Description Logics                                 in RTE                               Kilian Evang                      ...
Description LogicsDescription Logics                                                  in RTE                              ...
Description LogicsIndividuals, Concepts, Roles          in RTE                                    Kilian Evang            ...
Description LogicsSHOIN (D)                                                       in RTE                                  ...
Description LogicsExpressions in SHOIN (D)                                                 in RTE                         ...
Description LogicsInterpretations                                          in RTE                                         ...
Description LogicsIndividual Names                                              in RTE                                    ...
Description LogicsAtomic Roles                                              in RTE                                        ...
Description LogicsInverse Roles                                              in RTE                                       ...
Description LogicsAtomic Concepts                                in RTE                                             Kilian...
Description LogicsConjunction                                           in RTE                                            ...
Description LogicsDisjunction                                                  in RTE                                     ...
Description LogicsNegation                                                      in RTE                                    ...
Description LogicsExists Restriction                                                    in RTE                            ...
Description LogicsNumber Restrictions                                                in RTE                               ...
Description LogicsValue Restriction                                              in RTE                                   ...
Description LogicsNominals                                                         in RTE                                 ...
Description LogicsThe Universal Concept and the Bottom Concept        in RTE                                              ...
Description LogicsInclusions                                                 in RTE                                       ...
Description LogicsEqualities                                                    in RTE                                    ...
Description LogicsTransitive Roles                                                 in RTE                                 ...
Description LogicsConcept Assertions                                          in RTE                                      ...
Description LogicsRole Assertions                                  in RTE                                               Ki...
Description LogicsConcrete Domains                                                  in RTE                                ...
Description LogicsComparison of Four DLs                                            in RTE                                ...
Description LogicsKnowledge Bases                                                 in RTE                                  ...
Description LogicsAn Example Knowledge Base                                               in RTE                          ...
Description LogicsModelhood                                                          in RTE                               ...
Description LogicsReasoning Tasks for Concepts                                        in RTE                              ...
Description LogicsReasoning Tasks for Knowledge Bases                                   in RTE                            ...
Description Logics[Bedaride, 2003]: RTE in Four Steps                                 in RTE                              ...
Description LogicsStep 1: Represent T and H as Two ABoxes                          in RTE                                 ...
Description LogicsStep 2: TBox with Background Knowledge                                    in RTE                        ...
Description LogicsStep 3: Saturate ABoxes with TBox                                  in RTE                               ...
Description LogicsStep 4: Subgraph-Detect H in T                                         in RTE                           ...
Description LogicsReferences                                                         in RTE                               ...
Description LogicsReferences                                                    in RTE                                    ...
Description Logics                                           in RTE                                         Kilian Evang  ...
