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All But Not NothingLeft-Hand Side Universals for OWL EL
David Carral1 Adila A. Krisnadhi1,2
Sebastian Rudolph3 Pascal Hitzler1
1Wright State University, Dayton, OH, USA
2Universitas Indonesia, Depok, Indonesia
3Technical Universitat Dresden, Germany
OWLED 2014
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 1 / 18
Acknowledgements
The “La Caixa” Foundation
The National Science Foundation (NSF):
Award 1017225 “III: Small: TROn V Tractable Reasoning withOntologies” for the support of this work.ISWC 2014 student travel grant for presentation of this work.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 2 / 18
Motivation (1)
“All OWLED 2004 papers were written by George Lucas.”
Is the above sentence true?
Yes. Because there was no OWLED conference in 2004 – the seriesonly started in 2005, hence all of them, namely none, were written byGeorge Lucas.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 3 / 18
Motivation (1)
“All OWLED 2004 papers were written by George Lucas.”
Is the above sentence true?
Yes. Because there was no OWLED conference in 2004 – the seriesonly started in 2005, hence all of them, namely none, were written byGeorge Lucas.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 3 / 18
Motivation (1)
“All OWLED 2004 papers were written by George Lucas.”
Is the above sentence true?
Yes. Because there was no OWLED conference in 2004 – the seriesonly started in 2005, hence all of them, namely none, were written byGeorge Lucas.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 3 / 18
Motivation (2)
Semantics of the universal quantifier in first-order logic leads to aneat mathematical theory.
But, the reading is rather unintuitive and easily misunderstood bypeople not familiar with logic.
In natural language, if Alice says that “all my children are female”,then we assume that she does in fact have children, all of whom arefemale.
If we found out later that she never had any children, we may concludethat she was lying, instead of understanding her statement as avacuous truth.
Intuitively, Alice belongs to the class
∀hasChild.Female u ∃hasChild.Female
and not just ∀hasChild.Female.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 4 / 18
The Proposal
Witnessed universal/closure restriction: a syntactic sugar for the classexpression of the form ∀R.C u ∃R.D — originally disallowed in OWLEL.
Proposal
Allow occurrences of witnessed universal on the subclass part (i.e.,left-hand side) of a subclass axiom for OWL EL.
Reason 1 (Nguyen et al, 2013): adding LHS witnessed universal toHorn-SROIQ does not increase data complexity — via amodification of the SROIQ tableau algorithm.
Reason 2 (this work): there is a linear KB transformation to reducesatisfiability with LHS witnessed universal to the case without it.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 5 / 18
The Proposal
Witnessed universal/closure restriction: a syntactic sugar for the classexpression of the form ∀R.C u ∃R.D — originally disallowed in OWLEL.
Proposal
Allow occurrences of witnessed universal on the subclass part (i.e.,left-hand side) of a subclass axiom for OWL EL.
Reason 1 (Nguyen et al, 2013): adding LHS witnessed universal toHorn-SROIQ does not increase data complexity — via amodification of the SROIQ tableau algorithm.
Reason 2 (this work): there is a linear KB transformation to reducesatisfiability with LHS witnessed universal to the case without it.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 5 / 18
The Proposal
Witnessed universal/closure restriction: a syntactic sugar for the classexpression of the form ∀R.C u ∃R.D — originally disallowed in OWLEL.
Proposal
Allow occurrences of witnessed universal on the subclass part (i.e.,left-hand side) of a subclass axiom for OWL EL.
Reason 1 (Nguyen et al, 2013): adding LHS witnessed universal toHorn-SROIQ does not increase data complexity — via amodification of the SROIQ tableau algorithm.
Reason 2 (this work): there is a linear KB transformation to reducesatisfiability with LHS witnessed universal to the case without it.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 5 / 18
The Rewriting – by Example (1)
To model: “a Pizza, all of whose toppings are vegetarian, is in fact avegetarian Pizza.”
Not sufficient: Pizza u ∀hasTopping.VegTopping v VegPizza.(What’s a Pizza without a topping?)
Better:
Pizza u ∃hasTopping.VegTopping u ∀hasTopping.VegTopping v VegPizza
Can we get rid of the universal quantifier?
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 6 / 18
The Rewriting – by Example (1)
To model: “a Pizza, all of whose toppings are vegetarian, is in fact avegetarian Pizza.”
Not sufficient: Pizza u ∀hasTopping.VegTopping v VegPizza.(What’s a Pizza without a topping?)
