Consequence-Based Reasoning for Description Logic Ontologies by nyut545e2

VIEWS: 25 PAGES: 225

• pg 1
```									 C ONSEQUENCE -BASED R EASONING
FOR D ESCRIPTION L OGIC O NTOLOGIES

Yevgeny Kazakov

Oxford University Computing Laboratory

July 15, 2010
OVERVIEW
Introduction to Description Logic
Reasoning problems
Hierarchy of DLs
Related formalisms
Tableau-based reasoning procedures
Key reasoning phases
Practical limitations
Consequence-based reasoning procedures
Reasoning in the DL EL
Extension to Horn SHIQ
Related methods
Hyper-resolution
Ordered resolution
Automata-based methods
Conclusions
Yevgeny Kazakov   Consequence-Based Reasoning for DL Ontologies   2/36
Introduction

O UTLINE

1   I NTRODUCTION

2   TABLEAU -BASED R EASONING

3   C ONSEQUENCE -BASED R EASONING

4   R ELATED M ETHODS

5   C ONCLUSIONS

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   3/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The syntax

Heart     Organ       ∃ isComponentOf.CirculatorySystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The syntax
Atomic concepts [Classes]

Heart     Organ       ∃ isComponentOf.CirculatorySystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The syntax
Atomic concepts [Classes]
Atomic roles [Properties]

Heart     Organ       ∃ isComponentOf.CirculatorySystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The syntax
Atomic concepts [Classes]
Atomic roles [Properties]
Constructors

Heart     Organ       ∃ isComponentOf.CirculatorySystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics

Heart     Organ      ∃ isComponentOf.CirculatorySystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )

Heart     Organ      ∃ isComponentOf.CirculatorySystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets
Atomic roles ⇒ binary relations

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets
Atomic roles ⇒ binary relations
Constructors ⇒ set operators

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets
Atomic roles ⇒ binary relations
Constructors ⇒ set operators

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets
Atomic roles ⇒ binary relations
Constructors ⇒ set operators

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
Atomic concepts ⇒ sets
Atomic roles ⇒ binary relations
Constructors ⇒ set operators

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
I is a model iff all axioms are satisﬁed

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

S YNTAX AND S EMANTICS OF DL S
The semantics
Interpretation I = (∆I , ·I )
∆I is an interpretation domain (non-empty set)
·I is an interpretation function
I is a model iff all axioms are satisﬁed

Heart     Organ       ∃ isComponentOf.CirculatorySystem

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   4/36
Introduction

H IERARCHY OF DL S
Name                    DL syntax                      First-Order syntax
intersection            C1 C2                           C1 (x) ∧ C2 (x)
union                   C1 C2                           C1 (x) ∨ C2 (x)                    =A
complement                ¬C                                 ¬C(x)                          L
value restriction        ∀r.C                         ∀y.[r(x, y) → C(y)]                   C
existential restr.       ∃r.C                         ∃y.[r(x, y) ∧ C(y)]
concept inclusion       C1 C2                         ∀x.[C1 (x) → C2 (x)]

Basic DL ALC [Schmidt-Schauß, Smolka; 1991]:
is a syntactic variant of Kn :
∀r.C ⇒ r C
∃r.C ⇒ ♦r C
is a subset of GF 2
has tree-model property
has ﬁnite model property
satisﬁability problem is ExpTime-complete

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies        5/36
Introduction

H IERARCHY OF DL S
Name                    DL syntax                     First-Order syntax
intersection            C1 C2                 C1 (x) ∧ C2 (x)
union                   C1 C2                 C1 (x) ∨ C2 (x)           =A
complement                   ¬C                      ¬C(x)                L
value restriction          ∀r.C            ∀y.[r(x, y) → C(y)]            C
existential restr.         ∃r.C             ∃y.[r(x, y) ∧ C(y)]
concept inclusion       C1 C2              ∀x.[C1 (x) → C2 (x)]
transitivity              Tra(r)     ∀xyz.[r(x, y) ∧ r(y, z) → r(x, z)] = S
functionality            Fun(r)      ∀xyz.[r(x, y) ∧ r(x, z) → y z] +F
role inclusion           r1 r2          ∀xy.[r1 (x, y) → r2 (x, y)]     +H
inverse roles         [. . . r −...]          [. . . r(y, x) . . . ]    +I

SHIF :
has a generalized tree-model property (transitivity)
has no ﬁnite-model property (because of functionality)
satisﬁability problem is ExpTime-complete

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   5/36
Introduction

H IERARCHY OF DL S
Name                 DL syntax                     First-Order syntax
intersection         C1 C2                 C1 (x) ∧ C2 (x)
union                C1 C2                 C1 (x) ∨ C2 (x)           =A
complement                ¬C                      ¬C(x)                L
value restriction       ∀r.C            ∀y.[r(x, y) → C(y)]            C
existential restr.      ∃r.C             ∃y.[r(x, y) ∧ C(y)]
concept inclusion    C1 C2              ∀x.[C1 (x) → C2 (x)]
transitivity           Tra(r)     ∀xyz.[r(x, y) ∧ r(y, z) → r(x, z)] = S
functionality         Fun(r)      ∀xyz.[r(x, y) ∧ r(x, z) → y z] +F
role inclusion        r1 r2          ∀xy.[r1 (x, y) → r2 (x, y)]     +H
inverse roles      [. . . r −...]          [. . . r(y, x) . . . ]    +I
number restriction        n r.C        ∃≤n y.[r(x, y) ∧ C(y)]        +Q
nominals                   o                      x o                +O
SHOIQ:
no tree-model property (even generalized)
satisﬁability is NExpTime-complete (can be translated to C 2 )
Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   5/36
Introduction

B IO -M EDICAL O NTOLOGIES
SNOMED CT, GALEN, OBO, FMA, NCI Thesaurus, . . .

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   6/36
Introduction

B IO -M EDICAL O NTOLOGIES
SNOMED CT, GALEN, OBO, FMA, NCI Thesaurus, . . .
Simple inclusions:
Heart      Organ ∃isPartOf.Chest
Myocardium      Muscle ∃isPartOf.Heart
Myocarditis     Disorder ∃affects.Myocardium

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   6/36
Introduction

B IO -M EDICAL O NTOLOGIES
SNOMED CT, GALEN, OBO, FMA, NCI Thesaurus, . . .
Simple inclusions:
Heart Organ ∃isPartOf.Chest
Myocardium Muscle ∃isPartOf.Heart
Myocarditis Disorder ∃affects.Myocardium
Concept deﬁnitions:
MuscularOrgan ≡ Organ ∃hasPart.Muscle
HeartDisease ≡ Disorder ∃affects.∃isPartOf.Heart
KidneyExamination ≡ ClinicalAct
∃hasSubprocess.(Examination ∃involves.Kidney)

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   6/36
Introduction

B IO -M EDICAL O NTOLOGIES
SNOMED CT, GALEN, OBO, FMA, NCI Thesaurus, . . .
