Post on 05-Dec-2014
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Next Steps in Propositional Horn Contraction
Richard Booth Tommie Meyer Ivan Jose Varzinczak
Mahasarakham UniversityThailand
Meraka Institute, CSIRPretoria, South Africa
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 1 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 2 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 2 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 2 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 3 / 26
Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ? ϕ, with K ? ϕ |= ϕ and K ? ϕ 6|= ⊥Contraction: K − ϕLevi Identity: K ? ϕ = K − ¬ϕ+ ϕ
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 4 / 26
Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ? ϕ, with K ? ϕ |= ϕ and K ? ϕ 6|= ⊥Contraction: K − ϕLevi Identity: K ? ϕ = K − ¬ϕ+ ϕ
Also meaningful for ontologies
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 4 / 26
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) Horn Contraction 5 / 26
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) Horn Contraction 5 / 26
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
Full meet: Pick all remainder setsMaxichoice: Pick a single remainder set
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 6 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 7 / 26
Horn Clauses and Theories
A Horn clause is a sentence p1 ∧ p2 ∧ . . . ∧ pn → q, n ≥ 0
q may be ⊥pi may be >A Horn theory is a set of Horn clauses
Same semantics as PL
Horn belief sets: closed Horn theories, containing only Horn clauses
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 8 / 26
Horn Clauses and Theories
A Horn clause is a sentence p1 ∧ p2 ∧ . . . ∧ pn → q, n ≥ 0q may be ⊥pi may be >A Horn theory is a set of Horn clausesSame semantics as PLHorn belief sets: closed Horn theories, containing only Horn clauses
Example
H = Cn({p → q, q → r})
p → q
>q → r
p ∧ r → q
p → r
p ∧ q → r
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 8 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 9 / 26
Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H with ΦI we want H 6|= ΦI Some clause in Φ should not follow from H anymore
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 10 / 26
Delgrande’s Approach
Definition (Horn e-Remainder Sets [Delgrande, KR’2008])
For a belief set H, X ∈ H ↓e Φ iff
X ⊆ H
X 6|= Φ
for every X ′ s.t. X ⊂ X ′ ⊆ H, X ′ |= Φ.
We call H ↓e Φ the Horn e-remainder sets of H w.r.t. Φ
Definition (Horn e-Selection Functions)
A Horn e-selection function σ is a function from P(P(LH)) toP(P(LH)) s.t. σ(H ↓e Φ) = {H} if H ↓e Φ = ∅, and σ(H ↓e Φ) ⊆ H ↓e Φotherwise.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 11 / 26
Delgrande’s Approach
Definition (Horn e-Remainder Sets [Delgrande, KR’2008])
For a belief set H, X ∈ H ↓e Φ iff
X ⊆ H
X 6|= Φ
for every X ′ s.t. X ⊂ X ′ ⊆ H, X ′ |= Φ.
We call H ↓e Φ the Horn e-remainder sets of H w.r.t. Φ
Definition (Horn e-Selection Functions)
A Horn e-selection function σ is a function from P(P(LH)) toP(P(LH)) s.t. σ(H ↓e Φ) = {H} if H ↓e Φ = ∅, and σ(H ↓e Φ) ⊆ H ↓e Φotherwise.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 11 / 26
Delgrande’s Approach
Definition (Partial Meet Horn e-Contraction)
Given a Horn e-selection function σ, −σ is a partial meet Horne-contraction iff H −σ Φ =
⋂σ(H ↓e Φ).
Definition (Maxichoice and Full Meet)
Given a Horn e-selection function σ, −σ is a maxichoice Horne-contraction iff σ(H ↓Φ) is a singleton set. It is a full meet Horne-contraction iff σ(H ↓e Φ) = H ↓e Φ when H ↓e Φ 6= ∅.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 12 / 26
Delgrande’s Approach
Definition (Partial Meet Horn e-Contraction)
Given a Horn e-selection function σ, −σ is a partial meet Horne-contraction iff H −σ Φ =
⋂σ(H ↓e Φ).
