Making Repairs in Description Logics More Gentle Franz Baader 1 - - PowerPoint PPT Presentation

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Making Repairs in Description Logics More Gentle Franz Baader 1 - - PowerPoint PPT Presentation

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Making Repairs in Description Logics More Gentle Franz Baader 1 Francesco Kriegel 1 Adrian Nuradiansyah 1 Rafael Pealoza 2 1 TU Dresden 2 Free University of Bolzano November 1, 2018 Adrian Nuradiansyah KR 2018 November 1, 2018 1 / 17

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Making Repairs in Description Logics More Gentle

Franz Baader 1 Francesco Kriegel 1 Adrian Nuradiansyah 1 Rafael Peñaloza 2

1TU Dresden

2Free University of Bolzano

November 1, 2018

Adrian Nuradiansyah KR 2018 November 1, 2018 1 / 17

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Motivation

Reasoning in large ontologies O may provide unintended consequences α ⇒ O contains errors. In privacy setting, some (correct) consequences α should be hidden from attackers. If O | = α and α is unwanted, then let us repair O to O′ such that O′ | = α

Adrian Nuradiansyah KR 2018 November 1, 2018 2 / 17

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Motivation

Reasoning in large ontologies O may provide unintended consequences α ⇒ O contains errors. In privacy setting, some (correct) consequences α should be hidden from attackers. If O | = α and α is unwanted, then let us repair O to O′ such that O′ | = α

What people already did:

In (Schlobach et al. 2003), (Kalyanpur et al. 2007), (Meyer et al. 2006), etc Understand the reasons why O | = α ⇒ Justifications. Using those reasons and deleting a minimal number of axioms to repair O.

Adrian Nuradiansyah KR 2018 November 1, 2018 2 / 17

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Motivation

Reasoning in large ontologies O may provide unintended consequences α ⇒ O contains errors. In privacy setting, some (correct) consequences α should be hidden from attackers. If O | = α and α is unwanted, then let us repair O to O′ such that O′ | = α

What people already did:

In (Schlobach et al. 2003), (Kalyanpur et al. 2007), (Meyer et al. 2006), etc Understand the reasons why O | = α ⇒ Justifications. Using those reasons and deleting a minimal number of axioms to repair O.

What we want to do:

Instead of removing axioms, we propose axiom weakenings. Addressed in the context of Description Logic Ontologies

Adrian Nuradiansyah KR 2018 November 1, 2018 2 / 17

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EL Ontologies

EL-concepts C, D ::= ⊤ | A | C ⊓ D | ∃r.C. Inexpressive, but reasoning can be done in polynomial time. Mainly used in medical ontologies, e.g., SNOMED, GeneOntology, etc.

Adrian Nuradiansyah KR 2018 November 1, 2018 3 / 17

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EL Ontologies

EL-concepts C, D ::= ⊤ | A | C ⊓ D | ∃r.C. Inexpressive, but reasoning can be done in polynomial time. Mainly used in medical ontologies, e.g., SNOMED, GeneOntology, etc. An ontology O consists of TBox T and ABox A. A TBox T is a finite set of General Concept Inclusions (GCIs) C ⊑ D → Background knowledge An ABox A is a finite set of concept assertions C(a) and role assertions r(a, b) → Knowledge about individuals

Adrian Nuradiansyah KR 2018 November 1, 2018 3 / 17

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Ontology Repair

Assumptions:

O = Os ∪ Or, where Os is a static ontology and Or is a refutable ontology. Only the refutable part may be changed and Os | = α

Adrian Nuradiansyah KR 2018 November 1, 2018 4 / 17

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Ontology Repair

Assumptions:

O = Os ∪ Or, where Os is a static ontology and Or is a refutable ontology. Only the refutable part may be changed and Os | = α

Ontology Repair

Let Con(O) := {α | O | = α} be the set of all consequences of O.

