Work presented at the IJCAI'09 Workshop ARCOE.

1. First Steps in EL Contraction
Richard Booth Tommie Meyer Ivan Jos´ Varzinczak
e
Mahasarakham University Meraka Institute, CSIR
Thailand Pretoria, South Africa
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 1 / 24

2. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 2 / 24

3. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 2 / 24

4. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 2 / 24

5. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 3 / 24

6. Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ϕ, with K ϕ |= ϕ and K ϕ |= ⊥
Contraction: K − ϕ |= ϕ
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 4 / 24

7. Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ϕ, with K ϕ |= ϕ and K ϕ |= ⊥
Contraction: K − ϕ |= ϕ
Also meaningful for ontologies
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 4 / 24

8. AGM Approach
Contraction described on the knowledge level
Rationality Postulates
(K−1) K − ϕ = Cn(K − ϕ)
(K−2) K − ϕ ⊆ K
(K−3) If ϕ ∈ K , then K − ϕ = K
/
(K−4) If |= ϕ, then ϕ ∈ K − ϕ
/
(K−5) If ϕ ≡ ψ, then K − ϕ = K − ψ
(K−6) If ϕ ∈ K , then (K − ϕ) + ϕ = K
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 5 / 24

9. AGM Approach
Contraction described on the knowledge level
Rationality Postulates
(K−1) K − ϕ = Cn(K − ϕ)
(K−2) K − ϕ ⊆ K
(K−3) If ϕ ∈ K , then K − ϕ = K
/
(K−4) If |= ϕ, then ϕ ∈ K − ϕ
/
(K−5) If ϕ ≡ ψ, then K − ϕ = K − ψ
(K−6) If ϕ ∈ K , then (K − ϕ) + ϕ = K
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 5 / 24

10. AGM Approach
Construction method:
Identify the maximally consistent subsets that do not entail ϕ
(remainder sets)
Pick some non-empty subset of remainder sets
Take their intersection: Partial meet
Pick all remainder sets: Full meet
Pick a single remainder set: Maxichoice
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 6 / 24

11. AGM Approach
Example
Contraction of {p → r } from Horn theory K = Cn({p → q, q → r })
p∧r →q p∧q →r
p→q q→r
p→r
Maxichoice?
Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 7 / 24

12. AGM Approach
Example
Contraction of {p → r } from Horn theory K = Cn({p → q, q → r })
p∧r →q p∧q →r
p→q q→r
p→r
1 2
Maxichoice? Hmc = Cn({p → q}) or Hmc = Cn({q → r , p ∧ r → q})
Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 7 / 24

13. AGM Approach
Example
Contraction of {p → r } from Horn theory K = Cn({p → q, q → r })
p∧r →q p∧q →r
p→q q→r
p→r
1 2
Maxichoice? Hmc = Cn({p → q}) or Hmc = Cn({q → r , p ∧ r → q})
Full meet? Hfm = Cn({p ∧ r → q})
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 7 / 24

14. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 8 / 24

15. Description Logic EL [Baader, 2003] with ⊥
Concepts
C ::= A | |⊥|C C | ∃R.C
Interpretations I = ∆I , ·I
AI ⊆ ∆I , R I ⊆ ∆I × ∆I ,
I = ∆I , ⊥I = ∅,
(C D)I = C I ∩ D I ,
(∃R.C )I = {a ∈ ∆I | ∃b.(a, b) ∈ R I and b ∈ C I }
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 9 / 24

16. Description Logic EL [Baader, 2003]
Axioms C D
I |= C D iff C I ⊆ D I
TBox T: set of axioms
I |= T iff I satisfies every axiom in T
T |= C D iff for all I if I |= T then I |= C D
Cn(T) = {C D | T |= C D}
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 10 / 24

17. Description Logic EL [Baader, 2003]
Example
T = Cn({A B, B ∃R.A})
A ∃R.A B
A B ∃R.A
A B
B ∃R.A
A ∃R.A
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 11 / 24

18. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 12 / 24

19. Motivation
Let T be a TBox and Φ be a set of axioms
Contract T with Φ
we want T |= Φ
Some axiom in Φ should not follow from T anymore
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 13 / 24

20. Following Delgrande’s Approach [KR’2008]
Definition (Remainder Sets)
For a belief set T, X ∈ T ↓ Φ iff
X ⊆T
X |= Φ
for every X s.t. X ⊂ X ⊆ T, X |= Φ.
We call T ↓ Φ the remainder sets of T w.r.t. Φ
Do they exist?
EL is compact and has a Tarskian consequence relation
Definition (Selection Functions)
A selection function σ is a function from P(P(LEL )) to P(P(LEL ))
s.t. σ(T ↓ Φ) = {T} if T ↓ Φ = ∅, and σ(T ↓ Φ) ⊆ T ↓ Φ otherwise.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 14 / 24