Definition (Maxichoice and Full Meet)
Given a Horn e-selection function σ, −σ is a maxichoice Horne-contraction iff σ(H ↓Φ) is a singleton set. It is a full meet Horne-contraction iff σ(H ↓e Φ) = H ↓e Φ when H ↓e Φ 6= ∅.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 12 / 26
Delgrande’s Approach
Example
e-contraction of {p → r} from H = 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) Horn Contraction 13 / 26
Delgrande’s Approach
Example
e-contraction of {p → r} from H = 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) Horn Contraction 13 / 26
Delgrande’s Approach
Example
e-contraction of {p → r} from H = 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) Horn Contraction 13 / 26
Delgrande’s Approach
Example
e-contraction of {p → r} from H = 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})What about H′ = Cn({p ∧ r → q, p ∧ q → r})?
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 13 / 26
Delgrande’s Approach
Example
e-contraction of {p → r} from H = 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})What about H′ = Cn({p ∧ r → q, p ∧ q → r})?Hfm ⊆ H′ ⊆ H2
mc , but there is no partial meet e-contraction yielding H′!
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 13 / 26
Beyond Partial Meet
Definition (Infra e-Remainder Sets)
For belief sets H and X , X ∈ H ⇓e Φ iff there is some X ′ ∈ H ↓e Φ s.t.(⋂
H ↓e Φ) ⊆ X ⊆ X ′.
We call H ⇓e Φ the infra e-remainder sets of H w.r.t. Φ.Infra e-remainder sets contain all belief sets between some Horne-remainder set and the intersection of all Horn e-remainder sets
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 14 / 26
Beyond Partial Meet
Definition (Horn e-Contraction)
An infra e-selection function τ is a function from P(P(LH)) to P(LH)s.t. τ(H ⇓e Φ) = H whenever |= Φ, and τ(H ⇓e Φ) ∈ H ⇓e Φ otherwise. Acontraction function −τ is a Horn e-contraction iff H −τ Φ = τ(H ⇓e Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 15 / 26
A Representation Result
Postulates for Horn e-contraction
(H −e 1) H −e Φ = Cn(H −e Φ)
(H −e 2) H −e Φ ⊆ H
(H −e 3) If Φ 6⊆ H then H −e Φ = H
(H −e 4) If 6|= Φ then Φ 6⊆ H −e Φ
(H −e 5) If Cn(Φ) = Cn(Ψ) then H −e Φ = H −e Ψ
(H −e 6) If ϕ ∈ H \ (H −e Φ) then there is a H ′ such that⋂(H ↓e Φ) ⊆ H ′ ⊆ H, H ′ 6|= Φ, and H ′ + {ϕ} |= Φ
(H −e 7) If |= Φ then H −e Φ = H
Theorem
Every Horn e-contraction satisfies (H −e 1)–(H −e 7). Conversely, everycontraction function satisfying (H −e 1)–(H −e 7) is a Horn e-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 16 / 26
A Representation Result
Postulates for Horn e-contraction
(H −e 1) H −e Φ = Cn(H −e Φ)
(H −e 2) H −e Φ ⊆ H
(H −e 3) If Φ 6⊆ H then H −e Φ = H
(H −e 4) If 6|= Φ then Φ 6⊆ H −e Φ
(H −e 5) If Cn(Φ) = Cn(Ψ) then H −e Φ = H −e Ψ
(H −e 6) If ϕ ∈ H \ (H −e Φ) then there is a H ′ such that⋂(H ↓e Φ) ⊆ H ′ ⊆ H, H ′ 6|= Φ, and H ′ + {ϕ} |= Φ
(H −e 7) If |= Φ then H −e Φ = H
Theorem
Every Horn e-contraction satisfies (H −e 1)–(H −e 7). Conversely, everycontraction function satisfying (H −e 1)–(H −e 7) is a Horn e-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 16 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 17 / 26
Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for ΦI we want H ′ + Φ 6|= ⊥I Delgrande’s notation: (H −i Φ) + Φ 6|= ⊥
Definition (Horn i -Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆ H
X + Φ 6|= ⊥for every X ′ s.t. X ⊂ X ′ ⊆ H, X ′ + Φ |= ⊥.