Adrian Nuradiansyah KR 2018 November 1, 2018 4 / 17

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Ontology Repair

Assumptions:

O = Os ∪ Or, where Os is a static ontology and Or is a refutable ontology. Only the refutable part may be changed and Os | = α

Ontology Repair

Let Con(O) := {α | O | = α} be the set of all consequences of O. Let O | = α and Os | = α. The ontology O’ is a repair of O w.r.t. α if Con(Os ∪ O′) ⊆ Con(O) \ {α}

Adrian Nuradiansyah KR 2018 November 1, 2018 4 / 17

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Ontology Repair

Assumptions:

O = Os ∪ Or, where Os is a static ontology and Or is a refutable ontology. Only the refutable part may be changed and Os | = α

Ontology Repair

Let Con(O) := {α | O | = α} be the set of all consequences of O. Let O | = α and Os | = α. The ontology O’ is a repair of O w.r.t. α if Con(Os ∪ O′) ⊆ Con(O) \ {α} Optimal repair O′ of O w.r.t. α: No Repair O′′ of O w.r.t. α having more consequences than O′.

Adrian Nuradiansyah KR 2018 November 1, 2018 4 / 17

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Ontology Repair

Assumptions:

O = Os ∪ Or, where Os is a static ontology and Or is a refutable ontology. Only the refutable part may be changed and Os | = α

Ontology Repair

Let Con(O) := {α | O | = α} be the set of all consequences of O. Let O | = α and Os | = α. The ontology O’ is a repair of O w.r.t. α if Con(Os ∪ O′) ⊆ Con(O) \ {α} Optimal repair O′ of O w.r.t. α: No Repair O′′ of O w.r.t. α having more consequences than O′.

Theorem (Existence of Optimal Repairs)

Optimal repairs need not exist!

Adrian Nuradiansyah KR 2018 November 1, 2018 4 / 17

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Ontology Repair

Assumptions:

O = Os ∪ Or, where Os is a static ontology and Or is a refutable ontology. Only the refutable part may be changed and Os | = α

Ontology Repair

Let Con(O) := {α | O | = α} be the set of all consequences of O. Let O | = α and Os | = α. The ontology O’ is a repair of O w.r.t. α if Con(Os ∪ O′) ⊆ Con(O) \ {α} Optimal repair O′ of O w.r.t. α: No Repair O′′ of O w.r.t. α having more consequences than O′.

Theorem (Existence of Optimal Repairs)

Optimal repairs need not exist!

Consider: T := {A ⊑ ∃r.A, ∃r.A ⊑ A} A := {A(a)} α = A(a) If Or := A, then an optimal repair must contain ((∃r.)n⊤)(a) for infinitely many n

Adrian Nuradiansyah KR 2018 November 1, 2018 4 / 17

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Optimal Classical Repair

Optimal Classical Repair

The repair O′ is a classical repair of O w.r.t. α if O′ ⊂ Or. Optimal classical repair O′ of O w.r.t. α: No classical repair O′′ having more axioms than O′.

Adrian Nuradiansyah KR 2018 November 1, 2018 5 / 17

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Optimal Classical Repair

Optimal Classical Repair

The repair O′ is a classical repair of O w.r.t. α if O′ ⊂ Or. Optimal classical repair O′ of O w.r.t. α: No classical repair O′′ having more axioms than O′. Optimal classical repairs always exist → Justification and Hitting Set. (Reiter, 1987)

Adrian Nuradiansyah KR 2018 November 1, 2018 5 / 17

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Optimal Classical Repair

Optimal Classical Repair

The repair O′ is a classical repair of O w.r.t. α if O′ ⊂ Or. Optimal classical repair O′ of O w.r.t. α: No classical repair O′′ having more axioms than O′. Optimal classical repairs always exist → Justification and Hitting Set. (Reiter, 1987) Let O | = α. A justification J of O w.r.t. α is a minimal subset of Or s.t. Os ∪ J | = α. Let J1, . . . , Jk be the justifications of O w.r.t. α. A hitting set H of J1, . . . , Jk is a set of axioms such that H ∩ Ji = ∅