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Description Logics in RTE

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Description Logics in RTE

  1. 1. Description Logics in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles ConceptsDescription Logics in RTE Terminological Axioms Assertions Concrete Domains Comparison Reasoning Kilian Evang for Concepts for Knowledge Bases [Bedaride, 2003] T and H Background 2009-07-20 Knowledge ABox Saturation Subgraph Detection Back Matter
  2. 2. Description LogicsDescription Logics in RTE Kilian Evang Introduction SHOIN (D) Individual Names a family of logics Roles Concepts Terminological origins in research on knowledge representation systems Axioms Assertions widely used in practice, notably in Semantic Web Concrete Domains Comparison technology Reasoning for Concepts address expressivity-tractability tradeoff: adequate for Knowledge Bases [Bedaride, 2003] knowledge representation, useful inferencing T and H Background basic standard DL called AL Knowledge ABox Saturation degree of expressivity of a DL can be expressed in terms Subgraph Detection Back Matter of additional constructs added to AL
  3. 3. Description LogicsIndividuals, Concepts, Roles in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Axioms Assertions Concrete Domains Comparison Reasoning for Concepts for Knowledge Bases [Bedaride, 2003] T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter[Horridge et al., 2007], p. 13
  4. 4. Description LogicsSHOIN (D) in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Axioms chosen here because the XML description language Assertions Concrete Domains OWL DL is based on it Comparison Reasoning OWL DL and its subset OWL Lite widely used in for Concepts for Knowledge Bases Semantic Web technology [Bedaride, 2003] extends ALC of [Bedaride, 2003] by several constructs T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter
  5. 5. Description LogicsExpressions in SHOIN (D) in RTE Kilian Evang Introduction individual names SHOIN (D) Individual Names example: paul Roles Concepts denote individuals aka objects Terminological Axioms concepts (aka classes) Assertions Concrete Domains example: Person Comparison Reasoning denote sets of individuals for Concepts roles (aka properties) for Knowledge Bases [Bedaride, 2003] example: hasChild T and H Background denote binary relations between individuals, i.e. sets of Knowledge ABox Saturation ordered pairs of individuals Subgraph Detection formulas Back Matter terminological axioms assertions
  6. 6. Description LogicsInterpretations in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological An interpretation I consists of Axioms Assertions a domain ∆I of individuals and Concrete Domains Comparison an interpretation function ·I that maps Reasoning for Concepts I individual names to elements of ∆ for Knowledge Bases concept descriptions to subsets of ∆I [Bedaride, 2003] T and H role descriptions to subsets of ∆I × ∆I Background Knowledge ABox Saturation Subgraph Detection Back Matter
  7. 7. Description LogicsIndividual Names in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Syntax: a Terminological Axioms Semantics: a I ∈ ∆I Assertions Concrete Domains Comparison Example: paul Reasoning for Concepts Understand: “the individual named paul” for Knowledge Bases [Bedaride, 2003]Unique name assumption: an interpretation assigns each T and H Background Knowledgeindividual name a different individual. ABox Saturation Subgraph Detection Back Matter
  8. 8. Description LogicsAtomic Roles in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Syntax: R Terminological Axioms Semantics: R I ⊆ ∆I × ∆I Assertions Concrete Domains Comparison Example: hasChild Reasoning for Concepts Understand: “the set of all parent-child pairs” for Knowledge Bases [Bedaride, 2003] Example: isChildOf T and H Background Knowledge Understand: “the set of all child-parent pairs” ABox Saturation Subgraph Detection Back Matter
  9. 9. Description LogicsInverse Roles in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Syntax: R− Concepts Terminological Axioms Semantics: {(x, y) | (y, x) ∈ R I } Assertions Concrete Domains Comparison Example: hasChild− Reasoning for Concepts Understand: “the set of all child-parent pairs” for Knowledge Bases [Bedaride, 2003] Example: isChildOf − T and H Background Knowledge Understand: “the set of all parent-child pairs” ABox Saturation Subgraph Detection Back Matter
  10. 10. Description LogicsAtomic Concepts in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Axioms Syntax: A Assertions Concrete Domains Semantics: AI ⊆ ∆I Comparison Reasoning for Concepts Example: Person for Knowledge Bases Understand: “the set of all persons” [Bedaride, 2003] T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter
  11. 11. Description LogicsConjunction in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Syntax: C D Axioms Assertions Semantics: (C D)I = C I ∩ D I Concrete Domains Comparison Reasoning for Concepts for Knowledge Bases Example: Person Female [Bedaride, 2003] Understand: “the set of all female persons” T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter
  12. 12. Description LogicsDisjunction in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Syntax: C D Axioms Assertions Semantics: (C D)I = C I ∪ D I Concrete Domains Comparison Reasoning for Concepts for Knowledge Bases Example: Doctor Gardener [Bedaride, 2003] Understand: “the set of all doctors and gardeners” T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter
  13. 13. Description LogicsNegation in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Syntax: ¬C Terminological Axioms Semantics: (¬C )I ∆I C I Assertions Concrete Domains Comparison Reasoning for Concepts Example: ¬Flower for Knowledge Bases [Bedaride, 2003] Understand: “the set of all individuals that aren’t T and H flowers” Background Knowledge ABox Saturation Subgraph Detection Back Matter
  14. 14. Description LogicsExists Restriction in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Syntax: ∃R.C Terminological Axioms Semantics: (∃R.C )I = {x | ∃y ((x, y ) ∈ R I ∧ y ∈ C I )} Assertions Concrete Domains Comparison Reasoning for Concepts Example: ∃hasChild.Person for Knowledge Bases [Bedaride, 2003] Understand: “the set of all individulals that have a T and H child which is a person” Background Knowledge ABox Saturation Subgraph Detection Back Matter
  15. 15. Description LogicsNumber Restrictions in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Syntax: nP, nP Concepts Terminological nP)I = {x | |{y | (x, y ) ∈ P I }| Axioms Semantics: ( n} Assertions nP)I = {x | |{y | (x, y ) ∈ P I }| Concrete Domains ( n} Comparison Reasoning for Concepts for Knowledge Bases Example: 3hasChild [Bedaride, 2003] T and H Understand: “the set of all individuals with at least Background Knowledge three children” ABox Saturation Subgraph Detection Back Matter
  16. 16. Description LogicsValue Restriction in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Syntax: ∀R.C Concepts Semantics: (∀R.C )I = Terminological Axioms Assertions {x | ∀y ((x, y ) ∈ R I → y ∈ C I )} Concrete Domains Comparison Reasoning for Concepts for Knowledge Bases Example: ∀hasChild.Female [Bedaride, 2003] Understand: “the set of all individuals all of whose T and H Background children are female (including all Knowledge ABox Saturation individuals without any children)” Subgraph Detection Back Matter
  17. 17. Description LogicsNominals in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Syntax: {o1 , . . . , on } Concepts Terminological Axioms where o1 , . . . , on are individual names Assertions {o1 , . . . , on }I = {o1 , . . . , on } I I Concrete Domains Semantics: Comparison Reasoning for Concepts for Knowledge Bases Example: {china, france, russia, uk, usa} [Bedaride, 2003] T and H Understand: “the set of the permanent members of Background Knowledge the UN security council” ABox Saturation Subgraph Detection Back Matter
  18. 18. Description LogicsThe Universal Concept and the Bottom Concept in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Axioms Syntax: Assertions Concrete Domains Semantics: I = ∆I Comparison Reasoning for Concepts Syntax: ⊥ for Knowledge Bases Semantics: ⊥I = ∅ [Bedaride, 2003] T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter
  19. 19. Description LogicsInclusions in RTE Kilian Evang Introduction Syntax: C D (R S) SHOIN (D) Individual Names Semantics: An interpretation I Roles Concepts satisfies C D (R S) Terminological Axioms iff C I ⊆ D I (R I ⊆ S I ). Assertions Concrete Domains Comparison Reasoning for Concepts Example: Apple Fruit for Knowledge Bases Understand: “Every apple is a fruit.” [Bedaride, 2003] T and H Background Knowledge ABox Saturation Subgraph Detection Example: hasTopping hasIngredient Back Matter Understand: “Having something as a topping also means having it as an ingredient.”
  20. 20. Description LogicsEqualities in RTE Kilian Evang Syntax: C ≡ D (R ≡ S) Introduction Semantics: An interpretation I SHOIN (D) Individual Names satisfies C D (R S) Roles Concepts iff C I = D I (R I = S I ). Terminological Axioms Assertions Concrete Domains Comparison Example: SpicyPizza ≡ Reasoning for Concepts Pizza ∃hasTopping.SpicyTopping for Knowledge Bases Understand: “A SpicyPizza is defined to be a pizza [Bedaride, 2003] T and H with a spicy topping.” Background Knowledge ABox Saturation Subgraph Detection Back Matter Example: isChildOf ≡ hasChild− Understand: “isChildOf is defined to be the inverse role of hasChild.”