Better:
Pizza u ∃hasTopping.VegTopping u ∀hasTopping.VegTopping v VegPizza
Can we get rid of the universal quantifier?
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 6 / 18
The Rewriting – by Example (1)
To model: “a Pizza, all of whose toppings are vegetarian, is in fact avegetarian Pizza.”
Not sufficient: Pizza u ∀hasTopping.VegTopping v VegPizza.(What’s a Pizza without a topping?)
Better:
Pizza u ∃hasTopping.VegTopping u ∀hasTopping.VegTopping v VegPizza
Can we get rid of the universal quantifier?
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 6 / 18
The Rewriting – by Example (1)
To model: “a Pizza, all of whose toppings are vegetarian, is in fact avegetarian Pizza.”
Not sufficient: Pizza u ∀hasTopping.VegTopping v VegPizza.(What’s a Pizza without a topping?)
Better:
Pizza u ∃hasTopping.VegTopping u ∀hasTopping.VegTopping v VegPizza
Can we get rid of the universal quantifier?
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 6 / 18
The Rewriting – by Example (2)
Pizza u ∃hasTopping.VegTopping u ∀hasTopping.VegTopping v VegPizza
can be rewritten into:
Pizza u ∃hasTopping.VegTopping v ∃RPizza∃hasTopping.> (1)
RPizza∃hasTopping v hasTopping (2)
∃RPizza∃hasTopping.VegTopping v VegPizza (3)
Rewriting is expressible in ELH.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 7 / 18
(Extended) EL++ and Horn-SROIQ
Horn-SROIQ syntax covers the following types of axiom where Ai ’s andB are class names, R, S ,V are property names, and a, b are individualnames:
A1 u · · · u An v B ∃R.A v B A v ∃S .B
A1 u ∃R.A2 u ∀R.A3 v B ∃R.Self v B A v ∃S .Self
ran(R) v B {a} v B A v {b}R v S R ◦ V v S
R v S− A v ≤1S .B
EL++ allows only the first four rows from the above list of types of axiom.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 8 / 18
Main Theorem
Let O be a EL++ or Horn-SROIQ ontology. Then, let O∀ be obtainedfrom O by replacing every axiom α of the form:
α = A1 u ∃R.A2 u ∀R.A3 v B
with 5 fresh axioms:
∃R.A2 v Xα, A1 u Xα v Yα, Yα v ∃Zα.>,Zα v R, ∃Zα.A3 v B
where A1,A2,A3,B,R are atomic, Xα,Yα are fresh class names, and Zα isa fresh property name.
Theorem
O∀ contains no universal quantifier, and O and O∀ are equisatisfiable.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 9 / 18
OWL Syntax Modification: Overview
We add a witnessed universal construct as a class expression to OWL2.
For OWL 2 DL, witnessed universal is only a syntactic sugar.
For OWL 2 EL, this allows us to express some universal restrictions.
Rewriting generates axioms not expressible in OWL 2 RL and OWL 2QL.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 10 / 18
OWL 2 DL Functional Syntax Modification
Add a production rule:
ObjectSomeAllValuesFrom := ’ObjectSomeAllValuesFrom’ ’(’
ObjectPropertyExpression ClassExpression ClassExpression’)’
Then, add ObjectSomeAllValuesFrom as one branch to the ClassExpression
production rule.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 11 / 18
OWL 2 EL Functional Syntax Modification
Need to distinguish subClassExpression fromsuperClassExpression.
subClassExpression := Class | subObjectIntersectionOf | ObjectOneOf |subObjectSomeValuesFrom | ... | ObjectSomeAllValuesFrom
subObjectIntersectionOf := ’ObjectIntersectionOf’ ’(’
subClassExpression subClassExpression { subClassExpression } ’)’
...
ObjectSomeAllValuesFrom := ’ObjectSomeAllValuesFrom’ ’(’
ObjectPropertyExpression subClassExpression subClassExpression ’)’
superClassExpression := Class | superObjectIntersectionOf | ObjectOneOf |superObjectSomeValuesFrom | ...
superObjectIntersectionOf := ...
superObjectSomeValuesFrom := ...
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 12 / 18
Manchester syntax & mapping to Turtle
Manchester syntax:
Use the keyword ’someall’ (or ’onlysome’?)Add as a branch to the restriction production rule:
objectPropertyExpression ’someall’ primary primary
For Turtle, mapObjectSomeAllValuesFrom( OPE CE1 CE2 )
to
_:x rdf:type owl:Restriction .