Adrian Nuradiansyah KR 2018 November 1, 2018 5 / 17

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Optimal Classical Repair

Optimal Classical Repair

The repair O′ is a classical repair of O w.r.t. α if O′ ⊂ Or. Optimal classical repair O′ of O w.r.t. α: No classical repair O′′ having more axioms than O′. Optimal classical repairs always exist → Justification and Hitting Set. (Reiter, 1987) Let O | = α. A justification J of O w.r.t. α is a minimal subset of Or s.t. Os ∪ J | = α. Let J1, . . . , Jk be the justifications of O w.r.t. α. A hitting set H of J1, . . . , Jk is a set of axioms such that H ∩ Ji = ∅ A hitting set Hmin is minimal if there is no H′ of J1, . . . , Jk such that H′ ⊂ Hmin.

Adrian Nuradiansyah KR 2018 November 1, 2018 5 / 17

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Optimal Classical Repair

Optimal Classical Repair

The repair O′ is a classical repair of O w.r.t. α if O′ ⊂ Or. Optimal classical repair O′ of O w.r.t. α: No classical repair O′′ having more axioms than O′. Optimal classical repairs always exist → Justification and Hitting Set. (Reiter, 1987) Let O | = α. A justification J of O w.r.t. α is a minimal subset of Or s.t. Os ∪ J | = α. Let J1, . . . , Jk be the justifications of O w.r.t. α. A hitting set H of J1, . . . , Jk is a set of axioms such that H ∩ Ji = ∅ A hitting set Hmin is minimal if there is no H′ of J1, . . . , Jk such that H′ ⊂ Hmin. O′ := Or \ Hmin is an optimal classical repair of O w.r.t. α such that Os ∪ O′ | = α

Adrian Nuradiansyah KR 2018 November 1, 2018 5 / 17

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Gentle Repair

Obtaining Classical Repairs → removing axioms from O. Instead, we want to weaken axioms in Hmin! Given axioms β, γ, an axiom γ is weaker than β if Con({γ}) ⊂ Con({β})

Adrian Nuradiansyah KR 2018 November 1, 2018 6 / 17

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Gentle Repair

Obtaining Classical Repairs → removing axioms from O. Instead, we want to weaken axioms in Hmin! Given axioms β, γ, an axiom γ is weaker than β if Con({γ}) ⊂ Con({β})

Illustration

Os := {∃owns.(GermanCar ⊓ Diesel) ⊑ ∃gets.Compensations} Or := {GermanTaxiDriver ⊑ ∃owns.(GermanCar ⊓ Diesel).}

Every German taxi driver gets compensation w.r.t. Os ∪ Or.

Adrian Nuradiansyah KR 2018 November 1, 2018 6 / 17

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Gentle Repair

Obtaining Classical Repairs → removing axioms from O. Instead, we want to weaken axioms in Hmin! Given axioms β, γ, an axiom γ is weaker than β if Con({γ}) ⊂ Con({β})

Illustration

Os := {∃owns.(GermanCar ⊓ Diesel) ⊑ ∃gets.Compensations} Or := {GermanTaxiDriver ⊑ ∃owns.(GermanCar ⊓ Diesel).}

Every German taxi driver gets compensation w.r.t. Os ∪ Or. Classical: Removes β ∈ Or. Removes the correct consequence: GermanTaxiDriver ⊑ ∃owns.GermanCar.

