We consider extensions of the lightweight description logic (DL) EL with numerical datatypes such as naturals, integers, rationals and reals equipped with relations such as equality and inequalities. It is well-known that the main reasoning problems for such DLs are decid- able in polynomial time provided that the datatypes enjoy the so-called convexity property. Unfortunately many combinations of the numerical relations violate convexity, which makes the usage of these datatypes rather limited in practice. In this paper, we make a more fine-grained complexity analysis of these DLs by considering restrictions not only on the kinds of relations that can be used in ontologies but also on their occurrences, such as allowing certain relations to appear only on the left- hand side of the axioms. To this end, we introduce a notion of safety for a numerical datatype with restrictions (NDR) which guarantees tractabil- ity, extend the EL reasoning algorithm to these cases, and provide a complete classification of safe NDRs for natural numbers, integers, ra- tionals and reals.

1. T RACTABLE E XTENSIONS OF THE D ESCRIPTION
L OGIC EL WITH N UMERICAL DATATYPES
Despoina Magka, Yevgeny Kazakov and Ian Horrocks
Oxford University Computing Laboratory
July 16, 2010

2. Introduction
O UTLINE
1 I NTRODUCTION
2 R EASONING IN EL⊥ (D)
3 C ONCLUSION
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 2/20

3. Introduction
O NTOLOGIES
Ontologies are formal descriptions of domains such as:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

4. Introduction
O NTOLOGIES
Ontologies are formal descriptions of domains such as:
Aerospace
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

5. Introduction
O NTOLOGIES
Ontologies are formal descriptions of domains such as:
Aerospace
Cellular biology
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

6. Introduction
O NTOLOGIES
Ontologies are formal descriptions of domains such as:
Aerospace
Cellular biology
Human anatomy
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

7. Introduction
O NTOLOGIES
Ontologies are formal descriptions of domains such as:
Aerospace
Cellular biology
Human anatomy
Drugs
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

8. Introduction
O NTOLOGIES
Ontologies are formal descriptions of domains such as:
Aerospace
Cellular biology
Human anatomy
Drugs
Applications:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

9. Introduction
O NTOLOGIES
Ontologies are formal descriptions of domains such as:
Aerospace
Cellular biology
Human anatomy
Drugs
Applications:
Automated decision support
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 3/20

10. Introduction
O NTOLOGIES
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

11. Introduction
O NTOLOGIES
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

12. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

13. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

14. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

15. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

16. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A patient who has age less than 6 years
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

17. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A patient who has age less than 6 years
Axioms formulate properties and restrictions using terms:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

18. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A patient who has age less than 6 years
Axioms formulate properties and restrictions using terms:
Panadol is a drug.
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

19. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A patient who has age less than 6 years
Axioms formulate properties and restrictions using terms:
Panadol is a drug.
Panadol contains paracetamol.
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

20. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A 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.
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

21. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A 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 be
prescribed anything which [contains more than 250 mg of
paracetamol per tablet].
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

22. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A 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 be
prescribed anything which [contains more than 250 mg of
paracetamol per tablet].
Restrictions specify ranges of data values:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

23. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A 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 be
prescribed anything which [contains more than 250 mg of
paracetamol per tablet].
Restrictions specify ranges of data values:
500 mg
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

24. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A 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 be
prescribed anything which [contains more than 250 mg of
paracetamol per tablet].
Restrictions specify ranges of data values:
500 mg
more than 250 mg
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

25. Introduction
O NTOLOGY AXIOMS
Terms describe entities from specific domains, e.g. for a
drug ontology:
Paracetamol
Patient
A drug that contains 500 mg of paracetamol per tablet
A 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 be
prescribed anything which [contains more than 250 mg of
paracetamol per tablet].
Restrictions specify ranges of data values:
500 mg
more than 250 mg
less than 6 years
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 4/20

26. Introduction
D ESCRIPTION L OGICS
Formally describe axioms:
E XAMPLE
Panadol Drug
Panadol ∃contains.Paracetamol
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

27. Introduction
D ESCRIPTION L OGICS
Formally describe axioms:
E XAMPLE
Panadol Drug
Panadol ∃contains.Paracetamol
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Atomic concepts
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

28. Introduction
D ESCRIPTION L OGICS
Formally describe axioms:
E XAMPLE
Panadol Drug
Panadol ∃contains.Paracetamol
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Atomic concepts
Atomic roles
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

29. Introduction
D ESCRIPTION L OGICS
Formally describe axioms:
E XAMPLE
Panadol Drug
Panadol ∃contains.Paracetamol
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Atomic concepts
Atomic roles
Datatype restrictions
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

