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Hybrid Unification in the Description Logic EL Franz Baader, Oliver - - PowerPoint PPT Presentation
Hybrid Unification in the Description Logic EL Franz Baader, Oliver - - PowerPoint PPT Presentation
Hybrid Unification in the Description Logic EL Franz Baader, Oliver Fern andez Gil and Barbara Morawska 9th International Symposium, FroCoS Nancy, September 20th, 2013 1/23 Outline 1 The Description Logic EL . 2 Unification in EL . 3 Hybrid
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The Description Logic EL. Syntax and Semantics.
EL concept descriptions
built from finite sets: NC := {Head injury, Severe} NR := {status} using concept constructors: ⊤, ⊓, ∃r.C Head injury ⊓ ∃status.Severe
Semantics
An interpretation I is a pair (△I, .I) that assigns:
- subsets C I of △I to concept names C and,
- binary relations r I on △I to role names r.
Semantics of the constructors: ⊤I := △I (C ⊓ D)I := C I ∩ DI (∃r.C)I := {x | ∃y : (x, y) ∈ r I ∧ y ∈ C I}
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The Description Logic EL. Syntax and Semantics.
EL concept descriptions
built from finite sets: NC := {Head injury, Severe} NR := {status} using concept constructors: ⊤, ⊓, ∃r.C Head injury ⊓ ∃status.Severe
Semantics
An interpretation I is a pair (△I, .I) that assigns:
- subsets C I of △I to concept names C and,
- binary relations r I on △I to role names r.
Semantics of the constructors: ⊤I := △I (C ⊓ D)I := C I ∩ DI (∃r.C)I := {x | ∃y : (x, y) ∈ r I ∧ y ∈ C I}
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The Description Logic EL. Terminological Axioms.
Concept definitions
A ≡ C for A ∈ NC and a concept description C. A TBox T is a finite set of concept definitions: . . . Severe injury ≡ Injury ⊓ ∃status.Severe . . . cyclic TBox: T = {A ≡ . . . ⊓ ∃r.B ⊓ . . . , B ≡ . . . ⊓ ∃r.A ⊓ . . .}
General Concept Inclusions (GCIs)
C ⊑ D for concept descriptions C, D. An ontology O is a finite set of GCIs: . . . Severe injury ⊑ Injury ⊓ ∃status.Severe Injury ⊓ ∃status.Severe ⊑ Severe injury . . .
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The Description Logic EL. Terminological Axioms.
Concept definitions
A ≡ C for A ∈ NC and a concept description C. A TBox T is a finite set of concept definitions: . . . Severe injury ≡ Injury ⊓ ∃status.Severe . . . cyclic TBox: T = {A ≡ . . . ⊓ ∃r.B ⊓ . . . , B ≡ . . . ⊓ ∃r.A ⊓ . . .}
General Concept Inclusions (GCIs)
C ⊑ D for concept descriptions C, D. An ontology O is a finite set of GCIs: . . . Severe injury ⊑ Injury ⊓ ∃status.Severe Injury ⊓ ∃status.Severe ⊑ Severe injury . . .
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The Description Logic EL. Reasoning.
Subsumption problem
C ⊑O D iff C I ⊆ DI for all models I of O.
Equivalence problem
C ≡O D iff C ⊑O D and D ⊑O C. Both problems are polynomial for EL.
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Unification in EL. Example.
Unification can be used to detect redundancies in ontologies. Two concept descriptions that are meant to represent the concept of a patient with severe head injury: Patient ⊓ ∃finding.(Head injury ⊓ ∃status.Severe) Patient ⊓ ∃finding.(Severe injury ⊓ ∃location.Head) They are not equivalent, but can be made equivalent w.r.t. the following TBox: Severe injury ≡ Injury ⊓ ∃status.Severe Head injury ≡ Injury ⊓ ∃location.Head
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Unification in EL. The decision problem.