_:x owl:onProperty T(OPE) .
_:x owl:someAllValuesFrom T(SEQ CE1 CE2) .
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 13 / 18
Summary
We can add a restricted form of universal quantifier to EL++ andHorn-SROIQ without jeopardizing the nice computationalproperties.
We provided a suggestion for an OWL syntax extension.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 14 / 18
References
Nguyen, L.A., Nguyen, T.B.L., Szalas, A.: HornDL: An expressive Horn descriptionlogic with PTime data complexity. In: Faber, W., Lembo, D. (eds.) WebReasoning and Rule Systems – 7th International Conference, RR 2013, Mannheim,Germany, July 27-29, 2013. Proceedings. Lecture Notes in Computer Science, vol.7994, pp. 259V264. Springer (2013)
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 15 / 18
Correctness of Rewriting (part 1)
Every model J of O∀ is a model of O.
Only need to show that every α of the form A1 u ∃R.A2 u ∀R.A3 v Bis satisfied by J .
Suppose x ∈ (A1 u ∃R.A2 u ∀R.A3)I
x
A1, ∃R.A2,∀R.A3
.
A3
R
.
A3
A2
R
.
A3
.
A3
.
A3
R
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 17 / 18
Correctness of Rewriting (part 1)
Every model J of O∀ is a model of O.
Only need to show that every α of the form A1 u ∃R.A2 u ∀R.A3 v Bis satisfied by J .
By ∃R.A2 v Xα
x
A1, ∃R.A2,∀R.A3, Xα
.
A3
R
.
A3
A2
R
.
A3
.
A3
.
A3
R
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 17 / 18
Correctness of Rewriting (part 1)
Every model J of O∀ is a model of O.
Only need to show that every α of the form A1 u ∃R.A2 u ∀R.A3 v Bis satisfied by J .
By A1 u Xα v Yα
x
A1, ∃R.A2,∀R.A3, Xα, Yα
.
A3
R
.
A3
A2
R
.
A3
.
A3
.
A3
R
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 17 / 18
Correctness of Rewriting (part 1)
Every model J of O∀ is a model of O.
Only need to show that every α of the form A1 u ∃R.A2 u ∀R.A3 v Bis satisfied by J .
Due to Yα v ∃Zα.>, Zα v R, and by assumption, x ∈ (∀R.A3)I ,there is a z s.t. (x , z) ∈ RI ∩ ZIα and z ∈ AI3
x
A1, ∃R.A2,∀R.A3, Xα,Yα
.
A3
R
.
A3
A2
R
.
A3
.
A3
.
A3
R
z
A3
Zα,R
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 17 / 18
Correctness of Rewriting (part 1)
Every model J of O∀ is a model of O.
Only need to show that every α of the form A1 u ∃R.A2 u ∀R.A3 v Bis satisfied by J .
So, x ∈ (∃Zα.A3)I , hence x ∈ BI
x
A1, ∃R.A2,∀R.A3, Xα,Yα, B
.
A3
R
.
A3
A2
R
.
A3
.
A3
.
A3
R
z
A3
Zα,R
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 17 / 18
Correctness of Rewriting (part 2)
From every model I of O, a model J of O∀ can be constructed.
1 Set ∆J = ∆I , aJ = aI , AJ = AI , and RJ = RI , for everyindividual name a, class name A, and property name R.
2 For every α = A1 u ∃R.A2 u ∀R.A3 v B:
(a) if x ∈ (∃R.A2)I , then x ∈ XJα
(b) if x ∈ AI1 ∩ (∃R.A2)I , then x ∈ Y J
α
(c) if x ∈ AI1 ∩ (∃R.A2)I ∩ (∀R.A3)I , then also choose some y for which
(x , y) ∈ RI , and set (x , y) ∈ ZJα
(d) if x ∈ AI1 ∩ (∃R.A2)I but x /∈ (∀R.A3)I , then pick any y for which
(x , y) ∈ RI and y /∈ AI3 , and set (x , y) ∈ ZJ
α .
3 Check that every axiom in Oα is satisfied by J .
(2a) and (2b) make ∃R.A2 v Xα and A1 u Xα v Yα satisfied.(2c) and (2d) make Yα v ∃Zα.> and Zα v R satisfied(2c) and (1) make ∃Zα v B satisfied.
Carral, Krisnadhi, Rudolph, Hitzler All But Not Nothing OWLED 2014 18 / 18