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Knowledge Representation and Reasoning using Description Logic

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  1. Knowledge Representation and Reasoning using Description Logic Presenter Shamima Mithun

  2. Overview • Introduction to Description Logics (DL) • DL Syntax and Semantic • DL Systems Architecture • Reasoning Techniques • KB Construction and Access using PowerLoom

  3. Introduction • Description Logics are a family of logic based knowledge representation formalisms based on conceptsand roles: • Concepts (classes) are interpreted as sets of objects e.g. person. • Roles are interpreted as binary relations on objects e.g. has-child. • Key features of DLs are • a well defined semantics • Inference services

  4. Background Description Logics: • Introduced to resolve the ambiguities of Semantic Networks • Descendants of Semantic Networks, frame based systems and KL-ONE • Decidable fragment of FOL [2] • Useful for Semantic Web Language (DAML +OIL) [4]

  5. DL Applications & Systems Application Areas • Terminological KR and Ontologies • Semantic Web • Software Information Systems • Database Applications Systems • LOOM/PowerLoom, RACER, CLASSIC, and others

  6. DL Constructors • DLs are characterized by a set of constructors. These are used to construct complex concepts and roles from simple ones. • Common constructors: Conjunction(П), disjunction (),negation () • Restricted forms of quantification (,  ) • Example in ALC: the concept Happy-Father Happy-Father ManПhas- child. Male

  7. DL Syntax and Semantic Figure is taken from Ian Horrocks [2]

  8. Other DL Constructors

  9. DL and Other Logical Formalism: FOPL • Syntactic feature of DL: variable free notation. • Most DLs are fragments of FOL, e.g. ACL. • ACL expressions can be translated into FOL: • A unary predicate ΦA is introducedfor each concept C, and a binary relation ρR for each role R. • Translation ACL →FOL: artist П( CREATES. song ) → x y: artist ( x ) Λ(CREATES ( x, y ) Λ song ( y )) • Why not use FOL? The expressive power is too high for having good computational properties and efficient inference procedures.

  10. DL and Other Logical Formalism:Modal Logics • DLs are notational variants of Modal Logics ALC  multi-modal K: C П D  C Λ D, C  D  C ν D  C  C,  R.C  <R>C,  R.C  [R]C transitive roles  transitive frames (e.g., K4) inverse roles  converse programs (e.g., C-PDL) number restrictions  deterministic programs (e.g., D-PDL) • No TBoxes available in modal logics: internalise" axioms using a universal role u: C  D  [u] (CD) • No ABoxes available in modal logics: Use nominals

  11. DL Systems Architecture Knowledge Base TBox (schema) Man ≡ Human П Male Happy-Father ≡ Man П has-child. Female Interface Inference System ABox (data) John : Happy-Father John, Mary> : has-child Figure after Horrocks [3]

  12. DL TBox

  13. DL ABox

  14. Knowledge to Reasoning Reasoning about the knowledge • Add new knowledge to the KB that follows logically. • Ask KB if a statement is valid. Knowledge Base TBox ABox Reasoning Figure is taken from Ian Horrocks [3]

  15. Reasoning / Inference Basic Inference Problems, for TBox T: • Consistency: “A concept C is consistent with respect to T, if there exists a model I of T with CI . [I is a model of C]”. Inconsistent: songwriter  artist П(CREATES. song ) song  IS _ CREATED _ BY. songwriter • Subsumption: “A concept C is subsumed by a concept D with respect to T if CI DIfor every model Iof T”. male person Definitions taken from [6], p. 66.

  16. Furniture Movable- Furniture Fixed- Furniture Door Window Chair Sofa Is-a Classification “Classificationis the task of inserting new concepts or instances in a taxonomy” [ 3]. • Classification of concepts • allows to structure the terminology in the form of a subsumption hierarchy • Classification of instances • determines whether an individual is an instance of a certain concept.

  17. Reasoning Algorithms • Structural subsumption algorithms • Subsumption of concepts can be computed. • They are complete for simple languages with little expressivity. • Used by KL-ONE, LOOM and other systems. • Tableau–based algorithms • Turned out to be very efficient reasoning algorithms. • Sound, complete and decidable. • Used e.g. in RACER.

  18. Structural Subsumption Algorithm Proceed in two phases • The descriptions to be tested for subsumption are normalized. • Then the syntactic structure of the normal forms is compared with each other. An FLo- concept description is in normal form iff it is of the form A1П… ПAmПR1.C1П… П Rn.Cn Where A1,.., Am are distinct concept names, R1,..., Rn are distinct roles names, and C1,…, Cn are concept descriptions in normal from.

  19. Structural Subsumption Algorithm(contd.) Given is the normal form of the concept description C A1П… ПAmПR1.C1П… П Rn. Cn and the normal form of the concept description D B1П… ПBkПS1.D1П… П Sl. Dl D subsumes C: C  D iff • i, 1≤ i ≤k,  j, 1 ≤j ≤m: Bi = Aj • i,1≤ i ≤l, j, 1 ≤j ≤n: Si = Rj and Cj Di

  20. Tableau-based Algorithms • Construct a model for the input concept description C0. • Model is represented by tree form. • The formula has been translated into Negation Normal Form (NNM). • To decide satisfiability of the concept C0 , start with the initial tree (root node). • Repeatedly apply expansion rules until find contradiction or expansion completed. • Satisfiable iff a complete and contradiction-free tree is found.

