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# Next Steps in Propositional Horn Contraction

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### Next Steps in Propositional Horn Contraction

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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