OWLED 2014 - All But Not Nothing: Left Hand Side Universals for OWL EL

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.


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.


Motivation (1)





Motivation (1)





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.


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.


The Proposal


Proposal





The Proposal


Proposal





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?





Better:







Better:







Better:






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.


(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.


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.


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.


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.


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 := ...


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) .


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.


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)


Thanks!


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





By ∃R.A2 v Xα

x

A1, ∃R.A2,∀R.A3, Xα

.

A3

R

.

A3

A2

R

.

A3

.

A3

.

A3

R





By A1 u Xα v Yα

x

A1, ∃R.A2,∀R.A3, Xα, Yα

.

A3

R

.

A3

A2

R

.

A3

.

A3

.

A3

R





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





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



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.


Technology

OWLED 2014 - All But Not Nothing: Left Hand Side Universals for OWL EL