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Description Logic

Introduction. Description Logic. Motivation for Description Logic (DL) DL and First Order Logic (FOL) What is Description Logic? DL Architecture Reasoning Structures Applications of DL. March-03-09. Description Logic. Motivation. Any area with information overload.

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Description Logic

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  1. Introduction Description Logic Motivation for Description Logic (DL) DL and First Order Logic (FOL) What is Description Logic? DL Architecture Reasoning Structures Applications of DL March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  2. Description Logic Motivation • Any area with information overload. • Where amount of declarative information to be processed manually exceeds that of human abilities. • Even if a large number of resources could be found, it would take an impractical number of human hours to process. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  3. Description Logic Motivation • Need for greater push for standardization of terminologies: • Semantic Web • Health Services • Bio-informatics March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  4. Description Logic Motivation • We would like to model complex concepts using modern modeling structures. • OWL, Web Ontology Language, is a computer processible syntax for expressive Description Logic. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  5. Description Logic Motivation • Propositional Logic (PL) has well defined operators and axioms creating atomic formulas. • LHS and RHS Equivalences, such as: March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  6. Description Logic Motivation • We can already do a lot with PL • But … • No access to the structure of atomic statements. • Consider Propositions • Mary-is-female • John-is-male • Mary-and-John-are-siblings • Consider Statements • Mary is female • John is male • Mary and John are siblings March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  7. Description Logic Motivation • First Order Logic (FOL) • The language of Predicate Logic • Atomic formulas are sets of predicates interpreted as relations between elements of a domain. • Consider Predicates • Female (Mary) • Male (John) • Siblings (Mary, John) March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  8. Description Logic Motivation • Predicate Logic and FOL allows for formulas, arguments, free and bounded variables, and constants. • Allows for literals, atomic formulas (relations), and negation. • Well formed formulas, truth connectives, quantifiers over bounded variables. • Quantifiers: March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  9. Description Logic Motivation • Represent relationships between objects. • More complex relationships can be defined in terms of more basic relationships or objects. verses March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  10. Description Logic … and First Order Logic • Description Logic (DL) is derived from FOL. • DL Relationships are limited to 1-argument predicates called concepts and 2-argument predicates called roles. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  11. Description Logic … and First Order Logic • Concept C and its translation π(C)(x) are equivalent in the interpretation I = (ΔI, . I ) and all a ΔI , where Δ is the domain, and I is the interpretation. • Consider: a is an instance of an interpretation of WOMAN iff a FOL translation of this concept is true for "a" in the interpretation. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  12. Description Logic … and First Order Logic • Translation of roles in FOL takes the form of a predicate with two arguments. • Roles relate two concepts to each other. • Roles cannot occur by themselves; only as a part of a complex concept. • This puts restrictions on syntax of FOL formulas that are allowed. • Important to guarantee decidability and good computational properties of DLs. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  13. Description Logic … and First Order Logic • Translations require mappings on two variables • : π(x) and π(y) Mother(x) is true iff x truth values match those of the definition of Woman with some y a child. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  14. Description Logic … and First Order Logic • Unary Concept translations • Binary Role translation March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  15. Description Logic … and First Order Logic • Counting Quantifier translations • Some x with at least n y’s • A simple existential quantifier is insufficient: March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  16. Description Logic What is Description Logic? • Sub-language of First Order Logic • Model hierarchical and complex relationships. • Collection of these statements is called TBox. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  17. Description Logic What is Description Logic? • The TBox is known as an ONTOLOGY (from Greek: science of beings, or science of things that exist). • Ontology used to be a sub-discipline of philosophy. • It is now at forefront of real world applications. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  18. Description Logic What is Description Logic? • Models systems which are complex through size and cyclic definitions. • “Man who has Only Sons” (Mos) • “Man who has Only Male Offspring” (Momo) March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  19. Description Logic What is Description Logic? • DL handles cyclic definitions through fixpoints • Least fixpoint is the lowest x value where f(x) = x • Greatest fixpoint is the greatest x value where f(x) = x • Halting depends on the domain of x • Example: a binary tree’s least-point are the nodes with only terminal nodes at the branches. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  20. Description Logic What is Description Logic? • DL has variable-free syntax • variables are implicitly understood • There can be only two of them: x or y, but they can be reused. • DL concepts denote sets of individuals. • For concept C, and translation I. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  21. Description Logic DL Architecture • Basic description languageAL • AL = attribute language • A and B are atomic concepts • C and D are concept descriptions • R represents atomic roles • Other Languages in this family are extensions of AL. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  22. Description Logic DL Architecture • Basic description languageAL March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  23. Description Logic DL Architecture • Atomic Concepts and Roles March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  24. Description Logic DL Architecture • Limited to two-variable structures. • Handles acyclic, as well as cyclic statements. • Acyclic: LHS has no terms which appear in RHS • Cyclic: LHS has a term which appears in RHS March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  25. Description Logic DL Architecture • DL defines statements by stating assertions about a subject using its vocabulary. • Type of statements which make up a DL Ontology are the ABox and the TBox formalisms. • TBox = terminology used in a domain. • ABox = assertions which make statements about particular object instances in that domain, using the terminology in TBox. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  26. Description Logic DL Architecture Knowledgebase TBox Description Language Reasoning ABox Application Programs Rules March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  27. Description Logic DL Architecture TBox ABox March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  28. Description Logic Reasoning Structures • DL Knowledgebase stores more then definitions and assertions. • DL contains implicit knowledge which can be made explicit through inferences. • Inferences are made through a set of rules of inference. • TBox contains these statement. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  29. Description Logic Reasoning Structures • Modeling a domain requires defining a terminology, say T, which satisfies all previously defined statements about that domain. • Any new interpretation must satisfy the axioms of T, and result in a non-empty set of concepts. • Otherwise it is unsatisfiable. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  30. Description Logic Reasoning Structures Satisfiability: A concept C is satisfiable with respect to T if there exists a model I of T such that CI is nonempty: I|= C Subsumption: A concept C is subsumed by a concept D with respect to T if CIDI for every model I of T: C T D T|= (C D) |Π |Π March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  31. Description Logic Reasoning Structures • Subsumption con’t • LHS is subsumed by the more general terms in RHS: |Π |Π |Π |Π March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  32. Description Logic Reasoning Structures Equivalence: Two concepts C and D are equivalent if C subsumes D and D subsumes C, with respect to T if CI = DI for every model I of T: [ C ΞTD and T|=CΞ D ] [ C D and D C ] |Π |Π Disjointness: Two concepts C and D are disjoint with respect to T if: CI  DI =  March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  33. Description Logic Applications of DL • Different extensions to AL-languages. • AL in combination withU,ε, N, andFL- form 8 languages which are pairwise non-equivalent. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  34. Description Logic Applications of DL • The special extension ALUε , represented by ALC, describe a commonly used language with axioms for union and existential quantification. • TheALCextension is the basis for SHOIN, which in turn is the basis for the Web Ontology Language OWL • OWL is formally recommended by the World Wide Web Consortium, W3C. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  35. Description Logic Applications of DL • OWL is a markup language given the tasks of providing syntax used to model ontologies on the web. • The Semantic Web is an attempt to organize the plethora of information available on the internet March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  36. Description Logic Applications of DL • On the web, instead of showing all available links and infinite levels of sub-topics, a high level “description” is given. • Description identifies the subject of the content on the web. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  37. Description Logic Applications of DL • Cyclic self-referenced nodes are contained within a higher level concept. • Bound by fixpoints. • This abstraction inherited from DL ensures satisfiability in the system. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  38. Description Logic Applications of DL • OWL Example: “Daugther” (misspelling preserved) <owl:Class rdf:ID="Daugther"> <owl:equivalentClass> <owl:Class> <owl:intersectionOf rdf:parseType="Collection"> <owl:Class rdf:ID="Child"/> <owl:Class rdf:ID="Woman"/> </owl:intersectionOf> </owl:Class> </owl:equivalentClass> <owl:disjointWith> <owl:Class rdf:ID="Son"/> </owl:disjointWith> </owl:Class> - http://protege.cim3.net/file/pub/ontologies/family.swrl.owl/family.swrl.owl March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  39. Description Logic Applications of DL • OWL Example: “Daugther” (misspelling preserved) <owl:Class rdf:ID="Daugther"> <owl:equivalentClass> <owl:Class> <owl:intersectionOf rdf:parseType="Collection"> <owl:Class rdf:ID="Child"/> <owl:Class rdf:ID="Woman"/> </owl:intersectionOf> </owl:Class> </owl:equivalentClass> <owl:disjointWith> <owl:Class rdf:ID="Son"/> </owl:disjointWith> </owl:Class> - http://protege.cim3.net/file/pub/ontologies/family.swrl.owl/family.swrl.owl March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  40. Description Logic Applications of DL • Description Logic can be analyzed for satisfiability with Theorem Provers (Prover9 / Vampire / Otter) • Tableau Calculus • Tableaux Calculus is a decision procedure solving the problem of satisafiability. • If a formula is satisfiable, the procedure will constructively exhibit a model of the formula. March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  41. Description Logic Applications of DL • Tableau Calculus con’t • The basic idea is to incrementally build the model by looking at the formula, by decomposing it in a top/down fashion. • The procedure exhaustively looks at all the possibilities, so that it can eventually prove that no model could be found for unsatisfiable formulas. • Unsatifiable formula contains a contradiction such as: p(a) Π p(a) March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  42. Description Logic Applications of DL • Closed vs Open World Semantics • Connections to external databases require database schema mappings. • Communicating TBoxes need to be mapped in order for the data in ABoxes to be meaningful. • Closed systems represent internal databases where no mappings are required. • A complete theory in a closed system may not be complete in an open world semantic March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  43. Description Logic References Enrico Franconi, DESCRIPTION LOGICS - Tutorial Course Information, Faculty of Computer Science, Free University of Bozen-Bolzano, Italy; http://www.inf.unibz.it/~franconi/dl/course/ Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, Peter F. Patel-Schneider (Eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press 2003 Christine Golbreich, A SWRL/OWL Demo Ontology About Family Relationships, OWL Ontologies, ProtegeOntologiesLibrary; http://protege.cim3.net/file/pub/ontologies/family.swrl.owl/family.swrl.owl March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

  44. Description Logic The End Thank you Q & A Bart Gajderowicz bgajdero@ryerson.ca www.scs.ryerson.ca/~bgajdero March-03-09 Bart Gajderowicz, Ryerson University, Dept. of Computer Science

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