We call H ↓i Φ the Horn i-remainder sets of H w.r.t. Φ.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 18 / 26
Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for ΦI we want H ′ + Φ 6|= ⊥I Delgrande’s notation: (H −i Φ) + Φ 6|= ⊥
Definition (Horn i -Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆ H
X + Φ 6|= ⊥for every X ′ s.t. X ⊂ X ′ ⊆ H, X ′ + Φ |= ⊥.
We call H ↓i Φ the Horn i-remainder sets of H w.r.t. Φ.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 18 / 26
Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for ΦI we want H ′ + Φ 6|= ⊥I Delgrande’s notation: (H −i Φ) + Φ 6|= ⊥
Definition (Horn i -Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆ H
X + Φ 6|= ⊥for every X ′ s.t. X ⊂ X ′ ⊆ H, X ′ + Φ |= ⊥.
We call H ↓i Φ the Horn i-remainder sets of H w.r.t. Φ.
H ↓i Φ = ∅ iff Φ |= ⊥
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 18 / 26
Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for ΦI we want H ′ + Φ 6|= ⊥I Delgrande’s notation: (H −i Φ) + Φ 6|= ⊥
Definition (Horn i -Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆ H
X + Φ 6|= ⊥for every X ′ s.t. X ⊂ X ′ ⊆ H, X ′ + Φ |= ⊥.
We call H ↓i Φ the Horn i-remainder sets of H w.r.t. Φ.
H ↓i Φ = ∅ iff Φ |= ⊥Other definitions analogous to Horn e-contraction
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 18 / 26
Beyond Partial Meet
Definition (Infra i -Remainder Sets)
For belief sets H and X , X ∈ H ⇓i Φ iff there is some X ′ ∈ H ↓i Φ s.t.(⋂
H ↓i Φ) ⊆ X ⊆ X ′.
We call H ⇓i Φ the infra i-remainder sets of H w.r.t. Φ.
Definition (Horn i -Contraction)
An infra i-selection function τ is a function from P(P(LH)) to P(LH)s.t. τ(H ⇓i Φ) = H whenever Φ |= ⊥, and τ(H ⇓i Φ) ∈ H ⇓i Φ otherwise. Acontraction function −τ is a Horn i-contraction iff H −τ Φ = τ(H ⇓i Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 19 / 26
Beyond Partial Meet
Definition (Infra i -Remainder Sets)
For belief sets H and X , X ∈ H ⇓i Φ iff there is some X ′ ∈ H ↓i Φ s.t.(⋂
H ↓i Φ) ⊆ X ⊆ X ′.
We call H ⇓i Φ the infra i-remainder sets of H w.r.t. Φ.
Definition (Horn i -Contraction)
An infra i-selection function τ is a function from P(P(LH)) to P(LH)s.t. τ(H ⇓i Φ) = H whenever Φ |= ⊥, and τ(H ⇓i Φ) ∈ H ⇓i Φ otherwise. Acontraction function −τ is a Horn i-contraction iff H −τ Φ = τ(H ⇓i Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 19 / 26
A Representation Result
Postulates for Horn i-contraction
(H −i 1) H −i Φ = Cn(H −i Φ)
(H −i 2) H −i Φ ⊆ H
(H −i 3) If H + Φ 6|= ⊥ then H −i Φ = H
(H −i 4) If Φ 6|= ⊥ then (H −i Φ) + Φ 6|= ⊥(H −i 5) If Cn(Φ) = Cn(Ψ) then H −i Φ = H −i Ψ
(H −i 6) If ϕ ∈ H \ (H −i Φ), there is a H ′ s.t.⋂
(H ↓i Φ) ⊆ H ′ ⊆ H,H ′ + Φ 6|= ⊥, and H ′ + (Φ ∪ {ϕ}) |= ⊥
(H −i 7) If |= Φ then H −i Φ = H
Theorem
Every Horn i-contraction satisfies (H −i 1)–(H −i 7). Conversely, everycontraction function satisfying (H −i 1)–(H −i 7) is a Horn i-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 20 / 26
A Representation Result
Postulates for Horn i-contraction
(H −i 1) H −i Φ = Cn(H −i Φ)
(H −i 2) H −i Φ ⊆ H