Simple inclusions:
Heart Organ ∃isPartOf.Chest
Myocardium Muscle ∃isPartOf.Heart
Myocarditis Disorder ∃affects.Myocardium
Concept deﬁnitions:
MuscularOrgan ≡ Organ ∃hasPart.Muscle
HeartDisease ≡ Disorder ∃affects.∃isPartOf.Heart
KidneyExamination ≡ ClinicalAct
∃hasSubprocess.(Examination ∃involves.Kidney)
General concept inclusions:
Structure   ∃isPartOf.Heart
∃isComponentOf.CardiovascularSystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   6/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥
 Find unsatisﬁable atomic classes: ?- A : O |= A                    ⊥

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥
 Find unsatisﬁable atomic classes: ?- A : O |= A ⊥
 Compute subsumptions between all atomic classes:
?- A, B : O |= A B

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥
 Find unsatisﬁable atomic classes: ?- A : O |= A ⊥
 Compute subsumptions between all atomic classes:
?- A, B : O |= A B
The goal is to compute taxonomy, a.k.a. class hierarchy

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥
 Find unsatisﬁable atomic classes: ?- A : O |= A ⊥
 Compute subsumptions between all atomic classes:
?- A, B : O |= A B
The goal is to compute taxonomy, a.k.a. class hierarchy
All reasoning problems can be reduced to each other:

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥
 Find unsatisﬁable atomic classes: ?- A : O |= A ⊥
 Compute subsumptions between all atomic classes:
?- A, B : O |= A B
The goal is to compute taxonomy, a.k.a. class hierarchy
All reasoning problems can be reduced to each other:
O |= A   B        ⇔           O |= (A       ¬B)      ⊥

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥
 Find unsatisﬁable atomic classes: ?- A : O |= A ⊥
 Compute subsumptions between all atomic classes:
?- A, B : O |= A B
The goal is to compute taxonomy, a.k.a. class hierarchy
All reasoning problems can be reduced to each other:
O |= A   B        ⇔           O |= (A       ¬B) ⊥
O A      ⊥        ⇔           O∪{           ∃R.A} |= ⊥, R is fresh

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Introduction

R EASONING P ROBLEMS
Ontology Classiﬁcation:
 Check ontology consistency: ?- O |= ⊥
 Find unsatisﬁable atomic classes: ?- A : O |= A ⊥
 Compute subsumptions between all atomic classes:
?- A, B : O |= A B
The goal is to compute taxonomy, a.k.a. class hierarchy
All reasoning problems can be reduced to each other:
O |= A   B        ⇔           O |= (A      ¬B) ⊥
O A      ⊥        ⇔           O∪{           ∃R.A} |= ⊥, R is fresh
O |= ⊥            ⇔           O |= A       B, A, B are fresh

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   7/36
Tableau-Based Reasoning

O UTLINE

1   I NTRODUCTION

2   TABLEAU -BASED R EASONING

3   C ONSEQUENCE -BASED R EASONING

4   R ELATED M ETHODS

5   C ONCLUSIONS

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   8/36
Tableau-Based Reasoning

O UTLINE OF TABLEAU -BASED P ROCEDURES
Implemented in most ontologies reasoners:
FACT++, H ERMI T, P ELLET, R ACER.

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   9/36
Tableau-Based Reasoning

O UTLINE OF TABLEAU -BASED P ROCEDURES
Implemented in most ontologies reasoners:
FACT++, H ERMI T, P ELLET, R ACER.
Search / build model / model representation to satisfy a
given concept w.r.t. the ontology:

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   9/36
Tableau-Based Reasoning

O UTLINE OF TABLEAU -BASED P ROCEDURES
Implemented in most ontologies reasoners:
FACT++, H ERMI T, P ELLET, R ACER.
Search / build model / model representation to satisfy a
given concept w.r.t. the ontology:
1   To check O |= ⊥, build a model for

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   9/36
Tableau-Based Reasoning

O UTLINE OF TABLEAU -BASED P ROCEDURES
Implemented in most ontologies reasoners:
FACT++, H ERMI T, P ELLET, R ACER.
Search / build model / model representation to satisfy a
given concept w.r.t. the ontology:
1   To check O |= ⊥, build a model for
2   To check O |= A ⊥, build a model for A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   9/36
Tableau-Based Reasoning

O UTLINE OF TABLEAU -BASED P ROCEDURES
Implemented in most ontologies reasoners:
FACT++, H ERMI T, P ELLET, R ACER.
Search / build model / model representation to satisfy a
given concept w.r.t. the ontology:
1   To check O |= ⊥, build a model for
2   To check O |= A ⊥, build a model for A
3   To check O |= A B, build a model for A                ¬B.

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   9/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis Disorder ∃affects.Myocardium
Myocardium Muscle ∃isPartOf.Heart
HeartDisease ≡ Disorder ∃affects.∃isPartOf.Heart

?- Myocarditis    HeartDisease

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
 Myocarditis Disorder ∃affects.Myocardium
 Myocardium Muscle ∃isPartOf.Heart
 HeartDisease ≡ Disorder ∃affects.∃isPartOf.Heart

?- Myocarditis       HeartDisease

1    Normalization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
 Myocarditis         Disorder ∃affects.Myocardium
 Myocardium          Muscle ∃isPartOf.Heart
 HeartDisease        Disorder ∃affects.∃isPartOf.Heart
      Disorder       ∃affects.∃isPartOf.Heart HeartDisease
?- Myocarditis      HeartDisease

1    Normalization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
 Myocarditis         Disorder ∃affects.Myocardium
 Myocardium          Muscle ∃isPartOf.Heart
 HeartDisease        Disorder ∃affects.∃isPartOf.Heart
      Disorder       ∃affects.∃isPartOf.Heart HeartDisease
?- Myocarditis      HeartDisease

1    Normalization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
 Myocarditis         Disorder ∃affects.Myocardium
 Myocardium          Muscle ∃isPartOf.Heart
 HeartDisease        Disorder ∃affects.∃isPartOf.Heart
      Disorder       ¬∃affects.∃isPartOf.Heart HeartDisease
?- Myocarditis      HeartDisease

1    Normalization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
 Myocarditis         Disorder ∃affects.Myocardium
 Myocardium          Muscle ∃isPartOf.Heart
 HeartDisease        Disorder ∃affects.∃isPartOf.Heart
      Disorder       ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis      HeartDisease

1    Normalization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
 Myocarditis         Disorder ∃affects.Myocardium
 Myocardium          Muscle ∃isPartOf.Heart
 HeartDisease        Disorder ∃affects.∃isPartOf.Heart
     Disorder        ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis       HeartDisease

1    Normalization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization
2   Initialization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease ®

1   Normalization                  Myocarditis, ¬HeartDisease

2   Initialization

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
® Myocarditis         Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease

2   Initialization
3   Expansion

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
® Myocarditis         Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder

2   Initialization
3   Expansion

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
® Myocarditis         Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium
2   Initialization
3   Expansion

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium
2   Initialization
3   Expansion         affects
Myocardium

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
® Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium
2   Initialization
3   Expansion         affects
Myocardium

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
® Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium
2   Initialization
3   Expansion         affects
Myocardium, Muscle

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
® Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium
2   Initialization
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium
2   Initialization
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart

isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
® Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium
2   Initialization
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart

isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
® Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium,
2   Initialization                 ∀affects.∀isPartOf.¬Heart
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart

isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium,
2   Initialization                 ∀affects.∀isPartOf.¬Heart
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart,
∀isPartOf.¬Heart
isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium,
2   Initialization                 ∀affects.∀isPartOf.¬Heart
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart,
∀isPartOf.¬Heart
isPartOf
Heart, ¬Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium,
2   Initialization                 ∀affects.∀isPartOf.¬Heart
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart,
∀isPartOf.¬Heart
isPartOf
Heart, ¬Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
® Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium,
2   Initialization                 ∀affects.∀isPartOf.¬Heart
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart,
4   Backtracking
∀isPartOf.¬Heart
isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
® Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium, HeartDisease
2   Initialization
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart
4   Backtracking

isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium, HeartDisease
2   Initialization
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart
4   Backtracking

isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

E XAMPLE
Myocarditis        Disorder ∃affects.Myocardium
Myocardium          Muscle ∃isPartOf.Heart
HeartDisease         Disorder ∃affects.∃isPartOf.Heart
Disorder         ∀affects.∀isPartOf.¬Heart HeartDisease
?- Myocarditis        HeartDisease − Yes!

