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Consequence-Based Reasoning for Description Logic Ontologies

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Consequence-Based Reasoning for Description Logic Ontologies Powered By Docstoc
					 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
    Advantages
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
           [Hustadt, Motik, Saatler; JAR 2007]
                     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:

            cb-reasoner.googlecode.com

                                [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
Advantages over tableau-based procedures:
    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
Advantages over tableau-based procedures:
    Avoids non-determinism and backtracking
    Computationally optimal and “pay-as-you-go”
    Avoids enumerations of subsumption tests
    More goal-directed
Disadvantages:
    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
Advantages over tableau-based procedures:
    Avoids non-determinism and backtracking
    Computationally optimal and “pay-as-you-go”
    Avoids enumerations of subsumption tests
    More goal-directed
Disadvantages:
    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.

             Thank you for your attension!


            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

				
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