(H −i 3) If H + Φ 6|= ⊥ then H −i Φ = H
(H −i 4) If Φ 6|= ⊥ then (H −i Φ) + Φ 6|= ⊥(H −i 5) If Cn(Φ) = Cn(Ψ) then H −i Φ = H −i Ψ
(H −i 6) If ϕ ∈ H \ (H −i Φ), there is a H ′ s.t.⋂
(H ↓i Φ) ⊆ H ′ ⊆ H,H ′ + Φ 6|= ⊥, and H ′ + (Φ ∪ {ϕ}) |= ⊥
(H −i 7) If |= Φ then H −i Φ = H
Theorem
Every Horn i-contraction satisfies (H −i 1)–(H −i 7). Conversely, everycontraction function satisfying (H −i 1)–(H −i 7) is a Horn i-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 20 / 26
Outline
1 PreliminariesBelief ChangeHorn Logic
2 Propositional Horn ContractionEntailment-based ContractionInconsistency-based ContractionPackage Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 21 / 26
Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H so that none of the clauses in Φ follows from itI Removal of all sentences in Φ from H
Relates to repair of the subsumption hierarchy in EL
Definition (Horn p-Remainder Sets)
For a belief set H, X ∈ H ↓p Φ iff
X ⊆ H
Cn(X ) ∩ Φ = ∅for every X ′ s.t. X ⊂ X ′ ⊆ H, Cn(X ′) ∩ Φ 6= ∅
We call H ↓p Φ the Horn p-remainder sets of H w.r.t. Φ.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 22 / 26
Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H so that none of the clauses in Φ follows from itI Removal of all sentences in Φ from H
Relates to repair of the subsumption hierarchy in EL
Definition (Horn p-Remainder Sets)
For a belief set H, X ∈ H ↓p Φ iff
X ⊆ H
Cn(X ) ∩ Φ = ∅for every X ′ s.t. X ⊂ X ′ ⊆ H, Cn(X ′) ∩ Φ 6= ∅
We call H ↓p Φ the Horn p-remainder sets of H w.r.t. Φ.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 22 / 26
Beyond Partial Meet
Same counter-example
Definition (Infra p-Remainder Sets)
For belief sets H and X , X ∈ H ⇓p Φ iff there is some X ′ ∈ H ↓p Φ s.t.(⋂
H ↓p Φ) ⊆ X ⊆ X ′.
We call H ⇓p Φ the infra p-remainder sets of H w.r.t. Φ.
Definition (Horn p-contraction)
An infra p-selection function τ is a function from P(P(LH)) to P(LH)s.t. τ(H ⇓p Φ) = H whenever |=
∨Φ, and τ(H ⇓p Φ) ∈ H ⇓p Φ otherwise.
A contraction function −τ is a Horn p-contraction iff H −τ Φ = τ(H ⇓p Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 23 / 26
Beyond Partial Meet
Same counter-example
Definition (Infra p-Remainder Sets)
For belief sets H and X , X ∈ H ⇓p Φ iff there is some X ′ ∈ H ↓p Φ s.t.(⋂
H ↓p Φ) ⊆ X ⊆ X ′.
We call H ⇓p Φ the infra p-remainder sets of H w.r.t. Φ.
Definition (Horn p-contraction)
An infra p-selection function τ is a function from P(P(LH)) to P(LH)s.t. τ(H ⇓p Φ) = H whenever |=
∨Φ, and τ(H ⇓p Φ) ∈ H ⇓p Φ otherwise.
A contraction function −τ is a Horn p-contraction iff H −τ Φ = τ(H ⇓p Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 23 / 26
Beyond Partial Meet
Same counter-example
Definition (Infra p-Remainder Sets)
For belief sets H and X , X ∈ H ⇓p Φ iff there is some X ′ ∈ H ↓p Φ s.t.(⋂
H ↓p Φ) ⊆ X ⊆ X ′.
We call H ⇓p Φ the infra p-remainder sets of H w.r.t. Φ.
Definition (Horn p-contraction)
An infra p-selection function τ is a function from P(P(LH)) to P(LH)s.t. τ(H ⇓p Φ) = H whenever |=
∨Φ, and τ(H ⇓p Φ) ∈ H ⇓p Φ otherwise.
