blank node in the formula it occurs directly in. It isan arbitrary temporary name for a symbol which is existentially

quantified within the current formula (not the whole file). They canonly 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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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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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))

∀x : ((p(x , a)→ q(x , b))→(r(x , c)→ s(x , d)))

EYECwm

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Universals

{{?x :p :a.} => {?x :q :b.}.}

{{?x :r :c.} => {?x :s :d.}.}.

∀x : ((p(x , a)→ q(x , b))→(r(x , c)→ s(x , d)))

EYECwm

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Universals

{{?x :p :a.} => {?x :q :b.}.}

{{?x :r :c.} => {?x :s :d.}.}.

∀x : ((p(x , a)→ q(x , b))→(r(x , c)→ s(x , d)))

EYECwm

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Who is right?

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UniversalsThe team submission states:

“Apart from the set of statements, a formula also has a set of URIsof symbols which are universally quantified, and a set of URIs ofsymbols which are existentially quantified. Variables are then ingeneral symbols which have been quantified. There is a also a

shorthand syntax ?x which is the same as :x except that it impliesthat x is universally quantified not in the formula but in its

parent formula.”

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Universals

Which is the parent?

:Plato :says { :Socrates a Mortal.︸︷︷︸Formula

︸︷︷︸Parent formula

ÔThe parent formula p of a formula f is the formula containing {f }as a component.

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Universals

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Universals

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UniversalsBut: Universal quantification also counts for descendants

{?x :p :a.}=>{ :s :q { ?x :r :b.︸︷︷︸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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Is interpreted as:

∀x : p(x, a)→ q(s, r(x, b))

And not as

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Is interpreted as:

∀x : p(x, a)→ q(s, r(x, b))

And not as

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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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Is interpreted as:

∀x : p(x, a)→ q(s, r(x, b))

And not as

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Formal Semantics

Notation3 Logic

Formal SemanticsHandling variablesN3 context

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Handling variables

I The scope of an existential variable is always only theformula it occurs in, not its descendant

I The scope of a universal variable depends on its context,scoping is also valid on the descendants of a quantifiedformula

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Handling variables

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Handling variables

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Handling variables

We define two ways to apply a substitution σ:

1. Component wise application σc : replace only directcomponents of a formula

2. Total application σt : replace all direct components andnested components

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Handling variables

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Handling variables

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Handling variablesTo cope with the behavior of universal quantification we define thenesting level nf(?x) of a variable ?x in a formula f as the lowestlevel, counted from above, where ?x can be found:

f = ?x :p :o. → nf (?x) = 1f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2f = ?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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f = ?x :p :o.

→ nf (?x) = 1f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3

0 < nf(?x) ≤ 2

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f = ?x :p :o. → nf (?x) = 1

f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3

0 < nf(?x) ≤ 2

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f = ?x :p :o. → nf (?x) = 1f = ?x :p1 {?x :p2 :o2}.

→ nf (?x) = 1f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3

0 < nf(?x) ≤ 2

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f = ?x :p :o. → nf (?x) = 1f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1

f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3

0 < nf(?x) ≤ 2

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f = ?x :p :o. → nf (?x) = 1f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1f = ?y :p1 {?x :p2 :o2}.

→ nf (?x) = 2f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3

0 < nf(?x) ≤ 2

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f = ?x :p :o. → nf (?x) = 1f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2

f = ?y :p1 {?y :p2 {?x :p3 :o3}}. → nf (?x) = 3

0 < nf(?x) ≤ 2

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f = ?x :p :o. → nf (?x) = 1f = ?x :p1 {?x :p2 :o2}. → nf (?x) = 1f = ?y :p1 {?x :p2 :o2}. → nf (?x) = 2f = ?y :p1 {?y :p2 {?x :p3 :o3}}.

→ nf (?x) = 3

0 < nf(?x) ≤ 2

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0 < nf(?x) ≤ 2

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0 < nf(?x) ≤ 2

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N3 context

Our model definition is oriented on the RDF semantics with thefollowing additions:

I Ground implications have the usual first order meaningI Except if they occur in implications, graphs are handled as

simple URIsI 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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N3 context

I Ground implications have the usual first order meaning

I Except if they occur in implications, graphs are handled assimple URIs

I We add grounding steps for any formula f in the followingorder:1. If f contains accessible universal variables, it is valid iff f σt is

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N3 context

simple URIs

I We add grounding steps for any formula f in the followingorder:1. If f contains accessible universal variables, it is valid iff f σt is

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N3 context

order:

1. If f contains accessible universal variables, it is valid iff f σt isvalid for every ground substitution σ for those variables.

2. If f contains existentials on top level it is valid iff there exists aground substitution σ for those variables such that f σc is valid.

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N3 context

valid for every ground substitution σ for those variables.

2. If f contains existentials on top level it is valid iff there exists aground substitution σ for those variables such that f σc is valid.

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N3 context

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Notation3 Logic

Formal Semantics

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Conclusion

I It is possible to define the semantics as in the team submission

I It is difficult and rather unusual to define the scope ofuniversals as it is proposed in the team submission

I Implicit universal quantification could be handled easier, forexample on formula level

Let’s simplify this!

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Conclusion

I It is possible to define the semantics as in the team submissionI It is difficult and rather unusual to define the scope of

universals as it is proposed in the team submission

I Implicit universal quantification could be handled easier, forexample on formula level

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Conclusion

universals as it is proposed in the team submissionI Implicit universal quantification could be handled easier, for

example on formula level

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Conclusion

universals as it is proposed in the team submissionI Implicit universal quantification could be handled easier, for

example on formula level

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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).

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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).

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Example: Variable of not?

@forSome :y. :x :p :y. @forAll :x. :x:q :o.

Is the first :x quantified?

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Example: Variable of not?

@forSome :y. :x :p :y. @forAll :x. :x:q :o.

Is the first :x quantified?

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