1   Normalization                  Myocarditis, ¬HeartDisease, Disorder,
∃affects.Myocardium, HeartDisease
2   Initialization
3   Expansion         affects
Myocardium, Muscle, ∃isPartOf.Heart
4   Backtracking

isPartOf
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   10/36
Tableau-Based Reasoning

O BSERVATIONS
1   Classiﬁcation requires enumeration:
Every subsumption A B has to be checked separately
E.g., 300,000 atomic concepts (SNOMED CT) result in
90,000,000,000 subsumption tests
Over 99.99% of subsumptions do not hold

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   11/36
Tableau-Based Reasoning

O BSERVATIONS
1   Classiﬁcation requires enumeration:
Every subsumption A B has to be checked separately
E.g., 300,000 atomic concepts (SNOMED CT) result in
90,000,000,000 subsumption tests
Over 99.99% of subsumptions do not hold
2   Excessive non-determinism:
Concept deﬁnitions A ≡ B ∃R.C are very common
Normalization produces disjunctions: B A ∀R.¬C
Often B is a generic commonly-occuring concept:
HeartDisease ≡ Disorder           ∃affects.∃isPartOf.Heart
And so, the rules with Disorder          . . . apply very often

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   11/36
Tableau-Based Reasoning

O BSERVATIONS
1   Classiﬁcation requires enumeration:
Every subsumption A B has to be checked separately
E.g., 300,000 atomic concepts (SNOMED CT) result in
90,000,000,000 subsumption tests
Over 99.99% of subsumptions do not hold
2   Excessive non-determinism:
Concept deﬁnitions A ≡ B ∃R.C are very common
Normalization produces disjunctions: B A ∀R.¬C
Often B is a generic commonly-occuring concept:
HeartDisease ≡ Disorder           ∃affects.∃isPartOf.Heart
And so, the rules with Disorder          . . . apply very often
3   The models can be very very very large. . .
which makes every subsumption test very expensive

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   11/36
Tableau-Based Reasoning

R ECIPROCAL L INKS
E XAMPLE
Heart Organ
MuscularOrgan ≡ Organ ∃hasPart.Muscle
Myocardium Muscle ∃isPartOf.Heart

isPartOf      hasPart−
|= Heart      MuscularOrgan

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   12/36
Tableau-Based Reasoning

R ECIPROCAL L INKS
E XAMPLE
Heart Organ
MuscularOrgan ≡ Organ ∃hasPart.Muscle
Myocardium Muscle ∃isPartOf.Heart

isPartOf      hasPart−
|= Heart      MuscularOrgan

Heart
Organ

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   12/36
Tableau-Based Reasoning

R ECIPROCAL L INKS
E XAMPLE
Heart Organ
MuscularOrgan ≡ Organ ∃hasPart.Muscle
Myocardium Muscle ∃isPartOf.Heart
Heart ∃hasPart.Myocardium
isPartOf hasPart−
|= Heart MuscularOrgan

isPartOf

Heart
Myocardium
Organ
Muscle
hasPart        MuscularOrgan

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   12/36
Tableau-Based Reasoning

C YCLES IN O NTOLOGIES
E XAMPLE
Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

∆

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   13/36
Tableau-Based Reasoning

C YCLES IN O NTOLOGIES
E XAMPLE
® Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

Heart
isComponentOf
CirculatorySystem

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   13/36
Tableau-Based Reasoning

C YCLES IN O NTOLOGIES
E XAMPLE
Heart       ∃isComponentOf.CirculatorySystem
® CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

Heart
isComponentOf
CirculatorySystem
hasComponent
Lungs

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   13/36
Tableau-Based Reasoning

C YCLES IN O NTOLOGIES
E XAMPLE
Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
® Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

Heart
isComponentOf
CirculatorySystem
hasComponent
Lungs
isServedBy
PulmonaryArtery

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   13/36
Tableau-Based Reasoning

C YCLES IN O NTOLOGIES
E XAMPLE
Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
® PulmonaryArtery        ∃serves.Heart

Heart
isComponentOf
CirculatorySystem
hasComponent
Lungs
isServedBy
PulmonaryArtery
serves
Heart

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   13/36
Tableau-Based Reasoning

C YCLES IN O NTOLOGIES
E XAMPLE
® Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

Heart
isComponentOf
CirculatorySystem
hasComponent
Lungs
isServedBy
PulmonaryArtery
serves
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   13/36
Tableau-Based Reasoning

C YCLES IN O NTOLOGIES
E XAMPLE
Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

Heart
isComponentOf
CirculatorySystem
hasComponent
Lungs
isServedBy
PulmonaryArtery
serves
Heart

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   13/36
Tableau-Based Reasoning

B LOCKING IN P RACTICE
E XAMPLE
Heart−component − CirculatorySystem
PulmonaryArtery−component − CirculatorySystem
PulmonaryArtery−serve − Heart
ArterialOrgan ≡ Organ ∃isServedBy.Artery

Heart, Organ

isComponentOf
CirculatorySystem
hasComponent
PulmonaryArtery,
Artery
serves
Heart, Organ

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   14/36
Tableau-Based Reasoning

B LOCKING IN P RACTICE
E XAMPLE
Heart−component − CirculatorySystem
PulmonaryArtery−component − CirculatorySystem
PulmonaryArtery−serve − Heart
® ArterialOrgan ≡ Organ ∃isServedBy.Artery

Heart, Organ

isComponentOf
CirculatorySystem
hasComponent
PulmonaryArtery,
Artery
serves
Heart, Organ

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   14/36
Tableau-Based Reasoning

B LOCKING IN P RACTICE
E XAMPLE
Heart−component − CirculatorySystem
PulmonaryArtery−component − CirculatorySystem
PulmonaryArtery−serve − Heart
® ArterialOrgan ≡ Organ ∃isServedBy.Artery

Heart, Organ

isComponentOf
CirculatorySystem
hasComponent
PulmonaryArtery,
Artery
serves
Heart, Organ, ArterialOrgan

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   14/36
Tableau-Based Reasoning

B LOCKING IN P RACTICE
E XAMPLE
Heart−component − CirculatorySystem
PulmonaryArtery−component − CirculatorySystem
® PulmonaryArtery−serve − Heart
ArterialOrgan ≡ Organ ∃isServedBy.Artery

Heart, Organ

isComponentOf                isServedBy
PulmonaryArtery
CirculatorySystem
hasComponent
PulmonaryArtery,
Artery
serves
Heart, Organ, ArterialOrgan

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   14/36
Tableau-Based Reasoning

B LOCKING IN P RACTICE
E XAMPLE
Heart−component − CirculatorySystem
PulmonaryArtery−component − CirculatorySystem
PulmonaryArtery−serve − Heart
® ArterialOrgan ≡ Organ ∃isServedBy.Artery

Heart, Organ

isComponentOf         isServedBy
PulmonaryArtery,
CirculatorySystem
Artery
hasComponent
PulmonaryArtery,
Artery
serves
Heart, Organ, ArterialOrgan