A contraction function −τ is a Horn p-contraction iff H −τ Φ = τ(H ⇓p Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 23 / 26
A Representation Result
Postulates for Horn p-contraction
(H −p 1) H −p Φ = Cn(H −p Φ)
(H −p 2) H −p Φ ⊆ H
(H −p 3) If H ∩ Φ = ∅ then H −p Φ = H
(H −p 4) If 6|=∨
Φ then (H −p Φ) ∩ Φ = ∅(H −p 5) If
∨Φ ≡
∨Ψ then H −p Φ = H −p Ψ
(H −p 6) If ϕ ∈ H \ (H −p Φ), there is a H ′ s.t.⋂
(H ↓p Φ) ⊆ H ′ ⊆ H,Cn(H ′) ∩ Φ = ∅, and (H ′ + ϕ) ∩ Φ 6= ∅
(H −p 7) If |=∨
Φ then H −p Φ = H
Theorem
Every Horn p-contraction satisfies (H −p 1)–(H −p 7). Conversely, everycontraction function satisfying (H −p 1)–(H −p 7) is a Horn p-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 24 / 26
A Representation Result
Postulates for Horn p-contraction
(H −p 1) H −p Φ = Cn(H −p Φ)
(H −p 2) H −p Φ ⊆ H
(H −p 3) If H ∩ Φ = ∅ then H −p Φ = H
(H −p 4) If 6|=∨
Φ then (H −p Φ) ∩ Φ = ∅(H −p 5) If
∨Φ ≡
∨Ψ then H −p Φ = H −p Ψ
(H −p 6) If ϕ ∈ H \ (H −p Φ), there is a H ′ s.t.⋂
(H ↓p Φ) ⊆ H ′ ⊆ H,Cn(H ′) ∩ Φ = ∅, and (H ′ + ϕ) ∩ Φ 6= ∅
(H −p 7) If |=∨
Φ then H −p Φ = H
Theorem
Every Horn p-contraction satisfies (H −p 1)–(H −p 7). Conversely, everycontraction function satisfying (H −p 1)–(H −p 7) is a Horn p-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 24 / 26
p-contraction as i -contraction
Considering basic Horn clauses: p → q
Φ = {p1 → q1, . . . , pn → qn}i(Φ) = {p1, . . . , pn, q1 → ⊥, . . . , qn → ⊥}
Theorem
Let H be a Horn belief set and let Φ be a set of basic Horn clauses. ThenK −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A v B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26
p-contraction as i -contraction
Considering basic Horn clauses: p → q
Φ = {p1 → q1, . . . , pn → qn}i(Φ) = {p1, . . . , pn, q1 → ⊥, . . . , qn → ⊥}
Theorem
Let H be a Horn belief set and let Φ be a set of basic Horn clauses. ThenK −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A v B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26
p-contraction as i -contraction
Considering basic Horn clauses: p → q
Φ = {p1 → q1, . . . , pn → qn}i(Φ) = {p1, . . . , pn, q1 → ⊥, . . . , qn → ⊥}
Theorem
Let H be a Horn belief set and let Φ be a set of basic Horn clauses. ThenK −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A v B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26
p-contraction as i -contraction
Considering basic Horn clauses: p → q
Φ = {p1 → q1, . . . , pn → qn}i(Φ) = {p1, . . . , pn, q1 → ⊥, . . . , qn → ⊥}
Theorem
Let H be a Horn belief set and let Φ be a set of basic Horn clauses. ThenK −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A v B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26
Conclusion
Contribution:
Basic AGM account of e-, i- and p-contraction for Horn Logic
Weaker than partial meet contraction
Current and Future Work
Full AGM setting: extended postulates
Extension to ELProtege Plugin for repairing the subsumption hierarchy
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 26 / 26
Conclusion
Contribution:
Basic AGM account of e-, i- and p-contraction for Horn Logic
Weaker than partial meet contraction
Current and Future Work
Full AGM setting: extended postulates
Extension to ELProtege Plugin for repairing the subsumption hierarchy
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 26 / 26