Unification problem
NC is partitioned into a set of defined concepts Ndef and a set of primitive concepts Nprim. Instance: An ontology O. A finite set of subsumptions Γ = {C1 ⊑? D1, . . . , Cn ⊑? Dn}. Question: Is there an acyclic TBox T such that: C1 ⊑O∪T D1, . . . , Cn ⊑O∪T Dn Observations O contains only concept names from Nprim. The solution TBox T provides definitions for the concept names in Ndef .
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Unification in EL. The decision problem.
Unification problem
NC is partitioned into a set of defined concepts Ndef and a set of primitive concepts Nprim. Instance: An ontology O. A finite set of subsumptions Γ = {C1 ⊑? D1, . . . , Cn ⊑? Dn}. Question: Is there an acyclic TBox T such that: C1 ⊑O∪T D1, . . . , Cn ⊑O∪T Dn Observations O contains only concept names from Nprim. The solution TBox T provides definitions for the concept names in Ndef .
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Unification in EL. Previous Results.
[Baader and Morawska 2009]
Unification in EL without background ontology is NP-complete
An EL-unification problem that is solvable w.r.t. the empty ontology has a solution that is a local acyclic TBox.
Locality
Atoms of Γ (At): C, ∃r.D Non-variable atoms (Atnv): At \ Ndef A TBox T is local if each concept definition in T is of the following form: X ≡ D1 ⊓ . . . ⊓ Dn where Di ∈ Atnv for all i, 1 ≤ i ≤ Dn.
NP-decision procedure
Guesses a local acyclic TBox and then checks whether it is a unifier.
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Unification in EL. Previous Results.
[Baader and Morawska 2009]
Unification in EL without background ontology is NP-complete
An EL-unification problem that is solvable w.r.t. the empty ontology has a solution that is a local acyclic TBox.
Locality
Atoms of Γ (At): C, ∃r.D Non-variable atoms (Atnv): At \ Ndef A TBox T is local if each concept definition in T is of the following form: X ≡ D1 ⊓ . . . ⊓ Dn where Di ∈ Atnv for all i, 1 ≤ i ≤ Dn.
NP-decision procedure
Guesses a local acyclic TBox and then checks whether it is a unifier.
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Unification in EL. Previous Results.
[Baader and Morawska 2009]
Unification in EL without background ontology is NP-complete
An EL-unification problem that is solvable w.r.t. the empty ontology has a solution that is a local acyclic TBox.
Locality
Atoms of Γ (At): C, ∃r.D Non-variable atoms (Atnv): At \ Ndef A TBox T is local if each concept definition in T is of the following form: X ≡ D1 ⊓ . . . ⊓ Dn where Di ∈ Atnv for all i, 1 ≤ i ≤ Dn.
NP-decision procedure
Guesses a local acyclic TBox and then checks whether it is a unifier.
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Unification in EL. Previous Results.
Extending unification to non-empty ontologies
- The notion of locality does not work.
- Unification problems with solution, but no local unifiers.
- The algorithm is complete only for cycle restricted ontologies [Baader, Borgwardt,
Morawska 2011]. O | = C ⊑ ∃w.C
A different approach
Extend what is accepted as a solution to the unification problem:
- Allow the TBox T to be cyclic.
- Use greatest fixpoint semantics to interpret defined concepts in T .
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Unification in EL. Previous Results.
Extending unification to non-empty ontologies
- The notion of locality does not work.
- Unification problems with solution, but no local unifiers.
- The algorithm is complete only for cycle restricted ontologies [Baader, Borgwardt,
Morawska 2011]. O | = C ⊑ ∃w.C
A different approach
Extend what is accepted as a solution to the unification problem:
- Allow the TBox T to be cyclic.
- Use greatest fixpoint semantics to interpret defined concepts in T .
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Hybrid Unification. Greatest fixpoint semantics [Nebel 1991].
Example
- Nprim := {Node} and Ndef := {INode}
- T := {INode ≡ Node ⊓ ∃edge.INode}
- J is an interpretation of Node and edge:
. . . m1 m2 m3 n1
- How to extend J to a model I of T ?
classical models: {m1, m2, . . .} ∪ {n1}, {m1, m2, . . .}, {n1} or ∅. gfp model: {m1, m2, . . .} ∪ {n1}.
