First Steps in EL Contraction
Richard Booth Tommie Meyer Ivan Jose Varzinczak
Mahasarakham UniversityThailand
Meraka Institute, CSIRPretoria, South Africa
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 1 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 2 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 2 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 2 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 3 / 24
Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ? ϕ, with K ? ϕ |= ϕ and K ? ϕ 6|= ⊥Contraction: K − ϕ 6|= ϕ
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 4 / 24
Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ? ϕ, with K ? ϕ |= ϕ and K ? ϕ 6|= ⊥Contraction: K − ϕ 6|= ϕ
Also meaningful for ontologies
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 4 / 24
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 6|= 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
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 6|= 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
AGM Approach
Construction method:
Identify the maximally consistent subsets that do not entail ϕ(remainder sets)
Pick some non-empty subset of remainder setsI 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
AGM Approach
Example
Contraction of {p → r} from Horn theory K = Cn({p → q, q → r})
p → q
>q → r
p ∧ r → q
p → r
p ∧ q → r
Maxichoice?
Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 7 / 24
AGM Approach
Example
Contraction of {p → r} from Horn theory K = Cn({p → q, q → r})
p → q
>q → r
p ∧ r → q
p → r
p ∧ q → r
Maxichoice? H1mc = Cn({p → q}) or H2
mc = Cn({q → r , p ∧ r → q})Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 7 / 24
AGM Approach
Example
Contraction of {p → r} from Horn theory K = Cn({p → q, q → r})
p → q
>q → r
p ∧ r → q
p → r
p ∧ q → r
Maxichoice? H1mc = Cn({p → q}) or H2
mc = Cn({q → r , p ∧ r → q})Full meet? Hfm = Cn({p ∧ r → q})
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 7 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 8 / 24
Description Logic EL [Baader, 2003] with ⊥Concepts
C ::= A | > | ⊥ | C u C | ∃R.C
Interpretations I = 〈∆I , ·I〉
AI ⊆ ∆I , RI ⊆ ∆I ×∆I ,
>I = ∆I , ⊥I = ∅,
(C u D)I = CI ∩ DI ,
(∃R.C )I = {a ∈ ∆I | ∃b.(a, b) ∈ RI and b ∈ CI}
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 9 / 24
Description Logic EL [Baader, 2003]
Axioms C v DI |= C v D iff CI ⊆ DI
TBox T: set of axioms
I |= T iff I satisfies every axiom in T
T |= C v D iff for all I if I |= T then I |= C v D
Cn(T) = {C v D | T |= C v D}
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 10 / 24
Description Logic EL [Baader, 2003]
Example
T = Cn({A v B,B v ∃R.A})
A v B >
B v ∃R.A
A u ∃R.A v B
A v ∃R.A
A u B v ∃R.A
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 11 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 12 / 24
Motivation
Let T be a TBox and Φ be a set of axioms
Contract T with ΦI we want T 6|= ΦI Some axiom in Φ should not follow from T anymore
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 13 / 24
Following Delgrande’s Approach [KR’2008]
Definition (Remainder Sets)
For a belief set T, X ∈ T↓Φ iff
X ⊆ TX 6|= 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?I 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
Following Delgrande’s Approach [KR’2008]
Definition (Remainder Sets)
For a belief set T, X ∈ T↓Φ iff
X ⊆ TX 6|= 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?I 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
Following Delgrande’s Approach [KR’2008]
Definition (Remainder Sets)
For a belief set T, X ∈ T↓Φ iff
X ⊆ TX 6|= 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?I 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
Following Delgrande’s Approach
Definition (Partial Meet Contraction)
Given a selection function σ, −σ is a partial meet contraction iffT −σ Φ =
⋂σ(T ↓Φ).
Definition (Maxichoice and Full Meet)
Given a selection function σ, −σ is a maxichoice contraction iff σ(T ↓Φ) isa singleton set. It is a full meet contraction iff σ(T ↓Φ) = T ↓Φ whenT ↓Φ 6= ∅.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 15 / 24
Following Delgrande’s Approach
Definition (Partial Meet Contraction)
Given a selection function σ, −σ is a partial meet contraction iffT −σ Φ =
⋂σ(T ↓Φ).