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   14/36
Tableau-Based Reasoning

B LOCKING IN P RACTICE
E XAMPLE
Heart−component − CirculatorySystem
PulmonaryArtery−component − CirculatorySystem
PulmonaryArtery−serve − Heart
® ArterialOrgan ≡ Organ ∃isServedBy.Artery

Heart, Organ, ArterialOrgan

isComponentOf         isServedBy
PulmonaryArtery,
CirculatorySystem
Artery
hasComponent
PulmonaryArtery,
Artery
serves
Heart, Organ, ArterialOrgan

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   14/36
Tableau-Based Reasoning

O BSERVATIONS
1   Blocking is not persistent:
Blocking of nodes also depend on predecessor nodes
The “pairwise blocking” strategy is commonly used
Nodes are frequently blocked and unblocked
Highly dependent on the order of rule applications

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   15/36
Tableau-Based Reasoning

O BSERVATIONS
1   Blocking is not persistent:
Blocking of nodes also depend on predecessor nodes
The “pairwise blocking” strategy is commonly used
Nodes are frequently blocked and unblocked
Highly dependent on the order of rule applications
2   Models can be very large:
Contain similar nodes at different stages of expansion
The parts below the blocked are not discarded

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   15/36
Tableau-Based Reasoning

O BSERVATIONS
1   Blocking is not persistent:
Blocking of nodes also depend on predecessor nodes
The “pairwise blocking” strategy is commonly used
Nodes are frequently blocked and unblocked
Highly dependent on the order of rule applications
2   Models can be very large:
Contain similar nodes at different stages of expansion
The parts below the blocked are not discarded
3   Blocking conditions are hard to check
Required after every rule application

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   15/36
Consequence-Based Reasoning

O UTLINE

1   I NTRODUCTION

2   TABLEAU -BASED R EASONING

3   C ONSEQUENCE -BASED R EASONING

4   R ELATED M ETHODS

5   C ONCLUSIONS

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   16/36
Consequence-Based Reasoning

EL FAMILY OF DL S
Introduced by [Baader, Brandt, Lutz; IJCAI 2003, 2005]
Name                    DL syntax                    First-Order syntax
top
intersection             C1 C2                       C1 (x) ∧ C2 (x)                     =E
existential restr.        ∃r.C                     ∃y.[r(x, y) ∧ C(y)]                    L
concept inclusion        C1 C2                     ∀x.[C1 (x) → C2 (x)]

Redeﬁnes the basic DL: EL = ALC \ {⊥, ¬, ∀}
Reasoning problems are PTime-complete

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   17/36
Consequence-Based Reasoning

EL FAMILY OF DL S
Introduced by [Baader, Brandt, Lutz; IJCAI 2003, 2005]
Name                    DL syntax                      First-Order syntax
top
intersection             C1 C2                          C1 (x) ∧ C2 (x)                  =E
existential restr.        ∃r.C                       ∃y.[r(x, y) ∧ C(y)]                  L
concept inclusion        C1 C2                       ∀x.[C1 (x) → C2 (x)]
bottom                     ⊥                                    ⊥                        +⊥
role inclusion           r1 r2                     ∀xy.[r1 (x, y) → r2 (x, y)]           +H

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   17/36
Consequence-Based Reasoning

EL FAMILY OF DL S
Introduced by [Baader, Brandt, Lutz; IJCAI 2003, 2005]
Name                    DL syntax                    First-Order syntax
top
intersection        C1 C2               C1 (x) ∧ C2 (x)               =E
existential restr.    ∃r.C            ∃y.[r(x, y) ∧ C(y)]              L
concept inclusion   C1 C2            ∀x.[C1 (x) → C2 (x)]
bottom                 ⊥                       ⊥                      +⊥
role inclusion       r1 r2        ∀xy.[r1 (x, y) → r2 (x, y)]         +H
nominals                o                    x o                       +
complex RIAs      r1 ◦ r2 r3 ∀xyz.[r1 (x, y) ∧ r2 (y, z) → r3 (x, z)]  +

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   17/36
Consequence-Based Reasoning

EL FAMILY OF DL S
Introduced by [Baader, Brandt, Lutz; IJCAI 2003, 2005]
Name                    DL syntax                    First-Order syntax
top
intersection        C1 C2               C1 (x) ∧ C2 (x)               =E
existential restr.    ∃r.C            ∃y.[r(x, y) ∧ C(y)]              L
concept inclusion   C1 C2            ∀x.[C1 (x) → C2 (x)]
bottom                 ⊥                       ⊥                      +⊥
role inclusion       r1 r2        ∀xy.[r1 (x, y) → r2 (x, y)]         +H
nominals                o                    x o                       +
complex RIAs      r1 ◦ r2 r3 ∀xyz.[r1 (x, y) ∧ r2 (y, z) → r3 (x, z)]  +

EL++ :
has polynomial-model property
classiﬁcation can be computed in polynomial time
basis of the OWL 2 EL proﬁle

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   17/36
Consequence-Based Reasoning

ELH E XPRESSIVITY
Surprisingly useful:
SNOMED CT GO NCI Galen
         

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   18/36
Consequence-Based Reasoning

ELH E XPRESSIVITY
Surprisingly useful:
SNOMED CT GO NCI Galen
           
Simple inclusions:
Myocardium       Muscle ∃isPartOf.Heart
Myocarditis      Disorder ∃affects.Myocardium

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   18/36
Consequence-Based Reasoning

ELH E XPRESSIVITY
Surprisingly useful:
SNOMED CT GO NCI Galen
           
Simple inclusions:
Myocardium Muscle ∃isPartOf.Heart
Myocarditis Disorder ∃affects.Myocardium
Concept deﬁnitions:
MuscularOrgan ≡ Organ ∃hasPart.Muscle
KidneyExamination ≡ ClinicalAct
∃hasSubprocess.(Examination ∃involves.Kidney)

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   18/36
Consequence-Based Reasoning

ELH E XPRESSIVITY
Surprisingly useful:
SNOMED CT GO NCI Galen
           
Simple inclusions:
Myocardium Muscle ∃isPartOf.Heart
Myocarditis Disorder ∃affects.Myocardium
Concept deﬁnitions:
MuscularOrgan ≡ Organ ∃hasPart.Muscle
KidneyExamination ≡ ClinicalAct
∃hasSubprocess.(Examination ∃involves.Kidney)
General concept inclusions:
Structure    ∃isPartOf.Heart
∃isComponentOf.CardiovascularSystem

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   18/36
Consequence-Based Reasoning

ELH E XPRESSIVITY
Surprisingly useful:
SNOMED CT GO NCI Galen
            

E XAMPLE (G ALEN )
 BasilarArtery ∃hasBranch.VertebralArtery
 VertebralArtery ∃isBranchOf.BasilarArtery
     hasBranch isBranchOf−
               Fun(isBranchOf)
     hasBranch delimitingAttribute

Over 95% of axioms in Galen are in ELH