Adrian Nuradiansyah KR 2018 November 1, 2018 6 / 17

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Gentle Repair

Obtaining Classical Repairs → removing axioms from O. Instead, we want to weaken axioms in Hmin! Given axioms β, γ, an axiom γ is weaker than β if Con({γ}) ⊂ Con({β})

Illustration

Os := {∃owns.(GermanCar ⊓ Diesel) ⊑ ∃gets.Compensations} Or := {GermanTaxiDriver ⊑ ∃owns.(GermanCar ⊓ Diesel).}

Every German taxi driver gets compensation w.r.t. Os ∪ Or. Classical: Removes β ∈ Or. Removes the correct consequence: GermanTaxiDriver ⊑ ∃owns.GermanCar. Gentle: Weaken β to GermanTaxiDriver ⊑ ∃owns.GermanCar. But, this consequence GermanTaxiDriver ⊑ ∃owns.Diesel is also gone.

Adrian Nuradiansyah KR 2018 November 1, 2018 6 / 17

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Gentle Repair

Obtaining Classical Repairs → removing axioms from O. Instead, we want to weaken axioms in Hmin! Given axioms β, γ, an axiom γ is weaker than β if Con({γ}) ⊂ Con({β})

Illustration

Os := {∃owns.(GermanCar ⊓ Diesel) ⊑ ∃gets.Compensations} Or := {GermanTaxiDriver ⊑ ∃owns.(GermanCar ⊓ Diesel).}

Every German taxi driver gets compensation w.r.t. Os ∪ Or. Classical: Removes β ∈ Or. Removes the correct consequence: GermanTaxiDriver ⊑ ∃owns.GermanCar. Gentle: Weaken β to GermanTaxiDriver ⊑ ∃owns.GermanCar. But, this consequence GermanTaxiDriver ⊑ ∃owns.Diesel is also gone. More gentle: Weaken β to GermanTaxiDriver ⊑ ∃owns.GermanCar ⊓ ∃owns.Diesel

Adrian Nuradiansyah KR 2018 November 1, 2018 6 / 17

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Previous Works on Weakening Axioms

In (Horridge et.al., 2008) & (Du et.al., 2014), First, specific structural transformations are applied to axioms in O Then, repair this modified ontology using classical repairs It might blow up the size of O before repairing

Adrian Nuradiansyah KR 2018 November 1, 2018 7 / 17

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Previous Works on Weakening Axioms

In (Horridge et.al., 2008) & (Du et.al., 2014), First, specific structural transformations are applied to axioms in O Then, repair this modified ontology using classical repairs It might blow up the size of O before repairing In (Lam et.al., 2008) Using tracing tableau technique from (Baader & Hollunder, 1995) To identify which parts of the axioms involved in deriving α Their approach does not always yield a repair

Adrian Nuradiansyah KR 2018 November 1, 2018 7 / 17

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How do We Make it Gentle?

Gentle Repair Algorithm:

For each β ∈ Hmin and all J1, . . . , Jk containing β ,

Adrian Nuradiansyah KR 2018 November 1, 2018 8 / 17

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How do We Make it Gentle?

Gentle Repair Algorithm:

For each β ∈ Hmin and all J1, . . . , Jk containing β , replace β with exactly one γ, where γ is weaker than β such that

Adrian Nuradiansyah KR 2018 November 1, 2018 8 / 17

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How do We Make it Gentle?

Gentle Repair Algorithm:

For each β ∈ Hmin and all J1, . . . , Jk containing β , replace β with exactly one γ, where γ is weaker than β such that Os ∪ (Ji \ {β}) ∪ {γ} | = α for i = 1, . . . , k. (1)

Adrian Nuradiansyah KR 2018 November 1, 2018 8 / 17

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How do We Make it Gentle?

Gentle Repair Algorithm:

For each β ∈ Hmin and all J1, . . . , Jk containing β , replace β with exactly one γ, where γ is weaker than β such that Os ∪ (Ji \ {β}) ∪ {γ} | = α for i = 1, . . . , k. (1) Construct O’ obtained from Or by replacing each β ∈ Hmin with an appropriate weaker γ satisfying (1).

Adrian Nuradiansyah KR 2018 November 1, 2018 8 / 17

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How do We Make it Gentle?