Formally,
For two extensions I1, I2 of J : I1 ≺J I2 iff X I1 ⊆ X I2 for each X ∈ Ndef . The gfp model is the greatest element of ≺J that models T .
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Hybrid Unification. Greatest fixpoint semantics [Nebel 1991].
Example
- Nprim := {Node} and Ndef := {INode}
- T := {INode ≡ Node ⊓ ∃edge.INode}
- J is an interpretation of Node and edge:
. . . m1 m2 m3 n1
- How to extend J to a model I of T ?
classical models: {m1, m2, . . .} ∪ {n1}, {m1, m2, . . .}, {n1} or ∅. gfp model: {m1, m2, . . .} ∪ {n1}.
Formally,
For two extensions I1, I2 of J : I1 ≺J I2 iff X I1 ⊆ X I2 for each X ∈ Ndef . The gfp model is the greatest element of ≺J that models T .
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Hybrid Unification. Greatest fixpoint semantics [Nebel 1991].
Example
- Nprim := {Node} and Ndef := {INode}
- T := {INode ≡ Node ⊓ ∃edge.INode}
- J is an interpretation of Node and edge:
. . . m1 m2 m3 n1
- How to extend J to a model I of T ?
classical models: {m1, m2, . . .} ∪ {n1}, {m1, m2, . . .}, {n1} or ∅. gfp model: {m1, m2, . . .} ∪ {n1}.
Formally,
For two extensions I1, I2 of J : I1 ≺J I2 iff X I1 ⊆ X I2 for each X ∈ Ndef . The gfp model is the greatest element of ≺J that models T .
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Hybrid Unification. Hybrid EL-Ontologies.
Hybrid EL-ontology [Brandt and Model 2005]
A hybrid EL-ontology is a pair (O, T ) where O is an EL-ontology and T is a (cyclic) TBox. An interpretation I is hybrid model of a hybrid ontology (O, T ) iff:
- I is an extension of a model J of O.
- I is a gfp model of T .
Subsumption w.r.t. hybrid ontologies
C is subsumed by D w.r.t. (O, T ) (C ⊑gfp,O∪T D) iff: C I ⊆ DI, for every hybrid model I of (O, T ) Subsumption is decidable in polynomial time.[Brandt and Model 2005, Novakovic 2007]
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Hybrid Unification. Hybrid EL-Ontologies.
Hybrid EL-ontology [Brandt and Model 2005]
A hybrid EL-ontology is a pair (O, T ) where O is an EL-ontology and T is a (cyclic) TBox. An interpretation I is hybrid model of a hybrid ontology (O, T ) iff:
- I is an extension of a model J of O.
- I is a gfp model of T .
Subsumption w.r.t. hybrid ontologies
C is subsumed by D w.r.t. (O, T ) (C ⊑gfp,O∪T D) iff: C I ⊆ DI, for every hybrid model I of (O, T ) Subsumption is decidable in polynomial time.[Brandt and Model 2005, Novakovic 2007]
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Hybrid Unification. The decision problem.
Hybrid unification problem
Instance: An ontology O. A finite set of subsumptions Γ = {C1 ⊑? D1, . . . , Cn ⊑? Dn}. Question: Is there a (cyclic) TBox T such that: C1 ⊑gfp,O∪T D1, . . . , Cn ⊑gfp,O∪T Dn We call such a TBox T a hybrid unifier of Γ w.r.t. O. Remark Acyclic hybrid unifiers are classical unifiers.
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Hybrid Unification. The decision problem.
Hybrid unification problem
Instance: An ontology O. A finite set of subsumptions Γ = {C1 ⊑? D1, . . . , Cn ⊑? Dn}. Question: Is there a (cyclic) TBox T such that: C1 ⊑gfp,O∪T D1, . . . , Cn ⊑gfp,O∪T Dn We call such a TBox T a hybrid unifier of Γ w.r.t. O. Remark Acyclic hybrid unifiers are classical unifiers.
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Hybrid Unification. Example.