  21. 21. Description LogicsTransitive Roles in RTE Kilian Evang Introduction SHOIN (D) Syntax: R ∈ R+ Individual Names Semantics: R I = (R I )+ Roles Concepts Terminological Axioms Assertions Concrete Domains Example: isPartOf ∈ R+ Comparison Reasoning Understand: “If A is a part of B and B is a part for Concepts for Knowledge Bases of C, then A is also a part of C.” [Bedaride, 2003] T and H Background important for part-whole descriptions Knowledge ABox Saturation Subgraph Detection allows for defining concepts that have no finite model Back Matter [Sattler, 1996]
  22. 22. Description LogicsConcept Assertions in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Syntax: C (a) Terminological Axioms Semantics: An interpretation I satisfies C (a) iff Assertions Concrete Domains aI ∈ C I . Comparison Reasoning for Concepts for Knowledge Bases [Bedaride, 2003] Example: Father(peter) T and H Understand: “Peter is a father.” Background Knowledge ABox Saturation Subgraph Detection Back Matter
  23. 23. Description LogicsRole Assertions in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Syntax: R(a, b) Axioms Assertions Semantics: (a, b)I ∈ R I Concrete Domains Comparison Reasoning for Concepts for Knowledge Bases Example: hasChild(mary, paul) [Bedaride, 2003] Understand: “Paul is a child of Mary.” T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter
  24. 24. Description LogicsConcrete Domains in RTE Kilian Evang Introduction SHOIN (D) Individual Names RolesRouhgly and intuitively, concrete domains are a language Concepts Terminologicalextension that allows for “importing” Axioms √ Assertions Concrete Domains “individuals” such as 18, 2, "Zw¨lf Boxk¨mpfer", o a Comparison or "Zw¨" o Reasoning for Concepts “roles” such as greaterThan or startsWith for Knowledge Bases [Bedaride, 2003]from worlds such as arithmetic or string manipulation into T and H Backgroundthe logic. OWL DL uses this to assign Knowledge ABox Saturationnumeric/string/date/... properties to individuals. Subgraph Detection Back Matter
  25. 25. Description LogicsComparison of Four DLs in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles construct AL ALC S SHOIN (D) Concepts Terminological atomic negation Axioms Assertions conjunction Concrete Domains universal quantification Comparison existential quantification limited Reasoning for Concepts disjunction for Knowledge Bases transitive roles [Bedaride, 2003] number restrictions T and H Background role hierarchies Knowledge ABox Saturation inverse roles Subgraph Detection Back Matter
  26. 26. Description LogicsKnowledge Bases in RTE Kilian Evang Introduction SHOIN (D) a knowledge base is a set of formulas (explicit Individual Names Roles knowledge) Concepts Terminological Axioms sometimes divided up into two subsets: Assertions Concrete Domains TBox Comparison contains only terminological axioms Reasoning for Concepts provides a general terminology for Knowledge Bases ABox [Bedaride, 2003] T and H contains only assertions Background Knowledge provides a specific world description ABox Saturation Subgraph Detection also contains implicit knowledge Back Matter implicit knowledge can be made explicit by reasoning
  27. 27. Description LogicsAn Example Knowledge Base in RTE Kilian EvangTBox Introduction SHOIN (D) Woman ≡ Person Female Individual Names Roles Man ≡ Person ¬Woman Concepts Terminological Mother ≡ Woman ∃hasChild.Person Axioms Assertions Father ≡ Man ∃hasChild.Person Concrete Domains Comparison Parent ≡ Father Mother Reasoning for Concepts Grandmother ≡ Mother ∃hasChild.Parent for Knowledge Bases MotherWithManyChildren ≡ Mother 3hasChild [Bedaride, 2003] T and H MotherWithoutDaughter ≡ Mother ∀hasChild.¬Woman Background Knowledge Wife ≡ Woman ∃hasHusband.Man ABox Saturation Subgraph Detection Back MatterABoxhasChild(mary, paul), Father(paul)An example piece of implicit knowledgeGrandmother(mary)