Gentle Repair Algorithm:

For each β ∈ Hmin and all J1, . . . , Jk containing β , replace β with exactly one γ, where γ is weaker than β such that Os ∪ (Ji \ {β}) ∪ {γ} | = α for i = 1, . . . , k. (1) Construct O’ obtained from Or by replacing each β ∈ Hmin with an appropriate weaker γ satisfying (1). Check whether α is a consequence of Os ∪ O′. Using the algorithm above, α still can be a consequence of Os ∪ O′.

Adrian Nuradiansyah KR 2018 November 1, 2018 8 / 17

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How do We Make it Gentle?

Gentle Repair Algorithm:

For each β ∈ Hmin and all J1, . . . , Jk containing β , replace β with exactly one γ, where γ is weaker than β such that Os ∪ (Ji \ {β}) ∪ {γ} | = α for i = 1, . . . , k. (1) Construct O’ obtained from Or by replacing each β ∈ Hmin with an appropriate weaker γ satisfying (1). Check whether α is a consequence of Os ∪ O′. Using the algorithm above, α still can be a consequence of Os ∪ O′.

Adrian Nuradiansyah KR 2018 November 1, 2018 8 / 17

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Gentle Repairs Need Iterations

Theorem (Termination)

Obtaining gentle repairs needs Iterations until Os ∪ O′ | = α. There is an exponential upper bound on the required number

  • f iterations.

Adrian Nuradiansyah KR 2018 November 1, 2018 9 / 17

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Gentle Repairs Need Iterations

Theorem (Termination)

Obtaining gentle repairs needs Iterations until Os ∪ O′ | = α. There is an exponential upper bound on the required number

  • f iterations.

In (Troquard et.al., 2018) Weakening axioms via refinement operators (Lehmann & Hitzler, 2010). Realized that weakening axioms needs iterations. But, no termination proof.

Adrian Nuradiansyah KR 2018 November 1, 2018 9 / 17

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Weakening Relations

To obtain better bounds on the number of iterations, introduce weakening relations on axioms.

Weakening Relation

The binary relation ≻ on axioms is a weakening relation if β ≻ γ implies that γ is weaker than β;

Adrian Nuradiansyah KR 2018 November 1, 2018 10 / 17

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Weakening Relations

To obtain better bounds on the number of iterations, introduce weakening relations on axioms.

Weakening Relation

The binary relation ≻ on axioms is a weakening relation if β ≻ γ implies that γ is weaker than β; well-founded if there is no infinite ≻-chain β1 ≻ β2 ≻ β3 ≻ . . .;

Adrian Nuradiansyah KR 2018 November 1, 2018 10 / 17

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Weakening Relations

To obtain better bounds on the number of iterations, introduce weakening relations on axioms.

Weakening Relation

The binary relation ≻ on axioms is a weakening relation if β ≻ γ implies that γ is weaker than β; well-founded if there is no infinite ≻-chain β1 ≻ β2 ≻ β3 ≻ . . .; complete if for any axiom β that is not a tautology, there is a tautology γ such that β ≻ γ.

Adrian Nuradiansyah KR 2018 November 1, 2018 10 / 17

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Weakening Relations

To obtain better bounds on the number of iterations, introduce weakening relations on axioms.

Weakening Relation

The binary relation ≻ on axioms is a weakening relation if β ≻ γ implies that γ is weaker than β; well-founded if there is no infinite ≻-chain β1 ≻ β2 ≻ β3 ≻ . . .; complete if for any axiom β that is not a tautology, there is a tautology γ such that β ≻ γ. linear (polynomial) if for every axiom β, the length of the longest chain ≻- generated from β is linearly (polynomially) bounded by the size of β;

Adrian Nuradiansyah KR 2018 November 1, 2018 10 / 17

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Weakening Relations

To obtain better bounds on the number of iterations, introduce weakening relations on axioms.

