next steps in propositional horn contraction

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

Outline

3 Conclusion

Outline

3 Conclusion

Outline

3 Conclusion

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

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

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

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

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

Outline

3 Conclusion

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

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

Outline

3 Conclusion

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

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.

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.

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= ∅.

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= ∅.

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?

Example

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?

Example

p → q

>q → r

p ∧ r → q

p → r

p ∧ q → r

mc = Cn({q → r , p ∧ r → q})Full meet? Hfm = Cn({p ∧ r → q})

Example

p → q

>q → r

p ∧ r → q

p → r

p ∧ q → r

mc = Cn({q → r , p ∧ r → q})Full meet? Hfm = Cn({p ∧ r → q})What about H′ = Cn({p ∧ r → q, p ∧ q → r})?

Example

p → q

>q → r

p ∧ r → q

p → r

p ∧ q → r

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′!

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

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).

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.

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.

Outline

3 Conclusion

Motivation

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.

Motivation

X ⊆ H

Motivation

X ⊆ H

H ↓i Φ = ∅ iff Φ |= ⊥

Motivation

X ⊆ H

H ↓i Φ = ∅ iff Φ |= ⊥Other definitions analogous to Horn e-contraction

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).

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).

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.

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.

Outline

3 Conclusion

Motivation

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.

Motivation

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.

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).

Beyond Partial Meet

H ↓p Φ) ⊆ X ⊆ X ′.

Beyond Partial Meet

H ↓p Φ) ⊆ X ⊆ X ′.

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.

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.

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

Theorem

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

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

next steps in propositional horn contraction

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