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   18/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
E XAMPLE
A   ∃R.(B    C)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
E XAMPLE
A   ∃R.(B    C)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
E XAMPLE
A   ∃R.(B    C)               A    ∃R.D        D      B      C

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
E XAMPLE
A   ∃R.(B    C)               A    ∃R.D        D      B      C

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
E XAMPLE
A   ∃R.(B    C)               A    ∃R.D        D      B      D      C

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A    B    C    A     ∃R.B ∃R.B             C    R      S

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A    B    C    A     ∃R.B ∃R.B             C    R      S
2   Saturation / completion [Brandt; ECAI 2004]:

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A     B   C    A     ∃R.B ∃R.B               C    R      S
2   Saturation / completion [Brandt; ECAI 2004]:
IR1                            IR2
A   A                         A

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A     B   C    A     ∃R.B ∃R.B               C    R      S
2   Saturation / completion [Brandt; ECAI 2004]:
IR1                            IR2
A   A                         A

A   B   B   C
CR1
A   C

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A     B   C    A     ∃R.B ∃R.B               C      R    S
2   Saturation / completion [Brandt; ECAI 2004]:
IR1                            IR2
A   A                         A

A   B   B   C                 A     B     A       C    B     C      D
CR1                            CR2
A   C                                     A       D

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A     B   C    A     ∃R.B ∃R.B               C      R    S
2   Saturation / completion [Brandt; ECAI 2004]:
IR1                            IR2
A   A                         A

A   B   B   C                 A     B     A       C    B     C      D
CR1                            CR2
A   C                                     A       D

A   B   B ∃R.C
CR3
A   ∃R.C

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A     B   C    A     ∃R.B ∃R.B               C      R    S
2   Saturation / completion [Brandt; ECAI 2004]:
IR1                            IR2
A   A                         A

A   B   B   C                 A     B     A       C    B     C      D
CR1                            CR2
A   C                                     A       D

A   B   B ∃R.C                A     ∃R.B R            S
CR3                            CR4
A   ∃R.C                        A ∃S.B

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

ELH C LASSIFICATION P ROCEDURE
1   Normalization / structural transformation:
N ORMAL FORMS
A       B A     B   C    A     ∃R.B ∃R.B               C      R    S
2   Saturation / completion [Brandt; ECAI 2004]:
IR1                            IR2
A   A                         A

A   B   B   C                 A     B     A       C    B     C      D
CR1                            CR2
A   C                                     A       D

A   B   B ∃R.C                A     ∃R.B R            S
CR3                            CR4
A   ∃R.C                        A ∃S.B

A   ∃R.B    B    C ∃R.C            D
CR5
A    D
Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   19/36
Consequence-Based Reasoning

O BSERVATIONS
1   Procedure is more goal-directed:
Derives only subsumptions of the form A B or A ∃r.B
Only consequences of the axioms are derived
No enumeration: all subsumptions are derived in one pass

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   20/36
Consequence-Based Reasoning

O BSERVATIONS
1   Procedure is more goal-directed:
Derives only subsumptions of the form A B or A ∃r.B
Only consequences of the axioms are derived
No enumeration: all subsumptions are derived in one pass
2   Useful computational properties:
Polynomial worst-case complexity
No non-determinism, no backtracking
Relatively easy to implement
Easy to track dependencies for explanations
Can be made incremental, distributed, and parallel

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   20/36
Consequence-Based Reasoning

R ECIPROCAL L INKS AND C YCLES
E XAMPLE
Heart      ∃isComponentOf.CirculatorySystem
CirculatorySystem      ∃hasComponent.Lungs
Lungs      ∃isServedBy.PulmonaryArtery
PulmonaryArtery       ∃serves.Heart

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   21/36
Consequence-Based Reasoning

R ECIPROCAL L INKS AND C YCLES
E XAMPLE
Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

Inferences require matching existential restrictions:
A ∃R.B B C ∃R.C D
A D

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   21/36
Consequence-Based Reasoning

R ECIPROCAL L INKS AND C YCLES
E XAMPLE
Heart       ∃isComponentOf.CirculatorySystem
CirculatorySystem       ∃hasComponent.Lungs
Lungs       ∃isServedBy.PulmonaryArtery
PulmonaryArtery        ∃serves.Heart

Inferences require matching existential restrictions:
A ∃R.B B C ∃R.C D
A D
No inference is made for just positive existential restrictions
(FMA is trivially classiﬁed)

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   21/36
Consequence-Based Reasoning

B EYOND ELH
Galen uses two constructors that are outside of ELH:
inverse roles and role functionality:

E XAMPLE (G ALEN )
 BasilarArtery ∃hasBranch.VertebalArtery
 VertebalArtery ∃isBranchOf.BasilarArtery
    hasBranch isBranchOf−
              Fun(isBranchOf)
    hasBranch delimitingAttribute

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   22/36
Consequence-Based Reasoning

B EYOND ELH
Galen uses two constructors that are outside of ELH:
inverse roles and role functionality:

E XAMPLE (G ALEN )
 BasilarArtery ∃hasBranch.VertebalArtery
 VertebalArtery ∃isBranchOf.BasilarArtery
    hasBranch isBranchOf−
              Fun(isBranchOf)
    hasBranch delimitingAttribute

Adding either results in complexity increase
from PTime to ExpTime [Baader, Brandt, Lutz 2005; 2008]

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   22/36
Consequence-Based Reasoning

B EYOND ELH
Galen uses two constructors that are outside of ELH:
inverse roles and role functionality:

E XAMPLE (G ALEN )
 BasilarArtery ∃hasBranch.VertebalArtery
 VertebalArtery ∃isBranchOf.BasilarArtery
    hasBranch isBranchOf−
              Fun(isBranchOf)
    hasBranch delimitingAttribute

Adding either results in complexity increase
from PTime to ExpTime [Baader, Brandt, Lutz 2005; 2008]
We are not scared of the high complexity!

