TRACTABLE EXTENSIONS OF THE DESCRIPTION

LOGIC EL WITH NUMERICAL DATATYPES

Despoina Magka, Yevgeny Kazakov and Ian Horrocks

Oxford University Computing Laboratory

July 16, 2010

Introduction

OUTLINE

1 INTRODUCTION

2 REASONING IN EL⊥(D)

3 CONCLUSION

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 2/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGIES

Ontologies are formal descriptions of domains such as:

Aerospace

Cellular biology

Human anatomy

Drugs

Applications:

Automated decision support

Reference databases

. . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:

Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

Paracetamol

PatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:

Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatient

A drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:

Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tablet

A patient who has age less than 6 yearsAxioms formulate properties and restrictions using terms:

Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:

Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:

Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.

Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.Panadol contains paracetamol.

Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.

[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:

500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:500 mg

more than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:500 mgmore than 250 mg

less than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

ONTOLOGY AXIOMS

Terms describe entities from specific domains, e.g. for adrug ontology:

ParacetamolPatientA drug that contains 500 mg of paracetamol per tabletA patient who has age less than 6 years

Axioms formulate properties and restrictions using terms:Panadol is a drug.Panadol contains paracetamol.Panadol contains 500 mg of paracetamol per tablet.[A patient who has age less than 6 years] should not beprescribed anything which [contains more than 250 mg ofparacetamol per tablet].

Restrictions specify ranges of data values:500 mgmore than 250 mgless than 6 years

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

Introduction

DESCRIPTION LOGICS

Formally describe axioms:

EXAMPLEPanadol v DrugPanadol v ∃contains.ParacetamolPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])Patient u ∃hasAge.[< 6] u ∃hasPrescription.

∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Atomic conceptsAtomic rolesDatatype restrictionsConcept constructorsFoundations of W3C ontology languages (OWL and OWL 2)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

Introduction

DESCRIPTION LOGICS

Formally describe axioms:

EXAMPLEPanadol v DrugPanadol v ∃contains.ParacetamolPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])Patient u ∃hasAge.[< 6] u ∃hasPrescription.

∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Atomic concepts

Atomic rolesDatatype restrictionsConcept constructorsFoundations of W3C ontology languages (OWL and OWL 2)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

Introduction

DESCRIPTION LOGICS

Formally describe axioms:

EXAMPLEPanadol v DrugPanadol v ∃contains.ParacetamolPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])Patient u ∃hasAge.[< 6] u ∃hasPrescription.

∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Atomic conceptsAtomic roles

Datatype restrictionsConcept constructorsFoundations of W3C ontology languages (OWL and OWL 2)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

Introduction

DESCRIPTION LOGICS

Formally describe axioms:

EXAMPLEPanadol v DrugPanadol v ∃contains.ParacetamolPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])Patient u ∃hasAge.[< 6] u ∃hasPrescription.

∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Atomic conceptsAtomic rolesDatatype restrictions

Concept constructorsFoundations of W3C ontology languages (OWL and OWL 2)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

Introduction

DESCRIPTION LOGICS

Formally describe axioms:

EXAMPLEPanadol v DrugPanadol v ∃contains.ParacetamolPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])Patient u ∃hasAge.[< 6] u ∃hasPrescription.

∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Atomic conceptsAtomic rolesDatatype restrictionsConcept constructors

Foundations of W3C ontology languages (OWL and OWL 2)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

Introduction

DESCRIPTION LOGICS

Formally describe axioms:

EXAMPLEPanadol v DrugPanadol v ∃contains.ParacetamolPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])Patient u ∃hasAge.[< 6] u ∃hasPrescription.

∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Atomic conceptsAtomic rolesDatatype restrictionsConcept constructorsFoundations of W3C ontology languages (OWL and OWL 2)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

Introduction

DESCRIPTION LOGICS REASONING

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Can Panadol be prescribed to a 3-year-old patient?- Is X satisfiable?DL reasoning tasks:

Check satisfiability of a concept (O |= X v ⊥)Check satisfiability of an ontology (O |= ⊥ )Check subsumption (O |= A v B)Classification (compute all A v B such that O |= A v B) more general than all tasks above

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

Introduction

DESCRIPTION LOGICS REASONING

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

Can Panadol be prescribed to a 3-year-old patient?