30. Introduction
D ESCRIPTION L OGICS
Formally describe axioms:
E XAMPLE
Panadol Drug
Panadol ∃contains.Paracetamol
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Atomic concepts
Atomic roles
Datatype restrictions
Concept constructors
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 5/20

31. Introduction
D ESCRIPTION L OGICS
Formally describe axioms:
E XAMPLE
Panadol Drug
Panadol ∃contains.Paracetamol
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Atomic concepts
Atomic roles
Datatype restrictions
Concept 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

32. Introduction
D ESCRIPTION L OGICS R EASONING
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

33. Introduction
D ESCRIPTION L OGICS R EASONING
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
Can Panadol be prescribed to a 3-year-old patient?
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

34. Introduction
D ESCRIPTION L OGICS R EASONING
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X ≡ Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Can Panadol be prescribed to a 3-year-old patient?
- Is X satisfiable?
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

35. Introduction
D ESCRIPTION L OGICS R EASONING
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X ≡ Patient ∃hasAge.[= 3] ∃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 ⊥)
Check satisfiability of an ontology (O |= ⊥ )
Check subsumption (O |= A B)
Classification (compute all A B such that O |= A B)
more general than all tasks above
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 6/20

36. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

37. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

38. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

39. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
<, =
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

40. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
<, =
E XAMPLE
A ∃F.[< 2]
∃F.[= 1] B
∃F.[= 0] C
A B C
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

41. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
<, = >, = <, > ≤, = ≥, = ≤, ≥
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

42. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
<, = >, = <, > ≤, = ≥, = ≤, ≥
Baader et al. [IJCAI 2005] ensure polynomiality by
restrictions on datatype use
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

43. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
<, = >, = <, > ≤, = ≥, = ≤, ≥
Baader et al. [IJCAI 2005] ensure polynomiality by
restrictions on datatype use
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

44. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
<, = >, = <, > ≤, = ≥, = ≤, ≥
Baader et al. [IJCAI 2005] ensure polynomiality by
restrictions on datatype use
EL Profile of OWL 2 admits only equality
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 7/20

45. Introduction
EL AND DATATYPES
EL [Baader et al., IJCAI 2005] is a simple DL:
uses and ∃:
Panadol ≡ Drug ∃contains.Paracetamol
suitable for large-scale ontologies such as SNOMED or GO
reasoning in PTIME
EL with datatypes is EXPTIME-complete. Why?
EL extended with disjunction is EXPTIME-complete and
disjunction is expressible using datatypes:
<, = >, = <, > ≤, = ≥, = ≤, ≥
Baader et al. [IJCAI 2005] ensure polynomiality by
restrictions on datatype use
EL 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

46. Introduction
R ESULTS OVERVIEW
Relax the restrictions on datatype use by allowing
inequalities and remaining polynomial
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

47. Introduction
R ESULTS OVERVIEW
Relax the restrictions on datatype use by allowing
inequalities and remaining polynomial
Key idea
Distinguish negative (LHS of axiom) and positive (RHS of
axiom) occurrences of datatypes
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

48. Introduction
R ESULTS OVERVIEW
Relax the restrictions on datatype use by allowing
inequalities and remaining polynomial
Key idea
Distinguish negative (LHS of axiom) and positive (RHS of
axiom) occurrences of datatypes
Main results
Full classification of cases where datatypes are used and
tractability is preserved for N, Z, Q and R
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

49. Introduction
R ESULTS OVERVIEW
Relax the restrictions on datatype use by allowing
inequalities and remaining polynomial
Key idea
Distinguish negative (LHS of axiom) and positive (RHS of
axiom) occurrences of datatypes
Main results
Full classification of cases where datatypes are used and
tractability is preserved for N, Z, Q and R
Polynomial, sound and complete reasoning procedure for
extensions of EL⊥ with restricted numerical datatypes
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 8/20

50. Introduction
EL FAMILY OF D ESCRIPTION L OGICS
The EL language:
Syntax Semantics
Atomic concept C C(x)
Top
Conjunction C D C(x) ∧ D(x)
Existential restriction ∃R.C ∃y : R(x, y) ∧ C(y)
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

51. Introduction
EL FAMILY OF D ESCRIPTION L OGICS
The EL language:
Syntax Semantics
Atomic concept C C(x)
Top
Conjunction C 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
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

52. Introduction
EL FAMILY OF D ESCRIPTION L OGICS
The EL language:
Syntax Semantics
Atomic concept C C(x)
Top
Conjunction C 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 D ∀x : C(x) → D(x)
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