Ontology
Human ⊑ ∃parent.Human Horse ⊑ ∃parent.Horse
- Unification problem
Γ := {Human ⊑? X, Horse ⊑? X, X ⊑? ∃parent.X} Γ has a hybrid unifier, T := {X ≡ ∃parent.X} but it does not have a classical unifier.
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Hybrid Unification. Example.
Ontology
Human ⊑ ∃parent.Human Horse ⊑ ∃parent.Horse
- Unification problem
Γ := {Human ⊑? X, Horse ⊑? X, X ⊑? ∃parent.X} Γ has a hybrid unifier, T := {X ≡ ∃parent.X} but it does not have a classical unifier.
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Hybrid Unification is NP-complete. NP-membership.
Proposition
Γ has a hybrid unifier w.r.t. O iff Γ has a local hybrid unifier w.r.t. O.
Proof.
Assume that T is a hybrid unifier of Γ w.r.t. O. Define SX for each X ∈ Ndef as: SX = {D ∈ Atnv | X ⊑gfp,O∪T D} A local TBox T ′ is defined as: T ′ = {X ≡
- D∈SX
D | X ∈ Ndef } Show that T ′ is also a hybrid unifier of Γ w.r.t. O.
Theorem
Hybrid unification w.r.t. to an arbitrary EL-ontology is in NP.
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Hybrid Unification is NP-complete. NP-membership.
Proposition
Γ has a hybrid unifier w.r.t. O iff Γ has a local hybrid unifier w.r.t. O.
Proof.
Assume that T is a hybrid unifier of Γ w.r.t. O. Define SX for each X ∈ Ndef as: SX = {D ∈ Atnv | X ⊑gfp,O∪T D} A local TBox T ′ is defined as: T ′ = {X ≡
- D∈SX
D | X ∈ Ndef } Show that T ′ is also a hybrid unifier of Γ w.r.t. O.
Theorem
Hybrid unification w.r.t. to an arbitrary EL-ontology is in NP.
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Hybrid Unification is NP-complete. NP-hardness.
EL-matching problem modulo equivalence
C ≡? D, where C and D are concept descriptions s.t. C is ground.
Solution
A substitution σ such that C ≡ σ(D)? Matching in EL is NP-Complete. [K¨ uster 2001]
Reduction to the hybrid unification problem
Given C ≡? D, define Γ = {C ⊑? D, D ⊑? C} and O = ∅.
Proposition
C ≡? D has a solution iff Γ has a hybrid unifier w.r.t. O = ∅.
Theorem
Hybrid unification in EL is NP-hard.
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Hybrid Unification is NP-complete. NP-hardness.
EL-matching problem modulo equivalence
C ≡? D, where C and D are concept descriptions s.t. C is ground.
Solution
A substitution σ such that C ≡ σ(D)? Matching in EL is NP-Complete. [K¨ uster 2001]
Reduction to the hybrid unification problem
Given C ≡? D, define Γ = {C ⊑? D, D ⊑? C} and O = ∅.
Proposition
C ≡? D has a solution iff Γ has a hybrid unifier w.r.t. O = ∅.
Theorem
Hybrid unification in EL is NP-hard.
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Hybrid Unification is NP-complete. NP-hardness.
EL-matching problem modulo equivalence
C ≡? D, where C and D are concept descriptions s.t. C is ground.
Solution
A substitution σ such that C ≡ σ(D)? Matching in EL is NP-Complete. [K¨ uster 2001]
Reduction to the hybrid unification problem
Given C ≡? D, define Γ = {C ⊑? D, D ⊑? C} and O = ∅.
Proposition
C ≡? D has a solution iff Γ has a hybrid unifier w.r.t. O = ∅.
Theorem
Hybrid unification in EL is NP-hard.
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Proof Calculus HC [Novakovic 2007].
A sequent is of the form C ⊑n D where C and D are sub-descriptions in (O, T ) and n ≥ 0.
Rules of HC.