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   22/36
Consequence-Based Reasoning

SHIQ
Name                 DL syntax                 First-Order syntax
intersection         C1 C2                    C1 (x) ∧ C2 (x)
union                C1 C2                    C1 (x) ∨ C2 (x)           =A
complement                ¬C                         ¬C(x)                L
value restriction       ∀r.C               ∀y.[r(x, y) → C(y)]            C
existential restr.      ∃r.C                ∃y.[r(x, y) ∧ C(y)]
transitivity           Tra(r)        ∀xyz.[r(x, y) ∧ r(y, z) → r(x, z)] = S
functionality         Fun(r)         ∀xyz.[r(x, y) ∧ r(x, z) → y z] +F
role inclusion        r1 r2             ∀xy.[r1 (x, y) → r2 (x, y)]     +H
inverse roles      [. . . r− . . . ]          [. . . r(y, x) . . . ]    +I
number restriction        n r.C           ∃ ≤n y.[r(x, y) ∧ C(y)]       +Q
SHIQ:
has a generalized tree-model property (transitivity)
has no ﬁnite-model property (because of functionality)
satisﬁability problem is ExpTime-complete

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   23/36
Consequence-Based Reasoning

H ORN SHIQ
Name                       positive          negative        Horn-
intersection           ·    C1 C2 C1 C2 ·
union                        −          C1 C2 ·                =A
complement                 · ¬C                  −              L
value restriction         · ∀r.C                 −              C
existential restr.        · ∃r.C           ∃r.C ·
transitivity                       Tra(r)                       =S
functionality                     Fun(r)                        +F
role inclusion                    r1 r2                         +H
inverse roles                  [. . . r− . . . ]                +I
number restriction      ·     1 r.C              −              +Q
Horn SHIQ:
can be translated to the Horn fragment of ﬁrst-order logic
the reasoning problems are ExpTime-complete
data complexity (quiering assertions) is PTime-complete
Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   23/36
Consequence-Based Reasoning

N EW I NFERENCE RULES
A    ∃R.B A     ∀R.C
1
A ∃R.(B     C)

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies   24/36
Consequence-Based Reasoning

N EW I NFERENCE RULES
A    ∃R.B A     ∀R.C
1
A ∃R.(B     C)
A    ∃R.B ∃R− .A C
2                                         [(∃R− .A       C) ≡ (A         ∀R.C)]
A ∃R.(B C)

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies     24/36
Consequence-Based Reasoning

N EW I NFERENCE RULES
A    ∃R.B A         ∀R.C
1
A ∃R.(B         C)
A    ∃R.B ∃R− .A C
2                                             [(∃R− .A       C) ≡ (A         ∀R.C)]
A ∃R.(B C)
A    ∃R.B     A ∃R.C Fun(R)
3
A      ∃R.(B C)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies     24/36
Consequence-Based Reasoning

N EW I NFERENCE RULES
A    ∃R.B A         ∀R.C
1
A ∃R.(B         C)
A    ∃R.B ∃R− .A C
2                                             [(∃R− .A       C) ≡ (A         ∀R.C)]
A ∃R.(B C)
A    ∃R.B     A ∃R.C Fun(R)
3
A      ∃R.(B C)
A    ∃R.B     A      ∃R.C B D C                   D     A        1 R.D
4
A ∃R.(B C)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies     24/36
Consequence-Based Reasoning

N EW I NFERENCE RULES
A    ∃R.B A            ∀R.C
1
A ∃R.(B            C)
A        ∃R.B ∃R− .A C
2                                                  [(∃R− .A       C) ≡ (A         ∀R.C)]
A ∃R.(B C)
A       ∃R.B     A ∃R.C Fun(R)
3
A      ∃R.(B C)
A       ∃R.B     A      ∃R.C B D C                   D     A        1 R.D
4
A ∃R.(B C)
5     Old rules should be extended for new conjunctions:
A    ∃R.(B        C)    B     C      D ∃R.D            E
CR5
A     E

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies     24/36
Consequence-Based Reasoning

N EW I NFERENCE RULES
A    ∃R.B A            ∀R.C
1
A ∃R.(B            C)
A        ∃R.B ∃R− .A C
2                                                  [(∃R− .A       C) ≡ (A         ∀R.C)]
A ∃R.(B C)
A       ∃R.B     A ∃R.C Fun(R)
3
A      ∃R.(B C)
A       ∃R.B     A      ∃R.C B D C                   D     A        1 R.D
4
A ∃R.(B C)
5     Old rules should be extended for new conjunctions:
A    ∃R.(B        C)    B     C      D ∃R.D            E
CR5
A     E

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies     24/36
Consequence-Based Reasoning

N EW I NFERENCE RULES
M    ∃R.N M          ∀R.C
1
M ∃R.(N          C)
M    A ∃R.N ∃R− .A C
2
M A ∃R.(N C)
M       ∃R.N1 M ∃R.N2 Fun(R)
3
M ∃R.(N1 N2 )
M       ∃R.N1   M      ∃R.N2 N1 D N2                    D M              1 R.D
4
M ∃R.(N1 N2 )
5     Old rules should be extended for new conjunctions:
M    ∃R.N    M      D ∃R.D           E
CR5
M      E
M, N ∗ =     Ai                                                             all rules

Yevgeny Kazakov    Consequence-Based Reasoning for DL Ontologies          24/36
Consequence-Based Reasoning

O BSERVATIONS
1   Optimal complexity:
Derives only subsumptions of the form:
Ai   B   or        Ai    ∃R.       Bj
At most exponential number of inferences is possible

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   25/36
Consequence-Based Reasoning

O BSERVATIONS
1   Optimal complexity:
Derives only subsumptions of the form:
Ai   B   or        Ai    ∃R.       Bj
At most exponential number of inferences is possible
2   “Pay as you go" behaviour:
Remains polynomial for ELH
because the rules forming conjunctions never apply:
A    ∃R.B A        ∀R.C
A ∃R.(B        C)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   25/36
Consequence-Based Reasoning

O BSERVATIONS
1   Optimal complexity:
Derives only subsumptions of the form:
Ai   B   or        Ai    ∃R.       Bj
At most exponential number of inferences is possible
2   “Pay as you go" behaviour:
Remains polynomial for ELH
because the rules forming conjunctions never apply:
A    ∃R.B A        ∀R.C
A ∃R.(B        C)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   25/36
Consequence-Based Reasoning

E XPERIMENTAL R ESULTS
GO    NCI Galen v.0 Galen v.7 SNOMED CT
Concepts: 20465 27652      2748    23136       389472
FACT++     15.24   6.05  465.35         —       650.37
H ERMI T  199.52 169.47   45.72         —           —
P ELLET    72.02 26.47        —         —           —
CEL         1.84   5.76       —         —     1185.70
CB          1.17   3.57     0.32     9.58        49.44
Speed-Up: 1.57X 1.61X      143X         ∞      13.15X
The prototype reasoner CB implementing the procedure is
available open source from:

[Demo?]