Weakening Relation

The binary relation ≻ on axioms is a weakening relation if β ≻ γ implies that γ is weaker than β; well-founded if there is no infinite ≻-chain β1 ≻ β2 ≻ β3 ≻ . . .; complete if for any axiom β that is not a tautology, there is a tautology γ such that β ≻ γ. linear (polynomial) if for every axiom β, the length of the longest chain ≻- generated from β is linearly (polynomially) bounded by the size of β;

Theorem (Linearity/Polynomiality)

If ≻ is linear (polynomial) and complete, then the iterative algorithm stops after a linear (polynomial) number of iterations.

Adrian Nuradiansyah KR 2018 November 1, 2018 10 / 17

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Maximally Strong Weakenings

Maximally Strong Weakening Axioms

Let Os ∪ (Ji \ {β}) ∪ {γ} | = α

Adrian Nuradiansyah KR 2018 November 1, 2018 11 / 17

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Maximally Strong Weakenings

Maximally Strong Weakening Axioms

Let Os ∪ (Ji \ {β}) ∪ {γ} | = α γ is a maximally strong weakening (MSW) of β in Ji if Os ∪ (Ji \ {β}) ∪ {δ} | = α for all δ with β ≻ δ ≻ γ.

Adrian Nuradiansyah KR 2018 November 1, 2018 11 / 17

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Maximally Strong Weakenings

Maximally Strong Weakening Axioms

Let Os ∪ (Ji \ {β}) ∪ {γ} | = α γ is a maximally strong weakening (MSW) of β in Ji if Os ∪ (Ji \ {β}) ∪ {δ} | = α for all δ with β ≻ δ ≻ γ.

One-step generated

Let ≻ be a weakening relation. The one-step relation ≻1 of ≻ is: ≻1:= {(β, γ) ∈ ≻| there is no δ such that β ≻ δ ≻ γ}

Adrian Nuradiansyah KR 2018 November 1, 2018 11 / 17

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Maximally Strong Weakenings

Maximally Strong Weakening Axioms

Let Os ∪ (Ji \ {β}) ∪ {γ} | = α γ is a maximally strong weakening (MSW) of β in Ji if Os ∪ (Ji \ {β}) ∪ {δ} | = α for all δ with β ≻ δ ≻ γ.

One-step generated

Let ≻ be a weakening relation. The one-step relation ≻1 of ≻ is: ≻1:= {(β, γ) ∈ ≻| there is no δ such that β ≻ δ ≻ γ} If the transitive closure of ≻1 is again ≻, then ≻ is one-step generated.

Adrian Nuradiansyah KR 2018 November 1, 2018 11 / 17

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Maximally Strong Weakenings

Maximally Strong Weakening Axioms

Let Os ∪ (Ji \ {β}) ∪ {γ} | = α γ is a maximally strong weakening (MSW) of β in Ji if Os ∪ (Ji \ {β}) ∪ {δ} | = α for all δ with β ≻ δ ≻ γ.

One-step generated

Let ≻ be a weakening relation. The one-step relation ≻1 of ≻ is: ≻1:= {(β, γ) ∈ ≻| there is no δ such that β ≻ δ ≻ γ} If the transitive closure of ≻1 is again ≻, then ≻ is one-step generated. ≻ is effectively finitely branching if for all axioms β, the set {γ | β ≻1 γ} is finite.

Adrian Nuradiansyah KR 2018 November 1, 2018 11 / 17

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Maximally Strong Weakenings

Theorem (Computing MSWs)

To compute all MSWs, the weakening relation ≻ should be well-founded, complete, one-step generated, and effectively finitely branching.

Adrian Nuradiansyah KR 2018 November 1, 2018 12 / 17

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Maximally Strong Weakenings

Theorem (Computing MSWs)

To compute all MSWs, the weakening relation ≻ should be well-founded, complete, one-step generated, and effectively finitely branching.

Algorithm for Computing MSWs

There are only finitely many γ such that β ≻ γ. All these γ can be reached by following ≻1.