C ⊑n C (Ax) C ⊑n ⊤ (Top) C ⊑0 D (Start) C ⊑n E C ⊓ D ⊑n E (AndL1) D ⊑n E C ⊓ D ⊑n E (AndL2) C ⊑n D C ⊑n E C ⊑n D ⊓ E (AndR) C ⊑n D ∃r.C ⊑n ∃r.D (Ex) C ⊑n D X ⊑n D (DefL) D ⊑n C D ⊑n+1 X (DefR) for X ≡ C ∈ T C ⊑n E F ⊑n D C ⊑n D (GCI) for E ⊑ F ∈ O
C ⊑gfp,O∪T D iff C ⊑n D for all n ≥ 0 (C ⊑∞ D). There exists ℓ such that ⊑ℓ=⊑ℓ
+ 1= . . . 16/23
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Proof Calculus HC [Novakovic 2007].
A sequent is of the form C ⊑n D where C and D are sub-descriptions in (O, T ) and n ≥ 0.
Rules of HC.
C ⊑n C (Ax) C ⊑n ⊤ (Top) C ⊑0 D (Start) C ⊑n E C ⊓ D ⊑n E (AndL1) D ⊑n E C ⊓ D ⊑n E (AndL2) C ⊑n D C ⊑n E C ⊑n D ⊓ E (AndR) C ⊑n D ∃r.C ⊑n ∃r.D (Ex) C ⊑n D X ⊑n D (DefL) D ⊑n C D ⊑n+1 X (DefR) for X ≡ C ∈ T C ⊑n E F ⊑n D C ⊑n D (GCI) for E ⊑ F ∈ O
C ⊑gfp,O∪T D iff C ⊑n D for all n ≥ 0 (C ⊑∞ D). There exists ℓ such that ⊑ℓ=⊑ℓ
+ 1= . . . 16/23
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Proof Calculus HC [Novakovic 2007].
A sequent is of the form C ⊑n D where C and D are sub-descriptions in (O, T ) and n ≥ 0.
Rules of HC.
C ⊑n C (Ax) C ⊑n ⊤ (Top) C ⊑0 D (Start) C ⊑n E C ⊓ D ⊑n E (AndL1) D ⊑n E C ⊓ D ⊑n E (AndL2) C ⊑n D C ⊑n E C ⊑n D ⊓ E (AndR) C ⊑n D ∃r.C ⊑n ∃r.D (Ex) C ⊑n D X ⊑n D (DefL) D ⊑n C D ⊑n+1 X (DefR) for X ≡ C ∈ T C ⊑n E F ⊑n D C ⊑n D (GCI) for E ⊑ F ∈ O
C ⊑gfp,O∪T D iff C ⊑n D for all n ≥ 0 (C ⊑∞ D). There exists ℓ such that ⊑ℓ=⊑ℓ
+ 1= . . . 16/23
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Proof Calculus HC [Novakovic 2007].
A sequent is of the form C ⊑n D where C and D are sub-descriptions in (O, T ) and n ≥ 0.
Rules of HC.
C ⊑n C (Ax) C ⊑n ⊤ (Top) C ⊑0 D (Start) C ⊑n E C ⊓ D ⊑n E (AndL1) D ⊑n E C ⊓ D ⊑n E (AndL2) C ⊑n D C ⊑n E C ⊑n D ⊓ E (AndR) C ⊑n D ∃r.C ⊑n ∃r.D (Ex) C ⊑n D X ⊑n D (DefL) D ⊑n C D ⊑n+1 X (DefR) for X ≡ C ∈ T C ⊑n E F ⊑n D C ⊑n D (GCI) for E ⊑ F ∈ O
C ⊑gfp,O∪T D iff C ⊑n D for all n ≥ 0 (C ⊑∞ D). There exists ℓ such that ⊑ℓ=⊑ℓ
+ 1= . . . 16/23
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Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
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Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
17/23
SLIDE 36
Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
17/23
SLIDE 37
Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
17/23
SLIDE 38
Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
17/23
SLIDE 39
Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
17/23
SLIDE 40
Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
17/23
SLIDE 41
Goal-Oriented Unification Algorithm.
Main idea: Try to obtain proof trees for Ci ⊑?
ℓ Di while building T .
Applies two types of rules:
- eager rules (are always applied first) and nondeterministic rules.