53. Introduction
EL FAMILY OF D ESCRIPTION L OGICS
The EL language:
Syntax Semantics
Atomic concept C C(x)
Top
Conjunction C 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 D ∀x : C(x) → D(x)
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 9/20

54. ⊥
Reasoning in EL (D)
O UTLINE
1 I NTRODUCTION
2 R EASONING IN EL⊥ (D)
3 C ONCLUSION
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 10/20

55. ⊥
Reasoning in EL (D)
A LGORITHM S TAGES
1 Standard normalization of the axioms
Normal forms
NF1 A B
NF2 A1 A2 B
NF3 A ∃R.B
NF4 ∃R.B A
NF5 A ∃F.range
NF6 ∃F.range A
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 11/20

56. ⊥
Reasoning in EL (D)
A LGORITHM S TAGES
1 Standard normalization of the axioms
Normal forms
NF1 A B
NF2 A1 A2 B
NF3 A ∃R.B
NF4 ∃R.B A
NF5 A ∃F.range
NF6 ∃F.range A
2 Saturation of the axioms under inference rules, such as:
A B A C
B C D∈O
A D
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 11/20

57. ⊥
Reasoning in EL (D)
R EASONING RULES ( PART I)
Rules from EL⊥
IR1 IR2
A B
A A A CR1 B C∈O
A C
A B A C
CR2 B C D∈O
A D
A B
CR3 B ∃R.C ∈ O
A ∃R.C
A ∃R.B B C
CR4 ∃R.C D∈O
A D
A ∃R.B B ⊥
CR5
A ⊥
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 12/20

58. ⊥
Reasoning in EL (D)
R EASONING RULES ( PART II)
New rules for datatypes
A ∃F.[< m]
∃F.[< n] B ∈ O, m ≤ n
A B
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 13/20

59. ⊥
Reasoning in EL (D)
R EASONING RULES ( PART II)
New rules for datatypes
ID1 A ∃F.[< 0] ∈ O
A ⊥
A B
CD1 B ∃F.range ∈ O
A ∃F.range
A ∃F.[< m]
CD2(<,<) ∃F.[< n] B ∈ O, m ≤ n
A B
A ∃F.[= m]
CD2(=,<) ∃F.[< n] B ∈ O, m < n
A B
A ∃F.[= m]
CD2(=,=) ∃F.[= n] B ∈ O, m = n ...
A B
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 13/20

60. ⊥
Reasoning in EL (D)
T HE EL⊥ (D)A LGORITHM
The algorithm is:
sound: all rules derive logical consequences of premises
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

61. ⊥
Reasoning in EL (D)
T HE EL⊥ (D)A LGORITHM
The algorithm is:
sound: all rules derive logical consequences of premises
polynomial as only polynomially many axioms are possible
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

62. ⊥
Reasoning in EL (D)
T HE EL⊥ (D)A LGORITHM
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)
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

63. ⊥
Reasoning in EL (D)
T HE EL⊥ (D)A LGORITHM
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
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 14/20

64. ⊥
Reasoning in EL (D)
T HE EL⊥ (D)A LGORITHM
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

65. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR N
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Restrict positive and negative occurrences of datatype
relations so as disjunction is not expressible
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

66. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR N
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Restrict positive and negative occurrences of datatype
relations so as disjunction is not expressible
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

67. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR N
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Restrict positive and negative occurrences of datatype
relations so as disjunction is not expressible
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

68. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR N
Negative relations Positive relations
<, ≤, >, ≥, = =
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = >, ≥, =
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

69. ⊥
Reasoning in EL (D)
R ESTRICTIONS 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

70. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR N
Negative relations Positive relations
<, ≤, >, ≥, = = ,<
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = >, ≥, =
All restrictions are maximal:
E XAMPLE
A ∃F.[< 2]
∃F.[= 1] B
∃F.[= 0] C
A B C
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 15/20

71. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR Z
Negative relations Positive relations
<, ≤, >, ≥, = =
= <, ≤, >, ≥, =
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = >, ≥, =
>, ≥, = <, ≤, =
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 16/20

72. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR Z
Negative relations Positive relations
<, ≤, >, ≥, = =
= <, ≤, >, ≥, =
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = >, ≥, =
>, ≥, = <, ≤, =
Additional datatype restrictions: integers do not have a
minimal element such as 0.
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 16/20

73. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR Z
Negative relations Positive relations
<, ≤, >, ≥, = =
= <, ≤, >, ≥, =
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = >, ≥, =
>, ≥, = <, ≤, =
Additional datatype restrictions: integers do not have a
minimal element such as 0.
E XAMPLE
A ∃F.[< 2]
∃F.[= 1] B
∃F.[= 0] C
A B C
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 16/20

74. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR Q AND R
Negative relations Positive relations
<, ≤, >, ≥, = =
≤, = <, ≤, >, ≥, =
≥, = <, ≤, >, ≥, =
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = <, >, ≥, =
>, ≥, = <, ≤, >, =
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 17/20

75. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR Q AND R
Negative relations Positive relations
<, ≤, >, ≥, = =
≤, = <, ≤, >, ≥, =
≥, = <, ≤, >, ≥, =
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = <, >, ≥, =
>, ≥, = <, ≤, >, =
Between every two different numbers there exists a third
one:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 17/20

76. ⊥
Reasoning in EL (D)
R ESTRICTIONS FOR Q AND R
Negative relations Positive relations
<, ≤, >, ≥, = =
≤, = <, ≤, >, ≥, =
≥, = <, ≤, >, ≥, =
<, ≤ <, ≤, >, ≥, =
>, ≥ <, ≤, >, ≥, =
<, ≤, = <, >, ≥, =
>, ≥, = <, ≤, >, =
Between every two different numbers there exists a third
one:
E XAMPLE
A ∃F.[< 3]
∃F.[= 2] B
∃F.[< 2] C
A B C
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 17/20

77. Conclusion
O UTLINE
1 I NTRODUCTION
2 R EASONING IN EL⊥ (D)
3 C ONCLUSION
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 18/20

78. Conclusion
R ESULTS OVERVIEW
Polynomial, sound and complete reasoning procedure for
extensions of EL⊥ with “safe” numerical datatypes
Full classification of tractable cases for N, Z, Q and R
Common restriction for N, Z, Q and R:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

79. Conclusion
R ESULTS OVERVIEW
Polynomial, sound and complete reasoning procedure for
extensions of EL⊥ with “safe” numerical datatypes
Full classification of tractable cases for N, Z, Q and R
Common restriction for N, Z, Q and R:
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

80. Conclusion
R ESULTS OVERVIEW
Polynomial, sound and complete reasoning procedure for
extensions of EL⊥ with “safe” numerical datatypes
Full classification of tractable cases for N, Z, Q and R
Common restriction for N, Z, Q and R:
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

81. Conclusion
R ESULTS OVERVIEW
Polynomial, sound and complete reasoning procedure for
extensions of EL⊥ with “safe” numerical datatypes
Full classification of tractable cases for N, Z, Q and R
Common restriction for N, Z, Q and R:
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

82. Conclusion
R ESULTS OVERVIEW
Polynomial, sound and complete reasoning procedure for
extensions of EL⊥ with “safe” numerical datatypes
Full classification of tractable cases for N, Z, Q and R
Common restriction for N, Z, Q and R:
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃hasPrescription.Panadol
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 19/20

83. Conclusion
R ESULTS OVERVIEW
Polynomial, sound and complete reasoning procedure for
extensions of EL⊥ with “safe” numerical datatypes
Full classification of tractable cases for N, Z, Q and R
Common restriction for N, Z, Q and R:
Negative relations Positive relations
<, ≤, >, ≥, = =
E XAMPLE
Panadol ∃contains.(Paracetamol ∃mgPerTablet.[= 500])
Patient ∃hasAge.[< 6] ∃hasPrescription.
∃contains.(Paracetamol ∃mgPerTablet.[> 250]) ⊥
X Patient ∃hasAge.[= 3] ∃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

84. Conclusion
F UTURE WORK
Extend the reasoning algorithm:
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20

85. Conclusion
F UTURE WORK
Extend the reasoning algorithm:
complex role inclusions
functional data properties
nominals
domain and range restrictions
Horn SHIQ [Kazakov, IJCAI, 2009]
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20

86. Conclusion
F UTURE WORK
Extend the reasoning algorithm:
complex role inclusions
functional data properties
nominals
domain and range restrictions
Horn SHIQ [Kazakov, IJCAI, 2009]
More fine-grained analysis by also considering which data
properties are used with relations
Despoina Magka, Yevgeny Kazakov and Ian Horrocks Tractable Extensions of EL with Numerical Datatypes 20/20

87. Conclusion
F UTURE WORK
Extend the reasoning algorithm:
complex role inclusions
functional data properties
nominals
domain and range restrictions
Horn SHIQ [Kazakov, IJCAI, 2009]
More fine-grained analysis by also considering which data
properties 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

88. Conclusion
F UTURE WORK
Extend the reasoning algorithm:
complex role inclusions
functional data properties
nominals
domain and range restrictions
Horn SHIQ [Kazakov, IJCAI, 2009]
More fine-grained analysis by also considering which data
properties 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