The document discusses approaches to Horn contraction, which is a type of belief change where clauses are removed from a Horn theory or belief set. It presents Delgrande's approach to entailment-based Horn contraction using Horn e-remainder sets. An example is used to illustrate maxichoice, full meet, and limitations of the partial meet construction. The concept of infra e-remainder sets is introduced to address these limitations and define a more general Horn contraction operator.

1. Next Steps in Propositional Horn Contraction
Richard Booth Tommie Meyer Ivan Jos´ Varzinczak
e
Mahasarakham University Meraka Institute, CSIR
Thailand Pretoria, South Africa
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 1 / 26

2. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 2 / 26

3. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 2 / 26

4. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 2 / 26

5. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 3 / 26

6. Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ϕ, with K ϕ |= ϕ and K ϕ |= ⊥
Contraction: K − ϕ
Levi Identity: K ϕ = K − ¬ϕ + ϕ
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 4 / 26

7. Revision, Expansion and Contraction
Expansion: K + ϕ
Revision: K ϕ, with K ϕ |= ϕ and K ϕ |= ⊥
Contraction: K − ϕ
Levi Identity: K ϕ = K − ¬ϕ + ϕ
Also meaningful for ontologies
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 4 / 26

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) Horn Contraction 5 / 26

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) Horn Contraction 5 / 26

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
Full meet: Pick all remainder sets
Maxichoice: Pick a single remainder set
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 6 / 26

11. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 7 / 26

12. 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

13. 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
Example
H = Cn({p → q, q → r })
p∧r →q p∧q →r
p→q q→r
p→r
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 8 / 26

14. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 9 / 26

15. Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H with Φ
we want H |= Φ
Some clause in Φ should not follow from H anymore
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 10 / 26

16. Delgrande’s Approach
Definition (Horn e-Remainder Sets [Delgrande, KR’2008])
For a belief set H, X ∈ H ↓e Φ iff
X ⊆H
X |= 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 )) to
P(P(LH )) s.t. σ(H ↓e Φ) = {H} if H ↓e Φ = ∅, and σ(H ↓e Φ) ⊆ H ↓e Φ
otherwise.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 11 / 26

17. Delgrande’s Approach
Definition (Horn e-Remainder Sets [Delgrande, KR’2008])
For a belief set H, X ∈ H ↓e Φ iff
X ⊆H
X |= 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 )) to
P(P(LH )) s.t. σ(H ↓e Φ) = {H} if H ↓e Φ = ∅, and σ(H ↓e Φ) ⊆ H ↓e Φ
otherwise.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 11 / 26

18. Delgrande’s Approach
Definition (Partial Meet Horn e-Contraction)
Given a Horn e-selection function σ, −σ is a partial meet Horn
e-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 Horn
e-contraction iff σ(H ↓ Φ) is a singleton set. It is a full meet Horn
e-contraction iff σ(H ↓e Φ) = H ↓e Φ when H ↓e Φ = ∅.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 12 / 26

19. Delgrande’s Approach
Definition (Partial Meet Horn e-Contraction)
Given a Horn e-selection function σ, −σ is a partial meet Horn
e-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 Horn
e-contraction iff σ(H ↓ Φ) is a singleton set. It is a full meet Horn
e-contraction iff σ(H ↓e Φ) = H ↓e Φ when H ↓e Φ = ∅.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 12 / 26

20. Delgrande’s Approach
Example
e-contraction of {p → r } from H = 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) Horn Contraction 13 / 26

21. Delgrande’s Approach
Example
e-contraction of {p → r } from H = 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) Horn Contraction 13 / 26

22. Delgrande’s Approach
Example
e-contraction of {p → r } from H = 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) Horn Contraction 13 / 26

23. Delgrande’s Approach
Example
e-contraction of {p → r } from H = 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})
What about H = Cn({p ∧ r → q, p ∧ q → r })?
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 13 / 26

24. Delgrande’s Approach
Example
e-contraction of {p → r } from H = 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})
What about H = Cn({p ∧ r → q, p ∧ q → r })?
2
Hfm ⊆ H ⊆ Hmc , but there is no partial meet e-contraction yielding H !
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 13 / 26