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   26/36
Related Methods

O UTLINE

1   I NTRODUCTION

2   TABLEAU -BASED R EASONING

3   C ONSEQUENCE -BASED R EASONING

4   R ELATED M ETHODS

5   C ONCLUSIONS

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   27/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A      ∃R.B

B     A
∃R.A     C
?- A     C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A      ∃R.B

B     A
® ∃R.A     C
?- A     C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A      ∃R.B

B     A
A     ∀R− .C
® ?- A     C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A      ∃R.B

B     A
A     ∀R− .C
® ?- A     C

A, ¬C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
®A       ∃R.B

B     A
A     ∀R− .C
?- A     C

A, ¬C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
®A         ∃R.B

B       A
A       ∀R− .C
?- A       C

A, ¬C
R
B

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B

®B         A
A       ∀R− .C
?- A       C

A, ¬C
R
B

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B

®B         A
A       ∀R− .C
?- A       C

A, ¬C
R
B, A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B

B       A
®A         ∀R− .C
?- A       C

A, ¬C
R
B, A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B

B       A
®A         ∀R− .C
?- A       C

A, ¬C, C
R
B, A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B

B       A
A       ∀R− .C
?- A       C

A, ¬C, C
R
B, A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
®A         ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
®B         A                           ¬B(x) ∨ A(x)
®A         ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C
R
B, A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
® ?- A       C

A, ¬C, C                   A(c)
R                              ¬C(c)
B, A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C                   A(c)
R                              ¬C(c)
B, A

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C              ® A(c)
R                           ¬C(c)
B, A

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C              ® A(c)
R                           ¬C(c)
R(c, f (c))
B, A

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                         ¬A(x) ∨ R(x, f (x))
® ¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C              ® A(c)
R                           ¬C(c)
R(c, f (c))
B, A

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                         ¬A(x) ∨ R(x, f (x))
® ¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C              ® A(c)
R                           ¬C(c)
R(c, f (c))
B, A                    B(f (c))

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                          ® ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C                A(c)
R                           ¬C(c)
R(c, f (c))
B, A                  ® B(f (c))

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                          ® ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C                A(c)
R                           ¬C(c)
R(c, f (c))
B, A                  ® B(f (c))
A(f (c))

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                    ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                       ¬B(x) ∨ A(x)
A       ∀R− .C               ® ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C                A(c)
R                           ¬C(c)
® R(c, f (c))
B, A                    B(f (c))
® A(f (c))

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                    ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                       ¬B(x) ∨ A(x)
A       ∀R− .C               ® ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C                A(c)
R                           ¬C(c)
® R(c, f (c))
B, A                    B(f (c))
® A(f (c))
C(c)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C               A(c)
R                        ® ¬C(c)
R(c, f (c))
B, A                   B(f (c))
A(f (c))
® C(c)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A        ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

A, ¬C, C               A(c)
R                        ® ¬C(c)
R(c, f (c))
B, A                   B(f (c))
A(f (c))
® C(c)
⊥

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)
R                              ¬C(c)
R(c, f (c))
f (c)     B, A                       B(f (c))
A(f (c))
C(c)
⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
®A         ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)
R                              ¬C(c)
R(c, f (c))
f (c)     B, A                       B(f (c))
A(f (c))
C(c)
⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
®A         ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)
R                              ¬C(c)
R(c, f (c))
f (c)     B, A                       B(f (c))
R                               A(f (c))
B                          C(c)
⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
®B         A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)
R                              ¬C(c)
R(c, f (c))
f (c)     B, A                       B(f (c))
R                               A(f (c))
B                          C(c)
⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
®B         A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)
R                              ¬C(c)
R(c, f (c))
f (c)     B, A                       B(f (c))
R                               A(f (c))
B, A                       C(c)
⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
®A         ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)
R                              ¬C(c)
R(c, f (c))
f (c)     B, A                       B(f (c))
R                               A(f (c))
B, A                       C(c)
⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
®A         ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)
R                              ¬C(c)
R(c, f (c))
f (c)     B, A                       B(f (c))
R                               A(f (c))
B, A                       C(c)
R                               ⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                A(c)
R                           ¬C(c)
R(c, f (c))
f (c)     B, A                    B(f (c))
R                          ® A(f (c))
B, A                    C(c)
R                            ⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                A(c)        R(f (c), f (f (c)))
R                           ¬C(c)
R(c, f (c))
f (c)     B, A                    B(f (c))
R                          ® A(f (c))
B, A                    C(c)
R                            ⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                         ¬A(x) ∨ R(x, f (x))
® ¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                A(c)        R(f (c), f (f (c)))
R                           ¬C(c)
R(c, f (c))
f (c)     B, A                    B(f (c))
R                          ® A(f (c))
B, A                    C(c)
R                            ⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                         ¬A(x) ∨ R(x, f (x))
® ¬A(x) ∨ B(f (x))
B       A                            ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                A(c)        R(f (c), f (f (c)))
R                           ¬C(c)        B(f (f (c)))
R(c, f (c))
f (c)     B, A                    B(f (c))
R                          ® A(f (c))
f (f (c)) B, A                      C(c)
R                           ⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

TABLEAU VS . H YPER -R ESOLUTION
E XAMPLE
A       ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B       A                           ¬B(x) ∨ A(x)
A       ∀R− .C                     ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A       C

c    A, ¬C, C                   A(c)            R(f (c), f (f (c)))
R                              ¬C(c)            B(f (f (c)))
R(c, f (c))     ...
f (c)     B, A                       B(f (c))        No termination!
R                               A(f (c))
f (f (c)) B, A                         C(c)
R                              ⊥

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   28/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B

B   A
∃R.A   C
?- A   C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B

B    A
∃R.A    C
?- A    C
A   ∃R.B B     A ∃R.A         C
A     C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
®A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
®B     A                           ¬B(x) ∨ A(x)
® ∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A    ∃R.B B     A ∃R.A         C
A     C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B   A                           ¬B(x) ∨ A(x)
∃R.A   C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
® ?- A   C
A   ∃R.B B     A ∃R.A         C                A(c)
A     C                              ¬C(c)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C                A(c)
A     C                              ¬C(c)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                         ® ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C                A(c)
A     C                              ¬C(c)

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                         ® ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C             A(c)
A     C                          ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
® ¬A(x) ∨ B(f (x))
B    A                         ® ¬B(x) ∨ A(x)
∃R.A    C                         ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
® ¬A(x) ∨ B(f (x))
B    A                         ® ¬B(x) ∨ A(x)
∃R.A    C                         ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                           ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C             A(c)
A     C                          ¬C(c)
® ¬A(x) ∨ ¬A(f (x)) ∨ C(x)
® ¬A(x) ∨ A(f (x))

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                           ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C             A(c)
A     C                          ¬C(c)
® ¬A(x) ∨ ¬A(f (x)) ∨ C(x)
® ¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                           ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C             A(c)
A     C                          ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
® ¬A(x) ∨ A(f (x))
® ¬A(x) ∨ C(x)

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                           ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C             A(c)
A     C                          ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
® ¬A(x) ∨ A(f (x))
® ¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                           ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C          ® A(c)
A     C                          ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
® ¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                         ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                            ¬B(x) ∨ A(x)
∃R.A    C                           ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C          ® A(c)
A     C                          ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
® ¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))
C(c)

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))
C(c)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                      ® ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))
® C(c)

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                      ® ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))
® C(c)
⊥
Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))
C(c)
⊥
Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))
C(c)
⊥
Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                        ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A                           ¬B(x) ∨ A(x)
∃R.A    C                          ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C            A(c)
A     C                         ¬C(c)
¬A(x) ∨ ¬A(f (x)) ∨ C(x)
¬A(x) ∨ A(f (x))
¬A(x) ∨ C(x)
¬A(x) ∨ C(f (x))
C(c)
⊥
Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A
∃R.A    C                         ® ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C             A(c)
A     C                          ¬C(c)
® ¬A(x) ∨ ¬A(f (x)) ∨ C(x)

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

C.B. VS . O RDERED R ESOLUTION
E XAMPLE
A   ∃R.B                       ® ¬A(x) ∨ R(x, f (x))
¬A(x) ∨ B(f (x))
B    A
∃R.A    C                         ® ¬R(x, y) ∨ ¬A(y) ∨ C(x)
?- A    C
A   ∃R.B B     A ∃R.A         C             A(c)
A     C                          ¬C(c)
® ¬A(x) ∨ ¬A(f (x)) ∨ C(x)

Every pair of (unrelated) axioms
result in a resolution inference:
A1 ≡ B1 ∃R.C1
A2 ≡ B2 ∃R.C2

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   29/36
Related Methods

AUTOMATA -BASED P ROCEDURES

E XAMPLE
A   ∃R.B
B   A
∃R.A   C
?- A   C

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                      AC               ABC                B
∃R.A   C
?- A   C

C                BC

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
®A    ∃R.B
B   A                                      AC               ABC                B
∃R.A   C
?- A   C

C                BC

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
®B    A                                      AC               ABC                B
∃R.A   C
?- A   C

C                BC

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                      AC               ABC                B
® ∃R.A   C
?- A   C

C                BC

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                      AC               ABC                B
∃R.A   C
® ?- A   C

C                BC

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                         AC               ABC                B
∃R.A   C
?- A   C

Automata emptiness:                                     C                BC
is there a run not going
trough inconsistent
states and edges?