21. Following Delgrande’s Approach [KR’2008]
Definition (Remainder Sets)
For a belief set T, X ∈ T ↓ Φ iff
X ⊆T
X |= Φ
for every X s.t. X ⊂ X ⊆ T, X |= Φ.
We call T ↓ Φ the remainder sets of T w.r.t. Φ
Do they exist?
EL is compact and has a Tarskian consequence relation
Definition (Selection Functions)
A selection function σ is a function from P(P(LEL )) to P(P(LEL ))
s.t. σ(T ↓ Φ) = {T} if T ↓ Φ = ∅, and σ(T ↓ Φ) ⊆ T ↓ Φ otherwise.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 14 / 24

22. Following Delgrande’s Approach [KR’2008]
Definition (Remainder Sets)
For a belief set T, X ∈ T ↓ Φ iff
X ⊆T
X |= Φ
for every X s.t. X ⊂ X ⊆ T, X |= Φ.
We call T ↓ Φ the remainder sets of T w.r.t. Φ
Do they exist?
EL is compact and has a Tarskian consequence relation
Definition (Selection Functions)
A selection function σ is a function from P(P(LEL )) to P(P(LEL ))
s.t. σ(T ↓ Φ) = {T} if T ↓ Φ = ∅, and σ(T ↓ Φ) ⊆ T ↓ Φ otherwise.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 14 / 24

23. Following Delgrande’s Approach
Definition (Partial Meet Contraction)
Given a selection function σ, −σ is a partial meet contraction iff
T −σ Φ = σ(T ↓ Φ).
Definition (Maxichoice and Full Meet)
Given a selection function σ, −σ is a maxichoice contraction iff σ(T ↓ Φ) is
a singleton set. It is a full meet contraction iff σ(T ↓ Φ) = T ↓ Φ when
T ↓ Φ = ∅.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 15 / 24

24. Following Delgrande’s Approach
Definition (Partial Meet Contraction)
Given a selection function σ, −σ is a partial meet contraction iff
T −σ Φ = σ(T ↓ Φ).
Definition (Maxichoice and Full Meet)
Given a selection function σ, −σ is a maxichoice contraction iff σ(T ↓ Φ) is
a singleton set. It is a full meet contraction iff σ(T ↓ Φ) = T ↓ Φ when
T ↓ Φ = ∅.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 15 / 24

25. Following Delgrande’s Approach
Example
Contraction of {A ∃R.A} from T = Cn({A B, B ∃R.A})
A ∃R.A B
A B ∃R.A
A B
B ∃R.A
A ∃R.A
Maxichoice?
Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24

26. Following Delgrande’s Approach
Example
Contraction of {A ∃R.A} from T = Cn({A B, B ∃R.A})
A ∃R.A B
A B ∃R.A
A B
B ∃R.A
A ∃R.A
1
Maxichoice? Tmc = Cn({A 2
B}) or Tmc = Cn({B ∃R.A, A ∃R.A B})
Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24

27. Following Delgrande’s Approach
Example
Contraction of {A ∃R.A} from T = Cn({A B, B ∃R.A})
A ∃R.A B
A B ∃R.A
A B
B ∃R.A
A ∃R.A
1
Maxichoice? Tmc = Cn({A 2
B}) or Tmc = Cn({B ∃R.A, A ∃R.A B})
Full meet? Tfm = Cn({A ∃R.A B})
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24

28. Following Delgrande’s Approach
Example
Contraction of {A ∃R.A} from T = Cn({A B, B ∃R.A})
A ∃R.A B
A B ∃R.A
A B
B ∃R.A
A ∃R.A
1
Maxichoice? Tmc = Cn({A 2
B}) or Tmc = Cn({B ∃R.A, A ∃R.A B})
Full meet? Tfm = Cn({A ∃R.A B})
What about T = Cn({A ∃R.A B, A B ∃R.A})?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24

29. Following Delgrande’s Approach
Example
Contraction of {A ∃R.A} from T = Cn({A B, B ∃R.A})
A ∃R.A B
A B ∃R.A
A B
B ∃R.A
A ∃R.A
1
Maxichoice? Tmc = Cn({A 2
B}) or Tmc = Cn({B ∃R.A, A ∃R.A B})
Full meet? Tfm = Cn({A ∃R.A B})
What about T = Cn({A ∃R.A B, A B ∃R.A})?
2
Tfm ⊆ T ⊆ Tmc , but there is no partial meet contraction yielding T !
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24

30. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 17 / 24

31. Beyond Partial Meet [Booth et al., IJCAI’09]
Definition (Infra-Remainder Sets)
For belief sets T and X , X ∈ T ⇓ Φ iff there is some X ∈ T ↓ Φ s.t.
( T ↓ Φ) ⊆ X ⊆ X .
We call T ⇓ Φ the infra-remainder sets of T w.r.t. Φ.
Infra-remainder sets contain all belief sets between some remainder set and
the intersection of all remainder sets
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 18 / 24

32. Beyond Partial Meet [Booth et al., IJCAI’09]
Definition (EL Contraction)
An infra-selection function τ is a function from P(P(LEL )) to P(LEL )
s.t. τ (T ⇓ Φ) = T whenever |= Φ, and τ (T ⇓ Φ) ∈ T ⇓ Φ otherwise. A
contraction function −τ is an EL-contraction iff T −τ Φ = τ (T ⇓ Φ).
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 19 / 24

33. A Representation Result
Basic postulates for EL contraction
(T − 1) T − Φ = Cn(T − Φ)
(T − 2) T − Φ ⊆ T
(T − 3) If Φ ⊆ T then T − Φ = T
(T − 4) If |= Φ then Φ ⊆ T − Φ
(T − 5) If Cn(Φ) = Cn(Ψ) then T − Φ = T − Ψ
(T − 6) If ϕ ∈ T (T − Φ) then there is a T such that
(T ↓ Φ) ⊆ T ⊆ T, T |= Φ, and T + {ϕ} |= Φ
Conjecture
Every EL contraction satisfies (T − 1)–(T − 6). Conversely, every
contraction function satisfying (T − 1)–(T − 6) is an EL contraction.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 20 / 24

34. A Representation Result
Basic postulates for EL contraction
(T − 1) T − Φ = Cn(T − Φ)
(T − 2) T − Φ ⊆ T
(T − 3) If Φ ⊆ T then T − Φ = T
(T − 4) If |= Φ then Φ ⊆ T − Φ
(T − 5) If Cn(Φ) = Cn(Ψ) then T − Φ = T − Ψ
(T − 6) If ϕ ∈ T (T − Φ) then there is a T such that
(T ↓ Φ) ⊆ T ⊆ T, T |= Φ, and T + {ϕ} |= Φ
Conjecture
Every EL contraction satisfies (T − 1)–(T − 6). Conversely, every
contraction function satisfying (T − 1)–(T − 6) is an EL contraction.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 20 / 24

35. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 21 / 24

36. Other Types of Contraction
Inconsistency-based Contraction [Delgrande, KR’2008]
Let T be a TBox and Φ be a set of axioms
Contract T ‘making room’ for Φ
We want T + Φ |= ⊥
Package Contraction [Booth et al., IJCAI’09]
Let T be a TBox and Φ be a set of axioms
Contract T so that none of the axioms in Φ follows from it
Removal of all sentences in Φ from T
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 22 / 24

37. Other Types of Contraction
Inconsistency-based Contraction [Delgrande, KR’2008]
Let T be a TBox and Φ be a set of axioms
Contract T ‘making room’ for Φ
We want T + Φ |= ⊥
Package Contraction [Booth et al., IJCAI’09]
Let T be a TBox and Φ be a set of axioms
Contract T so that none of the axioms in Φ follows from it
Removal of all sentences in Φ from T
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 22 / 24

38. Outline
1 Preliminaries
Belief Change
Description Logic EL
2 Contraction in EL
First Attempt
A More Fine-grained Approach
Other Types of Contraction
3 Conclusion
Summary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 23 / 24

39. Summary, Open questions and Further Work
Summary
Basic AGM account of contraction for EL
Weaker than partial meet contraction
Open questions
Are infra-remainder sets enough?
Is Cn(.) what we really want?
Kernel contraction? (Renata knows the answer )
What about the syntax? (A, A ∃R.A, A ∃R.∃R.A,. . . )
Current and Future Work
Answer questions above
Full AGM setting: extended postulates
Relation to justifications in ontology repair
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 24 / 24

40. Summary, Open questions and Further Work
Summary
Basic AGM account of contraction for EL
Weaker than partial meet contraction
Open questions
Are infra-remainder sets enough?
Is Cn(.) what we really want?
Kernel contraction? (Renata knows the answer )
What about the syntax? (A, A ∃R.A, A ∃R.∃R.A,. . . )
Current and Future Work
Answer questions above
Full AGM setting: extended postulates
Relation to justifications in ontology repair
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 24 / 24

41. Summary, Open questions and Further Work
Summary
Basic AGM account of contraction for EL
Weaker than partial meet contraction
Open questions
Are infra-remainder sets enough?
Is Cn(.) what we really want?
Kernel contraction? (Renata knows the answer )
What about the syntax? (A, A ∃R.A, A ∃R.∃R.A,. . . )
Current and Future Work
Answer questions above
Full AGM setting: extended postulates
Relation to justifications in ontology repair
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 24 / 24