Adrian Nuradiansyah KR 2018 November 1, 2018 12 / 17

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Maximally Strong Weakenings

Theorem (Computing MSWs)

To compute all MSWs, the weakening relation ≻ should be well-founded, complete, one-step generated, and effectively finitely branching.

Algorithm for Computing MSWs

There are only finitely many γ such that β ≻ γ. All these γ can be reached by following ≻1. By a breadth-first search, we can compute the set of all γ such that there is a path β ≻1 δ1 ≻1 . . . ≻1 δn ≻1 γ with

Adrian Nuradiansyah KR 2018 November 1, 2018 12 / 17

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Maximally Strong Weakenings

Theorem (Computing MSWs)

To compute all MSWs, the weakening relation ≻ should be well-founded, complete, one-step generated, and effectively finitely branching.

Algorithm for Computing MSWs

There are only finitely many γ such that β ≻ γ. All these γ can be reached by following ≻1. By a breadth-first search, we can compute the set of all γ such that there is a path β ≻1 δ1 ≻1 . . . ≻1 δn ≻1 γ with Os ∪ (Ji \ {β}) ∪ {γ} | = α, but Os ∪ (Ji \ {β}) ∪ {δi} | = α ∀i ∈ {1, . . . , n}

Adrian Nuradiansyah KR 2018 November 1, 2018 12 / 17

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Maximally Strong Weakenings

Theorem (Computing MSWs)

To compute all MSWs, the weakening relation ≻ should be well-founded, complete, one-step generated, and effectively finitely branching.

Algorithm for Computing MSWs

There are only finitely many γ such that β ≻ γ. All these γ can be reached by following ≻1. By a breadth-first search, we can compute the set of all γ such that there is a path β ≻1 δ1 ≻1 . . . ≻1 δn ≻1 γ with Os ∪ (Ji \ {β}) ∪ {γ} | = α, but Os ∪ (Ji \ {β}) ∪ {δi} | = α ∀i ∈ {1, . . . , n} If this set contains comparable elements, then remove the weaker ones. The remaining set only consists of all MSWs of β in Ji.

Adrian Nuradiansyah KR 2018 November 1, 2018 12 / 17

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Weakening Axioms in EL

We define C ⊑ D ≻s C ′ ⊑ D′ if C ′ ⊑ C, D ⊑ D′, and {C ′ ⊑ D′} | = C ⊑ D ≻s is complete, but not well founded.

Adrian Nuradiansyah KR 2018 November 1, 2018 13 / 17

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Weakening Axioms in EL

We define C ⊑ D ≻s C ′ ⊑ D′ if C ′ ⊑ C, D ⊑ D′, and {C ′ ⊑ D′} | = C ⊑ D ≻s is complete, but not well founded. Specializing the left-hand side is not well-founded in EL. ⊤ ⊑ A ≻ ∃r.⊤ ⊑ A ≻ ∃r∃r.⊤ ⊑ A ≻ . . . Generalizing the right-hand side is well-founded in EL (Baader & Morawska, 2010). For assertions in A: D(a) is weakened by generalizing D r(a, b) is weakened to a tautological axiom

Adrian Nuradiansyah KR 2018 November 1, 2018 13 / 17

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A Weakening Relation ≻sub in EL

We define C ⊑ D ≻sub C ′ ⊑ D′ if C ′ = C and D ⊏ D′ and {C ′ ⊑ D′} | = C ⊑ D

Adrian Nuradiansyah KR 2018 November 1, 2018 14 / 17

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A Weakening Relation ≻sub in EL

We define C ⊑ D ≻sub C ′ ⊑ D′ if C ′ = C and D ⊏ D′ and {C ′ ⊑ D′} | = C ⊑ D It is well-founded, complete, one-step generated, finitely branching, but not polynomial. | D′ | can be exponential in | D |. Let Nn := {A1, . . . , A2n} be a set of 2n distinct concept names. ∃r. Nn ⊏

  • X⊆Nn∧|X|=n

∃r. X. Exponentially many ∃r. X that can be removed.