- rules represent the possible ways to derive Ci ⊑ℓ Di.
- A non-failing application of a rule does the following:
- it solves exactly one unsolved sequent,
- it may introduce new proof obligations, i.e.: a sequent C ′ ⊑?
ℓ D′,
- it may extend the current assignment S.
Estimation of the value for ℓ? we need to show Ci ⊑∞ Di.
- locality ⇒ maximal size of T is bounded w.r.t. the size of O and Γ.
- ℓ is polynomial on the size of O and Γ.
Run of the algorithm:
- Fails: if a rule application fails or there is an unsolved sequent to which no rule
applies.
- Succeeds: if there are no unsolved sequents.
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SLIDE 42
Goal-Oriented Unification Algorithm. Rules.
Eager Ground Solving: Applies to ground sequents: A ⊑?
n B −
→ fails if A ⊑O B Decomposition and Extension: ∃r.X1 ⊓ ∃r.X2 ⊑?
n ∃r.C
X1 ⊑?
n C
X2 ⊑?
n C
SX1 = {C} SX2 = {C} Mutation (using GCIs): ∃r.X ⊑?
n B
∃r.X ⊑?
n ∃r.A
B ⊑?
n B O = {∃r.A ⊑ B} 18/23
SLIDE 43
Goal-Oriented Unification Algorithm. Rules.
Eager Ground Solving: Applies to ground sequents: A ⊑?
n B −
→ fails if A ⊑O B Decomposition and Extension: ∃r.X1 ⊓ ∃r.X2 ⊑?
n ∃r.C
X1 ⊑?
n C
X2 ⊑?
n C
SX1 = {C} SX2 = {C} Mutation (using GCIs): ∃r.X ⊑?
n B
∃r.X ⊑?
n ∃r.A
B ⊑?
n B O = {∃r.A ⊑ B} 18/23
SLIDE 44
Goal-Oriented Unification Algorithm. Rules.
Eager Ground Solving: Applies to ground sequents: A ⊑?
n B −
→ fails if A ⊑O B Decomposition and Extension: ∃r.X1 ⊓ ∃r.X2 ⊑?
n ∃r.C
X1 ⊑?
n C
X2 ⊑?
n C
SX1 = {C} SX2 = {C} Mutation (using GCIs): ∃r.X ⊑?
n B
∃r.X ⊑?
n ∃r.A
B ⊑?
n B O = {∃r.A ⊑ B} 18/23
SLIDE 45
Goal-Oriented Unification Algorithm. Rules.
Eager Ground Solving: Applies to ground sequents: A ⊑?
n B −
→ fails if A ⊑O B Decomposition and Extension: ∃r.X1 ⊓ ∃r.X2 ⊑?
n ∃r.C
X1 ⊑?
n C
X2 ⊑?
n C
SX1 = {C} SX2 = {C} Mutation (using GCIs): ∃r.X ⊑?
n B
∃r.X ⊑?
n ∃r.A
B ⊑?
n B O = {∃r.A ⊑ B} 18/23
SLIDE 46
Goal-Oriented Unification Algorithm. Example 1.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = ∅ Mutation (Human ⊑ ∃par . . .)
19/23
SLIDE 47
Goal-Oriented Unification Algorithm. Example 1.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = ∅ Mutation (Human ⊑ ∃par . . .) X ⊑ℓ Human ∃parent.Human ⊑ℓ ∃parent.X
19/23
SLIDE 48
Goal-Oriented Unification Algorithm. Example 1.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = ∅ Mutation (Human ⊑ ∃par . . .) X ⊑ℓ Human ∃parent.Human ⊑ℓ ∃parent.X X ⊑ℓ Human SX = {Human} Extension
19/23
SLIDE 49
Goal-Oriented Unification Algorithm. Example 1.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = ∅ Mutation (Human ⊑ ∃par . . .) X ⊑ℓ Human ∃parent.Human ⊑ℓ ∃parent.X X ⊑ℓ Human SX = {Human} Extension Horse ⊑ℓ
− 1 Human
Expansion
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SLIDE 50
Goal-Oriented Unification Algorithm. Example 1.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = ∅ Mutation (Human ⊑ ∃par . . .) X ⊑ℓ Human ∃parent.Human ⊑ℓ ∃parent.X X ⊑ℓ Human SX = {Human} Extension Horse ⊑ℓ
− 1 Human
Expansion Horse ⊑ℓ
− 1 Human
Eager Ground Solving A failing run, Horse ⊑O Human.