- Is X satisfiable?DL reasoning tasks:

Check satisfiability of a concept (O |= X v ⊥)Check satisfiability of an ontology (O |= ⊥ )Check subsumption (O |= A v B)Classification (compute all A v B such that O |= A v B) more general than all tasks above

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

Introduction

DESCRIPTION LOGICS REASONING

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X ≡ Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Can Panadol be prescribed to a 3-year-old patient?- Is X satisfiable?

DL reasoning tasks:Check satisfiability of a concept (O |= X v ⊥)Check satisfiability of an ontology (O |= ⊥ )Check subsumption (O |= A v B)Classification (compute all A v B such that O |= A v B) more general than all tasks above

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

Introduction

DESCRIPTION LOGICS REASONING

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X ≡ Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Can Panadol be prescribed to a 3-year-old patient?- Is X satisfiable?DL reasoning tasks:

Check satisfiability of a concept (O |= X v ⊥)Check satisfiability of an ontology (O |= ⊥ )Check subsumption (O |= A v B)Classification (compute all A v B such that O |= A v B) more general than all tasks above

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?

EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

<,=

Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:uses u and ∃:

Panadol ≡ Drug u ∃contains.Paracetamolsuitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

<,=

EXAMPLE

A v ∃F.[< 2]∃F.[= 1] v B∃F.[= 0] v C

A v B t C

Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

<,= >,= <,> ≤,= ≥,= ≤,≥

Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

<,= >,= <,> ≤,= ≥,= ≤,≥Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype use

EL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

<,= >,= <,> ≤,= ≥,= ≤,≥Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype use

EL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

<,= >,= <,> ≤,= ≥,= ≤,≥Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equality

Absence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

EL AND DATATYPES

EL [Baader et al., IJCAI 2005] is a simple DL:

uses u and ∃:Panadol ≡ Drug u ∃contains.Paracetamol

suitable for large-scale ontologies such as SNOMED or GOreasoning in PTIME

EL with datatypes is EXPTIME-complete. Why?EL extended with disjunction is EXPTIME-complete anddisjunction is expressible using datatypes:

<,= >,= <,> ≤,= ≥,= ≤,≥Baader et al. [IJCAI 2005] ensure polynomiality byrestrictions on datatype useEL Profile of OWL 2 admits only equalityAbsence of inequalities reduces the utility of OWL 2 EL

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

Introduction

RESULTS OVERVIEW

Relax the restrictions on datatype use by allowinginequalities and remaining polynomial

Key ideaDistinguish negative (LHS of axiom) and positive (RHS ofaxiom) occurrences of datatypes

Main results

Full classification of cases where datatypes are used andtractability is preserved for N, Z, Q and R

Polynomial, sound and complete reasoning procedure forextensions of EL⊥ with restricted numerical datatypes

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

Introduction

RESULTS OVERVIEW

Relax the restrictions on datatype use by allowinginequalities and remaining polynomial

Key ideaDistinguish negative (LHS of axiom) and positive (RHS ofaxiom) occurrences of datatypes

Main results

Full classification of cases where datatypes are used andtractability is preserved for N, Z, Q and R

Polynomial, sound and complete reasoning procedure forextensions of EL⊥ with restricted numerical datatypes

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

Introduction

RESULTS OVERVIEW

Relax the restrictions on datatype use by allowinginequalities and remaining polynomial

Key ideaDistinguish negative (LHS of axiom) and positive (RHS ofaxiom) occurrences of datatypes

Main results

Full classification of cases where datatypes are used andtractability is preserved for N, Z, Q and R

Polynomial, sound and complete reasoning procedure forextensions of EL⊥ with restricted numerical datatypes

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

Introduction

RESULTS OVERVIEW

Relax the restrictions on datatype use by allowinginequalities and remaining polynomial

Key ideaDistinguish negative (LHS of axiom) and positive (RHS ofaxiom) occurrences of datatypes

Main results

Full classification of cases where datatypes are used andtractability is preserved for N, Z, Q and R

Polynomial, sound and complete reasoning procedure forextensions of EL⊥ with restricted numerical datatypes

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

Introduction

EL FAMILY OF DESCRIPTION LOGICS

The EL language:

Syntax SemanticsAtomic concept C C(x)Top > >Conjunction C u D C(x) ∧ D(x)Existential restriction ∃R.C ∃y : R(x, y) ∧ C(y)