Adrian Nuradiansyah KR 2018 November 1, 2018 14 / 17

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A Weakening Relation ≻sub in EL

We define C ⊑ D ≻sub C ′ ⊑ D′ if C ′ = C and D ⊏ D′ and {C ′ ⊑ D′} | = C ⊑ D It is well-founded, complete, one-step generated, finitely branching, but not polynomial. | D′ | can be exponential in | D |. Let Nn := {A1, . . . , A2n} be a set of 2n distinct concept names. ∃r. Nn ⊏

  • X⊆Nn∧|X|=n

∃r. X. Exponentially many ∃r. X that can be removed.

Complexity Results

The Algorithm for computing all maximally strong weakenings in EL w.r.t. ≻sub has non-elementary complexity. Deciding if γ is a maximally strong weakening w.r.t. ≻sub is coNP-hard.

Adrian Nuradiansyah KR 2018 November 1, 2018 14 / 17

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A Better Fragment ≻syn of ≻sub

Syntactic Generalizations

A concept D′ is a syntactic generalization of D, written D ⊏syn D′, iff some occurrences of subconcepts = ⊤ in D are replaced with ⊤.

We define C ⊑ D ≻syn C ′ ⊑ D′ if C ′ = C and D ⊏syn D′ and {C ′ ⊑ D′} | = C ⊑ D

Adrian Nuradiansyah KR 2018 November 1, 2018 15 / 17

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A Better Fragment ≻syn of ≻sub

Syntactic Generalizations

A concept D′ is a syntactic generalization of D, written D ⊏syn D′, iff some occurrences of subconcepts = ⊤ in D are replaced with ⊤.

We define C ⊑ D ≻syn C ′ ⊑ D′ if C ′ = C and D ⊏syn D′ and {C ′ ⊑ D′} | = C ⊑ D ≻syn is linear, complete, one-step generated, and finitely branching. | D | > | D′ |.

Adrian Nuradiansyah KR 2018 November 1, 2018 15 / 17

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A Better Fragment ≻syn of ≻sub

Syntactic Generalizations

A concept D′ is a syntactic generalization of D, written D ⊏syn D′, iff some occurrences of subconcepts = ⊤ in D are replaced with ⊤.

We define C ⊑ D ≻syn C ′ ⊑ D′ if C ′ = C and D ⊏syn D′ and {C ′ ⊑ D′} | = C ⊑ D ≻syn is linear, complete, one-step generated, and finitely branching. | D | > | D′ |.

Complexity Results

A single maximally strong weakening w.r.t. ≻syn can be computed in PTime. All maximally strong weakenings w.r.t. ≻syn can be computed in ExpTime. Deciding if γ is a maximally strong weakening w.r.t. ≻syn is coNP-complete.

Adrian Nuradiansyah KR 2018 November 1, 2018 15 / 17

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Conclusions and Future Work

Conclusions

Framework for repairing ontologies via weakening axioms rather than deleting Introduced weakening relations and maximally strong weakenings Applied the framework in Description Logic EL

Adrian Nuradiansyah KR 2018 November 1, 2018 16 / 17

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Conclusions and Future Work

Conclusions

Framework for repairing ontologies via weakening axioms rather than deleting Introduced weakening relations and maximally strong weakenings Applied the framework in Description Logic EL

Future Work

More complexity results for ≻sub – Finding better upper bound for deciding whether an axiom is an MSW w.r.t. ≻sub – Finding a better algorithm to compute MSWs w.r.t. ≻sub. Weakening relations for more expressive logics ⇒ ELO, ALC, etc. Choosing which axioms to be weakened and the maximally strong weakenings.

Adrian Nuradiansyah KR 2018 November 1, 2018 16 / 17

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Thank You

Adrian Nuradiansyah KR 2018 November 1, 2018 17 / 17