  28. 28. Description LogicsModelhood in RTE Kilian Evang Introduction SHOIN (D) Individual NamesAn interpretation I is a model of (satisifies) Roles Concepts Terminological a formula φ iff it satisfies φ. Axioms Assertions Concrete Domains a TBox T iff it is a model of every terminological axiom Comparison in T . Reasoning for Concepts an ABox A iff it is a model of every assertion in A. for Knowledge Bases [Bedaride, 2003] an ABox A with respect to a TBox T iff it is a model T and H Background of both A and T . Knowledge ABox Saturation a concept C iff C I is nonempty. Subgraph Detection Back Matter
  29. 29. Description LogicsReasoning Tasks for Concepts in RTE Kilian Evang IntroductionLet C , D concepts and T a TBox (e.g. see above). SHOIN (D) C is satisfiable wrt. T iff C and T have a common Individual Names Roles model. Concepts Terminological Axioms e.g. not satisfiable: Man Woman Assertions Concrete Domains C is subsumed by D wrt. T iff C I ⊆ D I for every Comparison model I of T . Reasoning for Concepts e.g. Mother is subsumed by Woman for Knowledge Bases C and D are equivalent wrt. T iff C I = D I for every [Bedaride, 2003] T and H model I of T . Background Knowledge e.g. ∃hasChild.Person is equivalent to Father Mother ABox Saturation Subgraph Detection C and D are disjoint wrt. T iff C I ∩ D I = ∅ for every Back Matter model I of T . e.g. Man and Woman are disjoint
  30. 30. Description LogicsReasoning Tasks for Knowledge Bases in RTE Kilian EvangLet K a knowledge base. Introduction consistency checking: K is consistent iff it has a SHOIN (D) Individual Names model. Roles Concepts e.g. above KB is consistent, adding Mother(paul) Terminological Axioms would make it inconsistent Assertions Concrete Domains instance checking: Given a concept C and an Comparison individual name a, K entails C (a) iff K ∪ {¬C (a)} is Reasoning for Concepts inconsistent. for Knowledge Bases e.g. Grandmother(mary) is entailed by above KB [Bedaride, 2003] T and H retrieval problem: Given a concept C , find all Background Knowledge individual names a such that K entails C (a). ABox Saturation Subgraph Detection e.g. the result for ∃hasChild.Person would be {mary} Back Matter realization problem: Given an individual name a, find the most specific concepts C such that K entails C (a). ...
  31. 31. Description Logics[Bedaride, 2003]: RTE in Four Steps in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts RTE in four steps: Terminological Axioms 1. represent T and H as two ABoxes Assertions Concrete Domains 2. make a TBox with background knowledge Comparison 3. saturate ABoxes with TBox Reasoning for Concepts 4. subgraph-detect ABox H in ABox T for Knowledge Bases Example T/H pair: [Bedaride, 2003] T and H T: “John buys a cat at the pet shop for 50 euros.” Background Knowledge H: “A shop sells an animal to John.” ABox Saturation Subgraph Detection Back Matter
  32. 32. Description LogicsStep 1: Represent T and H as Two ABoxes in RTE Kilian Evang Introduction ABox T = {CommercialTransaction(ct1), John(j1), SHOIN (D) Individual Names PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), Roles Concepts seller(ct1, ps1), goods(ct1, c1), money(ct1, p1)} Terminological Axioms Assertions ABox H = {CommercialTransaction(ct2), John(j2), Concrete Domains Comparison Shop(s2), Animal(a2), buyer(ct2, j2), Reasoning seller(ct2, s2), goods(ct2, a2)} for Concepts for Knowledge Bases Note: [Bedaride, 2003] T and H FrameNet frames and frame elements represented as Background Knowledge individuals, characterized by concept assertions ABox Saturation connected via frame-specific roles Subgraph Detection Back Matter no difference made between common/proper, definite/indefinite, singular/plural NP each ABox has its own set of individual names