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                         AC               ABC                B
∃R.A   C
?- A   C

Automata emptiness:                                     C                BC
is there a run not going
trough inconsistent
states and edges?
Solvable in polynomial
time by propagating
inconsistent states.

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                         AC               ABC                B
∃R.A   C
?- A   C

Automata emptiness:                                     C                BC
is there a run not going
trough inconsistent
states and edges?
Solvable in polynomial
time by propagating
inconsistent states.

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                         AC               ABC                B
∃R.A   C
?- A   C

Automata emptiness:                                     C                BC
is there a run not going
trough inconsistent
states and edges?
Solvable in polynomial
time by propagating
inconsistent states.

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                         AC               ABC                B
∃R.A   C
?- A   C

Automata emptiness:                                     C                BC
is there a run not going
trough inconsistent
states and edges?
Solvable in polynomial
time by propagating
inconsistent states.

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

AUTOMATA -BASED P ROCEDURES
A                AB
E XAMPLE
A   ∃R.B
B   A                                        AC               ABC                B
∃R.A   C
® ?- B   C

Automata emptiness:                                     C                BC
is there a run not going
trough inconsistent
states and edges?                           Note that other subsumption
Solvable in polynomial                      relations can be also
time by propagating                         determined
inconsistent states.

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies       30/36
Related Methods

O BSERVATIONS
1   Direct implementation is exponential even in the best case:
Builds exponentially-many states
Symbolic representation (BDDs, ZDDs) can be used to
reduce the complexity [Pan, Sattler, Vadi; 2006]

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   31/36
Related Methods

O BSERVATIONS
1   Direct implementation is exponential even in the best case:
Builds exponentially-many states
Symbolic representation (BDDs, ZDDs) can be used to
reduce the complexity [Pan, Sattler, Vadi; 2006]
2   Efﬁcinet implementations are already available:

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   31/36
Related Methods

O BSERVATIONS
1   Direct implementation is exponential even in the best case:
Builds exponentially-many states
Symbolic representation (BDDs, ZDDs) can be used to
reduce the complexity [Pan, Sattler, Vadi; 2006]
2   Efﬁcinet implementations are already available:
Tableau and hyper-resolution can be seen as bottom-up
procedures that search for a run

Yevgeny Kazakov     Consequence-Based Reasoning for DL Ontologies   31/36
Related Methods

O BSERVATIONS
1   Direct implementation is exponential even in the best case:
Builds exponentially-many states
Symbolic representation (BDDs, ZDDs) can be used to
reduce the complexity [Pan, Sattler, Vadi; 2006]
2   Efﬁcinet implementations are already available:
Tableau and hyper-resolution can be seen as bottom-up
procedures that search for a run
Consequence-based and ordered resolution can be seen as
top-down procedures that propagate inconsistent states:

A   ∃R.B   B     C     ∃R.C       D            {B, ¬C} is inconsistent
A    D                             {A, ¬D} is inconsistent

Yevgeny Kazakov      Consequence-Based Reasoning for DL Ontologies   31/36
Conclusions

O UTLINE

1   I NTRODUCTION

2   TABLEAU -BASED R EASONING

3   C ONSEQUENCE -BASED R EASONING

4   R ELATED M ETHODS

5   C ONCLUSIONS

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   32/36
Conclusions

C ONSEQUENCE -BASED R EASONING
Is a new kind of top-down reasoning procedure

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   33/36
Conclusions

C ONSEQUENCE -BASED R EASONING
Is a new kind of top-down reasoning procedure
Avoids non-determinism and backtracking
Computationally optimal and “pay-as-you-go”
Avoids enumerations of subsumption tests
More goal-directed

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   33/36
Conclusions

C ONSEQUENCE -BASED R EASONING
Is a new kind of top-down reasoning procedure
Avoids non-determinism and backtracking
Computationally optimal and “pay-as-you-go”
Avoids enumerations of subsumption tests
More goal-directed
Disconnected from the semantics of DLs
(model-theoretic, not proof-theoretic)
Difﬁcult to extend to disjunctions and counting constructors
(but we are working on it!)

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   33/36
Conclusions

C ONSEQUENCE -BASED R EASONING
Is a new kind of top-down reasoning procedure
Avoids non-determinism and backtracking
Computationally optimal and “pay-as-you-go”
Avoids enumerations of subsumption tests
More goal-directed
Disconnected from the semantics of DLs
(model-theoretic, not proof-theoretic)
Difﬁcult to extend to disjunctions and counting constructors
(but we are working on it!)
Tableau-based reasoners are catching up:
Hyper-tableau procedures reduce non-determinism
Smarter blocking: “core blocking”, “speculative blocking”
Reducing the number of subsumption tests by ﬁnding
non-subsumptions from the models

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   33/36
Conclusions

L ESSONS LEARNED
What is important:
Knowing the input (kinds of constructors, their usage)
Avoiding destructive transformations

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   34/36
Conclusions

L ESSONS LEARNED
What is important:
Knowing the input (kinds of constructors, their usage)
Avoiding destructive transformations
What is not that important:
Worst case complexity:
even O(n2 )-procedure can be impractical
Complying with standards:
not a big deal if nominals are not supported

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   34/36
Conclusions

L ESSONS LEARNED
What is important:
Knowing the input (kinds of constructors, their usage)
Avoiding destructive transformations
What is not that important:
Worst case complexity:
even O(n2 )-procedure can be impractical
Complying with standards:
not a big deal if nominals are not supported
Something to consider:
Things are not as easy as they may seem
Reductions (e.g., to general ATP) don’t work well in the end
Implementation makes huge difference: proﬁle a lot!

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies   34/36
R EFERENCES
Baader, F., Brandt, S., Lutz, C.: Pushing the EL Envelope.
IJCAI 2005: 364-369
Kazakov, Y.: Consequence-Driven Reasoning for Horn
SHIQ Ontologies. IJCAI 2009: 2040-2045
Pan, G., Sattler, U., Vardi, M. Y.: BDD-based decision
procedures for the modal logic K. Journal of Applied
Non-Classical Logics 16(1-2): 169-208 (2006)
Motik, B., Shearer, R., Horrocks, I.: Hypertableau
Reasoning for Description Logics. JAIR 36: 165-228 (2009)
Glimm, B., Horrocks, I., Motik, B.: Optimized Description
Logic Reasoning via Core Blocking. IJCAR 2010.

Yevgeny Kazakov   Consequence-Based Reasoning for DL Ontologies   35/36
T HE I NFERENCE RULES FOR H ORN SHIQ
n
M       A1 . . . M       An
:         Ai     C∈O
M     A      A           M                             M C
i=1

M     ∃R.N N              ⊥                M     ∃R1 .N M             ∀R2 .A
: R1        O   R2
M ⊥                                      M ∃R1 .(N             A)

M      ∃R1 .N       N      ∀R2 .A
: R1   O   R2 −
M         A

M     ∃R1 .N 1       N1   B                      M      ∃R1 .N 1 M B
M     ∃R2 .N 2       N2   B                      N1     ∃R2 .(N 2 A)
M       1 S.B                     R1   O    S    N1       1 S.B N 2 A                    B R1    O   S−
:                                                           :
M       ∃R1 .(N 1     N 2)        R2   O    S       M       A    M        ∃R− .N 1
2
R2   O   S

Where M, N =             Ai

Yevgeny Kazakov       Consequence-Based Reasoning for DL Ontologies        36/36

```
To top