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SLIDE 51
Goal-Oriented Unification Algorithm. Example 2.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = {∃parent.X} Extension
20/23
SLIDE 52
Goal-Oriented Unification Algorithm. Example 2.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = {∃parent.X} Extension Human ⊑ℓ
− 1 ∃parent.X
Expansion Horse ⊑ℓ
− 1 ∃parent.X 20/23
SLIDE 53
Goal-Oriented Unification Algorithm. Example 2.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = {∃parent.X} Extension Human ⊑ℓ
− 1 ∃parent.X
Expansion Horse ⊑ℓ
− 1 ∃parent.X
Human ⊑ℓ
− 1 ∃parent.X
Mutation
20/23
SLIDE 54
Goal-Oriented Unification Algorithm. Example 2.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = {∃parent.X} Extension Human ⊑ℓ
− 1 ∃parent.X
Expansion Horse ⊑ℓ
− 1 ∃parent.X
Human ⊑ℓ
− 1 ∃parent.X
Mutation Human ⊑ℓ
− 1 Human
∃parent.Human ⊑ℓ
− 1 ∃parent.X 20/23
SLIDE 55
Goal-Oriented Unification Algorithm. Example 2.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = {∃parent.X} Extension Human ⊑ℓ
− 1 ∃parent.X
Expansion Horse ⊑ℓ
− 1 ∃parent.X
Human ⊑ℓ
− 1 ∃parent.X
Mutation Human ⊑ℓ
− 1 Human
∃parent.Human ⊑ℓ
− 1 ∃parent.X
∃parent.Human ⊑ℓ
− 1 ∃parent.X
Decomposition
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SLIDE 56
Goal-Oriented Unification Algorithm. Example 2.
O = {Horse ⊑ ∃parent.Horse, Human ⊑ ∃parent.Human} Γ = {Human ⊑ℓ X, Horse ⊑ℓ X, X ⊑ℓ ∃parent.X} X ⊑ℓ ∃parent.X SX = {∃parent.X} Extension Human ⊑ℓ
− 1 ∃parent.X
Expansion Horse ⊑ℓ
− 1 ∃parent.X
Human ⊑ℓ
− 1 ∃parent.X
Mutation Human ⊑ℓ
− 1 Human
∃parent.Human ⊑ℓ
− 1 ∃parent.X
∃parent.Human ⊑ℓ
− 1 ∃parent.X
Decomposition Human ⊑ℓ
− 1 X
. . . Human ⊑0 X T = {X ≡ ∃parent.X} is a hybrid unifier of Γ w.r.t. O.
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SLIDE 57
Goal-Oriented Unification Algorithm. Results.
Theorem
The algorithm is sound and complete.
Theorem
The algorithm is an NP-decision procedure for hybrid unification in EL.
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SLIDE 58
Conclusions.
Our Results...
- Definition of the hybrid unification problem in EL w.r.t. arbitrary ontologies.
- Hybrid EL-unification is NP-complete.
- A sound and complete goal-oriented unification algorithm.
Future Work...
- Implementation of the algorithm. Optimizations (estimation of ℓ?).
- Decidability and complexity of classical EL-unification w.r.t. arbitrary ontologies.
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SLIDE 59
Conclusions.
Our Results...
- Definition of the hybrid unification problem in EL w.r.t. arbitrary ontologies.
- Hybrid EL-unification is NP-complete.
- A sound and complete goal-oriented unification algorithm.
Future Work...
- Implementation of the algorithm. Optimizations (estimation of ℓ?).
- Decidability and complexity of classical EL-unification w.r.t. arbitrary ontologies.
22/23
SLIDE 60