25. 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 Horn
e-remainder set and the intersection of all Horn e-remainder sets
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 14 / 26

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. A
contraction function −τ is a Horn e-contraction iff H −τ Φ = τ (H ⇓e Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 15 / 26

27. 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 ⊆ H then H −e Φ = H
(H −e 4) If |= Φ then Φ ⊆ 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 |= 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, every
contraction function satisfying (H −e 1)–(H −e 7) is a Horn e-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 16 / 26

28. 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 ⊆ H then H −e Φ = H
(H −e 4) If |= Φ then Φ ⊆ 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 |= 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, every
contraction function satisfying (H −e 1)–(H −e 7) is a Horn e-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 16 / 26

29. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 17 / 26

30. Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for Φ
we want H + Φ |= ⊥
Delgrande’s notation: (H −i Φ) + Φ |= ⊥
Definition (Horn i-Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆H
X + Φ |= ⊥
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

31. Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for Φ
we want H + Φ |= ⊥
Delgrande’s notation: (H −i Φ) + Φ |= ⊥
Definition (Horn i-Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆H
X + Φ |= ⊥
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

32. Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for Φ
we want H + Φ |= ⊥
Delgrande’s notation: (H −i Φ) + Φ |= ⊥
Definition (Horn i-Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆H
X + Φ |= ⊥
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

33. Motivation
Let H be a Horn theory and Φ be a set of clauses
Contract H ‘making room’ for Φ
we want H + Φ |= ⊥
Delgrande’s notation: (H −i Φ) + Φ |= ⊥
Definition (Horn i-Remainder Sets)
For a belief set H, X ∈ H ↓i Φ iff
X ⊆H
X + Φ |= ⊥
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

34. 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. A
contraction function −τ is a Horn i-contraction iff H −τ Φ = τ (H ⇓i Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 19 / 26

35. 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. A
contraction function −τ is a Horn i-contraction iff H −τ Φ = τ (H ⇓i Φ).
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 19 / 26

36. 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 |= ⊥ then H −i Φ = H
(H −i 4) If Φ |= ⊥ then (H −i Φ) + Φ |= ⊥
(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 |= ⊥, 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, every
contraction function satisfying (H −i 1)–(H −i 7) is a Horn i-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 20 / 26

37. 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 |= ⊥ then H −i Φ = H
(H −i 4) If Φ |= ⊥ then (H −i Φ) + Φ |= ⊥
(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 |= ⊥, 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, every
contraction function satisfying (H −i 1)–(H −i 7) is a Horn i-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 20 / 26

38. Outline
1 Preliminaries
Belief Change
Horn Logic
2 Propositional Horn Contraction
Entailment-based Contraction
Inconsistency-based Contraction
Package Contraction
3 Conclusion
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 21 / 26

39. 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 it
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 = ∅
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

40. 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 it
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 = ∅
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

41. 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

42. 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

43. 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

44. 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 |= 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 = ∅
(H −p 7) If |= Φ then H −p Φ = H
Theorem
Every Horn p-contraction satisfies (H −p 1)–(H −p 7). Conversely, every
contraction function satisfying (H −p 1)–(H −p 7) is a Horn p-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 24 / 26

45. 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 |= 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 = ∅
(H −p 7) If |= Φ then H −p Φ = H
Theorem
Every Horn p-contraction satisfies (H −p 1)–(H −p 7). Conversely, every
contraction function satisfying (H −p 1)–(H −p 7) is a Horn p-contraction.
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 24 / 26

46. 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. Then
K −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26

47. 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. Then
K −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26

48. 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. Then
K −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26

49. 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. Then
K −p Φ = K −i i(Φ).
Links to basic subsumption statements in EL: A B
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 25 / 26

50. 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 EL
Prot´g´ Plugin for repairing the subsumption hierarchy
e e
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 26 / 26

51. 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 EL
Prot´g´ Plugin for repairing the subsumption hierarchy
e e
Booth, Meyer, Varzinczak (MU/KSG) Horn Contraction 26 / 26