  21. Tableau-based Algorithms - Example Determine the satisfiability of the concept-definition: ( ( CHILD. Male ) П ( CHILD.  Male ) ) ( ( CHILD. Male ) П ( CHILD.  Male ) ) <x> ( CHILD. Male ) <x> П -rule ( CHILD.  Male ) <x> П –rule CHILD <x, y >  -rule  Male < y>  -rule Male <y >  -rule <CLASH >

  22. Reasoning(contd.) Reasoning services like subsumption and consistency • Speed-up the inference procedures for query. • Help to infer implicitly represented knowledge from the explicitly contained knowledge of KB. T-BoxA-Box Female  Male  Human Mary: Mother Mother  Female John: Father Father  Male Mary: parent.Child Child  has.Mother Пhas.Father John: parent.Child • Able to deduce implicit knowledge, like Mary is a Human.

  23. Reasoning: Decidability vs. Expressivity • KR system should • answer queries in a reasonable time. • The reasoning procedures should terminate. • Trade-off between the expressivity of DLs and the complexity of their reasoning. • Very expressive DLs can have inference problems of high complexity, they may even be undecidable. • Very Weak DLs my not be sufficiently expressive to represent the important concepts of an application. • Quest for a highly expressive DL with decidable/ practical inference problems.

  24. Conclusion • DL are logic based knowledge representation formalisms. • DL systems provide efficient inference services like consistency, subsumption. • DLs are effective in a range of applications.

  25. PowerLoom

  26. PowerLoom Commands • Defines concept, relation, function and rules using defconcept, defrelation, deffunction, defrule. • Add/Remove facts from KB with assert and retract • Query KB ask, retrieve Note: Relations have to defined, before they are used in assertions or queries.

  27. Model Domain Furniture Movable- Furniture Fixed- Furniture Age Color Door Window Chair Sofa Red Green Old New Size Small Big IS-A

  28. Define Concept |=(defconcept furniture) |c| FURNITURE |= (defconcept movable-furniture (?f furniture)) |c| MOVABLE-FURNITURE |= (defconcept fixed-furniture (?f furniture)) |c| FIXED-FURNITURE |=(defconcept chair (?f movable-furniture)) |c| CHAIR |= (defconcept sofa (?f movable-furniture)) |c| SOFA

  29. Define Concept(contd.) |= (defconcept window (?ff fixed-furniture)) |c| WINDOW |= (defconcept door (?ff fixed-furniture)) |c| DOOR |= (defconcept age (?a) :<=> (member-of ?a (setof new old))) |c| AGE |= (defconcept color (?c) :<=> (member-of ?c (setof green red blue))) |c| COLOR |= (defconcept size (?s) :<=> (member-of ?s (setof small big))) |c| SIZE

  30. Model Domain Furniture has-color has-age Movable- Furniture Fixed- Furniture Age Color Door Window Chair Sofa has-size Red Green Old New Size Small Big IS-A

  31. Define Relation and Function |= (defrelation has-age ((?f furniture) (?a age))) |r| HAS-AGE |= (defrelation has-color ((?f furniture) (?c color))) ) |r| HAS-COLOR |= (defrelation has-size ((?f furniture) (?s size))) |r| HAS-SIZE |= (deffunction has-price ((?f furniture)):-> (?n INTEGER)) |r| HAS-PRICE

  32. Model Domain Furniture has-color has-age Movable- Furniture Fixed- Furniture Age Color Door Window Chair Sofa Red Green Old New has-size Size Big Small IS-A

  33. Value Assertion |= (assert (has-size chair big)) |P|(HAS_SIZE CHAIR BIG) |= (assert (has-color sofa red)) |P|(HAS_COLOR SOFA RED) |= (assert (has-age chair new)) |P|(HAS_AGE CHAIR NEW) |= (assert (has-age door old)) |P|(HAS_AGE DOOR OLD) |= (assert (forall (?x ?y) (=> (has-color ?x ?y) (and (furniture ?x) (color ?y)))))

  34. Query the KB: Retrieve, Retract, and Ask |=(retrieve (has-color sofa ?x)) There is 1 solution so far: #1: ?X=RED |= (retract (has-color sofa red)) |P?|(HAS-COLOR SOFA RED) |= (retrieve (has-color sofa ?x)) No solutions. |= (ask (has-size chair big)) TRUE |= (ask (has-color door red)) UNKNOWN

  35. References 1.PowerLoom Homepage http://www.isi.edu/isd/LOOM/PowerLoom/ 2. Ian Horrocks and Ulrike Sattler: Description Logics - Basics, Applications, and More. Aavilable athttp://www.cs.man.ac.uk 3. Ian Horrocks: Reasoning with Expressive Description Logics: Theory and Practice. Available at: http://www.cs.man.ac.uk/~horrocks/Slides 4. Christel Kemke. Lecture Notes on Artificial Intelligence. Available at http://www.cs.umanitoba.ca/~ckemke/74.419-AI/ 5. Daniele Nardi and Ronald J. Brachman. An introduction to Description Logics. In Baader, Calvanese, McGuinnes Nardi and Patel-Schneider, (eds). The Description Logics Handbook, chapter 1. Cambridge University Press, 2003. 6. Franz Baader and Werner Nutt. Basic Description Logics. In Baader, Calvanese, McGuinnes Nardi and Patel-Schneider, (eds). The Description Logics Handbook, chapter 2. Cambridge University Press, 2003.

  36. Thanks