EL⊥ with Numerical Datatypes

Bottom ⊥ ⊥Datatype restriction ∃F.range ∃v : F(x, v) ∧ v ∈ range

range = [< n] | [≤ n] | [> n] | [≥ n] | [= n] ⊆ D = N,Z,R,QAxioms

Concept inclusion C v D ∀x : C(x)→ D(x)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

Introduction

EL FAMILY OF DESCRIPTION LOGICS

The EL language:

Syntax SemanticsAtomic concept C C(x)Top > >Conjunction C u D C(x) ∧ D(x)Existential restriction ∃R.C ∃y : R(x, y) ∧ C(y)

EL⊥ with Numerical Datatypes

Bottom ⊥ ⊥Datatype restriction ∃F.range ∃v : F(x, v) ∧ v ∈ range

range = [< n] | [≤ n] | [> n] | [≥ n] | [= n] ⊆ D = N,Z,R,Q

Axioms

Concept inclusion C v D ∀x : C(x)→ D(x)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

Introduction

EL FAMILY OF DESCRIPTION LOGICS

The EL language:

Syntax SemanticsAtomic concept C C(x)Top > >Conjunction C u D C(x) ∧ D(x)Existential restriction ∃R.C ∃y : R(x, y) ∧ C(y)

EL⊥ with Numerical Datatypes

Bottom ⊥ ⊥Datatype restriction ∃F.range ∃v : F(x, v) ∧ v ∈ range

range = [< n] | [≤ n] | [> n] | [≥ n] | [= n] ⊆ D = N,Z,R,QAxioms

Concept inclusion C v D ∀x : C(x)→ D(x)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

Introduction

EL FAMILY OF DESCRIPTION LOGICS

The EL language:

Syntax SemanticsAtomic concept C C(x)Top > >Conjunction C u D C(x) ∧ D(x)Existential restriction ∃R.C ∃y : R(x, y) ∧ C(y)

EL⊥ with Numerical Datatypes

Bottom ⊥ ⊥Datatype restriction ∃F.range ∃v : F(x, v) ∧ v ∈ range

range = [< n] | [≤ n] | [> n] | [≥ n] | [= n] ⊆ D = N,Z,R,QAxioms

Concept inclusion C v D ∀x : C(x)→ D(x)

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

Reasoning in EL⊥(D)

OUTLINE

1 INTRODUCTION

2 REASONING IN EL⊥(D)

3 CONCLUSION

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 10/20

Reasoning in EL⊥(D)

ALGORITHM STAGES

1 Standard normalization of the axioms

Normal formsNF1 A v BNF2 A1 u A2 v BNF3 A v ∃R.BNF4 ∃R.B v ANF5 A v ∃F.rangeNF6 ∃F.range v A

2 Saturation of the axioms under inference rules, such as:

A v B A v CA v D

B u C v D ∈ O

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 11/20

Reasoning in EL⊥(D)

ALGORITHM STAGES

1 Standard normalization of the axioms

Normal formsNF1 A v BNF2 A1 u A2 v BNF3 A v ∃R.BNF4 ∃R.B v ANF5 A v ∃F.rangeNF6 ∃F.range v A

2 Saturation of the axioms under inference rules, such as:

A v B A v CA v D

B u C v D ∈ O

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 11/20

Reasoning in EL⊥(D)

REASONING RULES (PART I)

Rules from EL⊥

IR1A v A

IR2A v > CR1

A v BA v C

B v C ∈ O

CR2A v B A v C

A v DB u C v D ∈ O

CR3A v B

A v ∃R.CB v ∃R.C ∈ O

CR4A v ∃R.B B v C

A v D∃R.C v D ∈ O

CR5A v ∃R.B B v ⊥

A v ⊥

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 12/20

Reasoning in EL⊥(D)

REASONING RULES (PART II)

New rules for datatypes

A v ∃F.[< m]

A v B∃F.[< n] v B ∈ O, m ≤ n

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 13/20

Reasoning in EL⊥(D)

REASONING RULES (PART II)

New rules for datatypes

ID1A v ⊥

A v ∃F.[< 0] ∈ O

CD1A v B

A v ∃F.rangeB v ∃F.range ∈ O

CD2(<,<)A v ∃F.[< m]

A v B∃F.[< n] v B ∈ O, m ≤ n

CD2(=,<)A v ∃F.[= m]

A v B∃F.[< n] v B ∈ O, m < n

CD2(=,=)A v ∃F.[= m]

A v B∃F.[= n] v B ∈ O, m = n . . .