  33. 33. Description LogicsStep 2: TBox with Background Knowledge in RTE Kilian Evang ABox T = {CommercialTransaction(ct1), John(j1), Introduction PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), SHOIN (D) Individual Names seller(ct1, ps1), goods(ct1, c1), money(ct1, p1)} Roles Concepts ABox H = {CommercialTransaction(ct2), John(j2), Terminological Axioms Shop(s2), Animal(a2), buyer(ct2, j2), Assertions Concrete Domains seller(ct2, s2), goods(ct2, a2)} Comparison Reasoning TBox BK = {PetShop Shop, Cat Animal} for Concepts for Knowledge Bases Note: [Bedaride, 2003] T and H atomic concepts mapped to WordNet synsets (how – Background Knowledge WSD?) ABox Saturation for each pair (Sh , St ) of synsets from H and T, check if Subgraph Detection Back Matter there is a relation and if so, add the appropriate axiom(s) to the TBox: Sh St for hyponymy, St Sh for hypernymy, Sh St and St Sh for synonymy, Sh ¬St and St ¬Sh for antonymy
  34. 34. Description LogicsStep 3: Saturate ABoxes with TBox in RTE Kilian Evang Introduction SHOIN (D) TBox BK = {PetShop Shop, Cat Animal} Individual Names Roles ABox T = {CommercialTransaction(ct1), John(j1), Concepts Terminological Axioms PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), Assertions Concrete Domains seller(ct1, ps1), goods(ct1, c1), money(ct1, p1), Comparison Shop(ps1), Animal(c1)} Reasoning for Concepts ABox H = {CommercialTransaction(ct2), John(j2), for Knowledge Bases [Bedaride, 2003] Shop(s2), Animal(a2), buyer(ct2, j2), T and H Background seller(ct2, s2), goods(ct2, a2)} Knowledge ABox Saturation Note: Subgraph Detection Back Matter T (H ) is T (H) saturated with BK , i.e. containing every assertion entailed by BK ∪ T (BK ∪ H)
  35. 35. Description LogicsStep 4: Subgraph-Detect H in T in RTE Kilian Evang Introduction SHOIN (D) Let σ = {ct2/ct1, j2/j1, a2/c1, s2/ps1} Individual Names Roles ABox T = {CommercialTransaction(ct1), John(j1), Concepts Terminological PetShop(ps1), Cat(c1), 50Euros(p1), buyer(ct1, j1), Axioms Assertions seller(ct1, ps1), goods(ct1, c1), money(ct1, p1), Concrete Domains Comparison Shop(ps1), Animal(c1)} Reasoning for Concepts ABox H σ = {CommercialTransaction(ct1), for Knowledge Bases John(j1), Shop(ps1), Animal(c1), buyer(ct1, j1), [Bedaride, 2003] T and H seller(ct1, ps1), goods(ct1, c1)} Background Knowledge ABox Saturation Note: Subgraph Detection We detect entailment iff we can find a individual name Back Matter substitution σ such that H σ ⊆ T , i.e. all information in H is also in T .
  36. 36. Description LogicsReferences in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Franz Baader, Diego Calvanese, Deborah L. McGuiness, Concepts Terminological Daniele Nardi and Peter F. Patel-Schneider (2003) Axioms Assertions The description logic handbook: theory, implementation, Concrete Domains Comparison and applications Reasoning Cambride University Press for Concepts for Knowledge Bases [Bedaride, 2003] Paul Bedaride (2003) T and H Using Description Logics for Recognising Textual Background Knowledge ABox Saturation Entailment Subgraph Detection In: Proceedings of the Twelfth ESSLLI Student Session Back Matter
  37. 37. Description LogicsReferences in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Matthew Horridge, Simon Jupp, Georgina Moulton, Terminological Axioms Alan Rector, Robert Stevens and Chris Wroe (2007) Assertions Concrete Domains A Practical Guide to Building OWL Ontologies Using Comparison Prot´g´ 4 and CO-ODE Tools, Edition 1.1 e e Reasoning for Concepts for Knowledge Bases Ulrike Sattler (1996) [Bedaride, 2003] A concept language extended with different kinds of T and H Background transitive roles Knowledge ABox Saturation Springer Subgraph Detection Back Matter
  38. 38. Description Logics in RTE Kilian Evang Introduction SHOIN (D) Individual Names Roles Concepts Terminological Axioms AssertionsRteClassMember ∃thanks− .{kilian} Concrete Domains Comparison Reasoning for Concepts for Knowledge Bases [Bedaride, 2003] T and H Background Knowledge ABox Saturation Subgraph Detection Back Matter
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