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

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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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 Cwm

I 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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I Blank nodes _:x are existentially quantified variablesI 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.




"There is someone whois loved by everyone."

3 7

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?x :loves _:y.




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

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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."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))

or

∀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.}.}.

(∀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)))


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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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

∀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

∀x1 : p(x1, a)→ (∀x2 : q(s, r(x2, b)))((((((((((((((((((hhhhhhhhhhhhhhhhhh

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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

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 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

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 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: Variable of not?

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

Is the first :x quantified?

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ruleml2015: semantics of notation3 logic: a solution for implicit quantification

Science

walle multimedia lab

formal semantics of

rule logic

w3c team submission

semantic web

quantied variables

tim bernerslee

dan connolly