Definition (Maxichoice and Full Meet)
Given a selection function σ, −σ is a maxichoice contraction iff σ(T ↓Φ) isa singleton set. It is a full meet contraction iff σ(T ↓Φ) = T ↓Φ whenT ↓Φ 6= ∅.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 15 / 24
Following Delgrande’s Approach
Example
Contraction of {A v ∃R.A} from T = Cn({A v B,B v ∃R.A})
A v B >
B v ∃R.A
A u ∃R.A v B
A v ∃R.A
A u B v ∃R.A
Maxichoice?
Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24
Following Delgrande’s Approach
Example
Contraction of {A v ∃R.A} from T = Cn({A v B,B v ∃R.A})
A v B >
B v ∃R.A
A u ∃R.A v B
A v ∃R.A
A u B v ∃R.A
Maxichoice? T 1mc = Cn({A v B}) or T 2
mc = Cn({B v ∃R.A,A u ∃R.A v B})Full meet?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24
Following Delgrande’s Approach
Example
Contraction of {A v ∃R.A} from T = Cn({A v B,B v ∃R.A})
A v B >
B v ∃R.A
A u ∃R.A v B
A v ∃R.A
A u B v ∃R.A
Maxichoice? T 1mc = Cn({A v B}) or T 2
mc = Cn({B v ∃R.A,A u ∃R.A v B})Full meet? Tfm = Cn({A u ∃R.A v B})
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24
Following Delgrande’s Approach
Example
Contraction of {A v ∃R.A} from T = Cn({A v B,B v ∃R.A})
A v B >
B v ∃R.A
A u ∃R.A v B
A v ∃R.A
A u B v ∃R.A
Maxichoice? T 1mc = Cn({A v B}) or T 2
mc = Cn({B v ∃R.A,A u ∃R.A v B})Full meet? Tfm = Cn({A u ∃R.A v B})What about T ′ = Cn({A u ∃R.A v B,A u B v ∃R.A})?
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24
Following Delgrande’s Approach
Example
Contraction of {A v ∃R.A} from T = Cn({A v B,B v ∃R.A})
A v B >
B v ∃R.A
A u ∃R.A v B
A v ∃R.A
A u B v ∃R.A
Maxichoice? T 1mc = Cn({A v B}) or T 2
mc = Cn({B v ∃R.A,A u ∃R.A v B})Full meet? Tfm = Cn({A u ∃R.A v B})What about T ′ = Cn({A u ∃R.A v B,A u B v ∃R.A})?Tfm ⊆ T ′ ⊆ T 2
mc , but there is no partial meet contraction yielding T ′!
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 16 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 17 / 24
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 andthe intersection of all remainder sets
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 18 / 24
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. Acontraction function −τ is an EL-contraction iff T −τ Φ = τ(T⇓Φ).
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 19 / 24
A Representation Result
Basic postulates for EL contraction
(T − 1) T − Φ = Cn(T − Φ)
(T − 2) T − Φ ⊆ T(T − 3) If Φ 6⊆ T then T − Φ = T(T − 4) If 6|= Φ then Φ 6⊆ 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 ′ 6|= Φ, and T ′ + {ϕ} |= Φ
Conjecture
Every EL contraction satisfies (T − 1)–(T − 6). Conversely, everycontraction function satisfying (T − 1)–(T − 6) is an EL contraction.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 20 / 24
A Representation Result
Basic postulates for EL contraction
(T − 1) T − Φ = Cn(T − Φ)
(T − 2) T − Φ ⊆ T(T − 3) If Φ 6⊆ T then T − Φ = T(T − 4) If 6|= Φ then Φ 6⊆ 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 ′ 6|= Φ, and T ′ + {ϕ} |= Φ
Conjecture
Every EL contraction satisfies (T − 1)–(T − 6). Conversely, everycontraction function satisfying (T − 1)–(T − 6) is an EL contraction.
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 20 / 24
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 21 / 24
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 6|= ⊥
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
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 6|= ⊥
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
Outline
1 PreliminariesBelief ChangeDescription Logic EL
2 Contraction in ELFirst AttemptA More Fine-grained ApproachOther Types of Contraction
3 ConclusionSummary, Open questions and Further Work
Booth, Meyer, Varzinczak (MU/KSG) EL Contraction 23 / 24
Summary, Open questions and Further Work
Summary
Basic AGM account of contraction for ELWeaker 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 u ∃R.A, A u ∃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
Summary, Open questions and Further Work
Summary
Basic AGM account of contraction for ELWeaker 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 u ∃R.A, A u ∃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
Summary, Open questions and Further Work
Summary
Basic AGM account of contraction for ELWeaker 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 u ∃R.A, A u ∃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