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 13/20

Reasoning in EL⊥(D)

THE EL⊥(D)ALGORITHM

The algorithm is:

sound: all rules derive logical consequences of premises

polynomial as only polynomially many axioms are possible

not complete in general (EL⊥(D) is EXPTIME-complete)

complete under certain restrictions on datatypes

What type of restrictions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

Reasoning in EL⊥(D)

THE EL⊥(D)ALGORITHM

The algorithm is:

sound: all rules derive logical consequences of premises

polynomial as only polynomially many axioms are possible

not complete in general (EL⊥(D) is EXPTIME-complete)

complete under certain restrictions on datatypes

What type of restrictions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

Reasoning in EL⊥(D)

THE EL⊥(D)ALGORITHM

The algorithm is:

sound: all rules derive logical consequences of premises

polynomial as only polynomially many axioms are possible

not complete in general (EL⊥(D) is EXPTIME-complete)

complete under certain restrictions on datatypes

What type of restrictions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

Reasoning in EL⊥(D)

THE EL⊥(D)ALGORITHM

The algorithm is:

sound: all rules derive logical consequences of premises

polynomial as only polynomially many axioms are possible

not complete in general (EL⊥(D) is EXPTIME-complete)

complete under certain restrictions on datatypes

What type of restrictions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

Reasoning in EL⊥(D)

THE EL⊥(D)ALGORITHM

The algorithm is:

sound: all rules derive logical consequences of premises

polynomial as only polynomially many axioms are possible

not complete in general (EL⊥(D) is EXPTIME-complete)

complete under certain restrictions on datatypes

What type of restrictions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR N

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Restrict positive and negative occurrences of datatyperelations so as disjunction is not expressible

All restrictions are maximal:

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR N

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Restrict positive and negative occurrences of datatyperelations so as disjunction is not expressible

All restrictions are maximal:

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR N

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Restrict positive and negative occurrences of datatyperelations so as disjunction is not expressible

All restrictions are maximal:

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR N

Negative relations Positive relations<,≤, >,≥,= =

<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= >,≥,=

All restrictions are maximal:

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR N

Negative relations Positive relations<,≤, >,≥,= =

<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= >,≥,=

All restrictions are maximal:

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR N

Negative relations Positive relations<,≤, >,≥,= = ,<

<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= >,≥,=

All restrictions are maximal:

EXAMPLE

A v ∃F.[< 2]∃F.[= 1] v B∃F.[= 0] v C

A v B t C

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR ZNegative relations Positive relations

<,≤, >,≥,= == <,≤, >,≥,=

<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= >,≥,=>,≥,= <,≤,=

Additional datatype restrictions: integers do not have aminimal element such as 0.

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 16/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR ZNegative relations Positive relations

<,≤, >,≥,= == <,≤, >,≥,=

<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= >,≥,=>,≥,= <,≤,=

Additional datatype restrictions: integers do not have aminimal element such as 0.

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 16/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR ZNegative relations Positive relations

<,≤, >,≥,= == <,≤, >,≥,=

<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= >,≥,=>,≥,= <,≤,=

Additional datatype restrictions: integers do not have aminimal element such as 0.

EXAMPLE

A v ∃F.[< 2]∃F.[= 1] v B∃F.[= 0] v C

A 6v B t C

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 16/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR Q AND RNegative relations Positive relations

<,≤, >,≥,= =≤,= <,≤, >,≥,=≥,= <,≤, >,≥,=<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= <,>,≥,=>,≥,= <,≤, >,=

Between every two different numbers there exists a thirdone:

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 17/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR Q AND RNegative relations Positive relations

<,≤, >,≥,= =≤,= <,≤, >,≥,=≥,= <,≤, >,≥,=<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= <,>,≥,=>,≥,= <,≤, >,=

Between every two different numbers there exists a thirdone:

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 17/20

Reasoning in EL⊥(D)

