Ivan Varzinczak

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


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Revision, Expansion and Contraction

Expansion: K + ϕ

Revision: K ? ϕ, with K ? ϕ |= ϕ and K ? ϕ 6|= ⊥Contraction: K − ϕ 6|= ϕ


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


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 setsI Take their intersection: Partial meet

Pick all remainder sets: Full meet

Pick a single remainder set: Maxichoice


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?


AGM Approach

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?


AGM Approach

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? Hfm = Cn({p ∧ r → q})


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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}


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}


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


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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


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.





X ⊆ TX 6|= Φ










X ⊆ TX 6|= Φ







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= ∅.



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= ∅.



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?



Example


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?



Example


A v B >

B v ∃R.A

A u ∃R.A v B

A v ∃R.A

A u B v ∃R.A


mc = Cn({B v ∃R.A,A u ∃R.A v B})Full meet? Tfm = Cn({A u ∃R.A v B})



Example


A v B >

B v ∃R.A

A u ∃R.A v B

A v ∃R.A

A u B v ∃R.A


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})?



Example


A v B >

B v ∃R.A

A u ∃R.A v B

A v ∃R.A

A u B v ∃R.A


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 ′!


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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


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).


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.


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.


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Other Types of Contraction

Inconsistency-based Contraction [Delgrande, KR’2008]


Contract T ‘making room’ for Φ

We want T ′ + Φ 6|= ⊥

Package Contraction [Booth et al., IJCAI’09]


Contract T so that none of the axioms in Φ follows from it

Removal of all sentences in Φ from T


Other Types of Contraction

Inconsistency-based Contraction [Delgrande, KR’2008]


Contract T ‘making room’ for Φ

We want T ′ + Φ 6|= ⊥

Package Contraction [Booth et al., IJCAI’09]


Contract T so that none of the axioms in Φ follows from it

Removal of all sentences in Φ from T


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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



Summary


Open questions











Summary


Open questions










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Ivan Varzinczak