RESTRICTIONS FOR Q AND RNegative relations Positive relations

<,≤, >,≥,= =≤,= <,≤, >,≥,=≥,= <,≤, >,≥,=<,≤ <,≤, >,≥,=>,≥ <,≤, >,≥,=

<,≤,= <,>,≥,=>,≥,= <,≤, >,=

Between every two different numbers there exists a thirdone:

EXAMPLE

A v ∃F.[< 3]∃F.[= 2] v B∃F.[< 2] v C

A 6v B t C

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 17/20

Conclusion

OUTLINE

1 INTRODUCTION

2 REASONING IN EL⊥(D)

3 CONCLUSION

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 18/20

Conclusion

RESULTS OVERVIEWPolynomial, sound and complete reasoning procedure forextensions of EL⊥ with “safe” numerical datatypesFull classification of tractable cases for N, Z, Q and RCommon restriction for N, Z, Q and R:

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Candidate for an OWL 2 EL Profile extension

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

Conclusion

RESULTS OVERVIEWPolynomial, sound and complete reasoning procedure forextensions of EL⊥ with “safe” numerical datatypesFull classification of tractable cases for N, Z, Q and RCommon restriction for N, Z, Q and R:

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Candidate for an OWL 2 EL Profile extension

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

Conclusion

RESULTS OVERVIEWPolynomial, sound and complete reasoning procedure forextensions of EL⊥ with “safe” numerical datatypesFull classification of tractable cases for N, Z, Q and RCommon restriction for N, Z, Q and R:

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Candidate for an OWL 2 EL Profile extension

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

Conclusion

RESULTS OVERVIEWPolynomial, sound and complete reasoning procedure forextensions of EL⊥ with “safe” numerical datatypesFull classification of tractable cases for N, Z, Q and RCommon restriction for N, Z, Q and R:

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Candidate for an OWL 2 EL Profile extension

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

Conclusion

RESULTS OVERVIEWPolynomial, sound and complete reasoning procedure forextensions of EL⊥ with “safe” numerical datatypesFull classification of tractable cases for N, Z, Q and RCommon restriction for N, Z, Q and R:

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Candidate for an OWL 2 EL Profile extension

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

Conclusion

RESULTS OVERVIEWPolynomial, sound and complete reasoning procedure forextensions of EL⊥ with “safe” numerical datatypesFull classification of tractable cases for N, Z, Q and RCommon restriction for N, Z, Q and R:

Negative relations Positive relations<,≤, >,≥,= =

EXAMPLEPanadol v ∃contains.(Paracetamol u ∃mgPerTablet.[= 500])

Patient u ∃hasAge.[< 6] u ∃hasPrescription.∃contains.(Paracetamol u ∃mgPerTablet.[> 250]) v ⊥

X v Patient u ∃hasAge.[= 3] u ∃hasPrescription.Panadol

Candidate for an OWL 2 EL Profile extension

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

Conclusion

FUTURE WORK

Extend the reasoning algorithm:

complex role inclusionsfunctional data propertiesnominalsdomain and range restrictionsHorn SHIQ [Kazakov, IJCAI, 2009]

More fine-grained analysis by also considering which dataproperties are used with relations

Thank you for your attention! Questions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20

Conclusion

FUTURE WORK

Extend the reasoning algorithm:complex role inclusionsfunctional data propertiesnominalsdomain and range restrictionsHorn SHIQ [Kazakov, IJCAI, 2009]

More fine-grained analysis by also considering which dataproperties are used with relations

Thank you for your attention! Questions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20

Conclusion

FUTURE WORK

Extend the reasoning algorithm:complex role inclusionsfunctional data propertiesnominalsdomain and range restrictionsHorn SHIQ [Kazakov, IJCAI, 2009]

More fine-grained analysis by also considering which dataproperties are used with relations

Thank you for your attention! Questions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20

Conclusion

FUTURE WORK

Extend the reasoning algorithm:complex role inclusionsfunctional data propertiesnominalsdomain and range restrictionsHorn SHIQ [Kazakov, IJCAI, 2009]

More fine-grained analysis by also considering which dataproperties are used with relations

Thank you for your attention! Questions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20

Conclusion

FUTURE WORK

Extend the reasoning algorithm:complex role inclusionsfunctional data propertiesnominalsdomain and range restrictionsHorn SHIQ [Kazakov, IJCAI, 2009]

More fine-grained analysis by also considering which dataproperties are used with relations

Thank you for your attention! Questions?

Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20