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

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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5 Description Logic Motivation Propositional Logic (PL) has well defined operators and axioms creating atomic formulas. LHS and RHS Equivalences, such as: Commutativity ϕ ∨ ψ Ξ Ψ ∨ϕ Tautology ϕ ∨Τϕ ∨Τ Ξ Τ Unsatisfiability ϕ ∧ ϕ ∧ Ξ Implication ϕ -> ψ Ξ ϕ ∨ ψ Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science Consider Predicates Female (Mary) Male (John) Siblings (Mary, John) March-03-09

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8 Description Logic Motivation Bart Gajderowicz, Ryerson University, Dept. of Computer Science Quantifiers: Existential x. ϕ Ξ For some x, ϕ is true Universal x. ϕ Ξ For all x, ϕ is true 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. March-03-09

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9 Description Logic Motivation Represent relationships between objects. More complex relationships can be defined in terms of more basic relationships or objects. WomanΞPerson Π Female ManΞ Person Π Woman Bart Gajderowicz, Ryerson University, Dept. of Computer Science Woman-is-a-Person-who-is-a- Female Man-is-a-Person-who-is-not-a- Woman verses March-03-09

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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. FOL Predicates DL Relationships Description unary conceptsWoman( x ) binary rolesWoman( x ).hasChild( y, x ) Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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Unary Concept translations Binary Role translation 14 Description Logic … and First Order Logic Description LogicFirst Order Logic π x(A) A(x) π x(C Π D) π x(C) ∧ π x(D) π x(C D) π x(C) ∨ π x(D) π x( R.C) y.R(x; y) ∧ π y(C) π x( R.C) y.R(x; y) π y(C) Description LogicFirst Order Logic π y(A) A(y) π y(C Π D) π y(C) ∧ π y(D) π y(C D) π y(C) ∨ π y(D) π y( R.C) y.R(y, x) ∧ π x(C) π y( R.C) y.R(y, x) π x(C) Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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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. ManΞ Person Π Woman FatherΞ (Person Π Woman) Π hasChild:Person FatherΞ (Man) Π hasChild:Person Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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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. BinaryTreeΞ Tree Π (≤ 2 has-branch) Π ( has-branch.BinaryTree) BinaryEndTreeΞ Tree Π (≤ 2 has-branch) Π ( has-branch.BinaryTree) Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. a C I iff I |= π (C)(a). Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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21 Description Logic DL Architecture Basic description language AL 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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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22 Description Logic DL Architecture Basic description language AL C, D Aatomic concept C, D Τuniversal concept C, D bottom concept C, D AA atomic negation C, D C Π Dintersection C, D R. value restriction C, D R.Τ Limited existential quantification Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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23 Description Logic DL Architecture Atomic Concepts and Roles Person Π Female Person who is not a female Person Π hasChild.Τ Person with at least 1 child Person Π hasChild. Person with no children Person Π hasChild.Female Person with at least one daughter Person Π hasChild.Female Person with only daughters Mother Father Person who is a Parent Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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 MomoΞ Man Π hasChild.Momo Cyclic: LHS has a term which appears in RHS MosΞ Man Π hasChild.Man Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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27 Description Logic DL Architecture TBox ABox MotherWithoutDaughter(MARY)Father(PETER) hasChild(MARY, PETER)hasChild(PETER, HARRY) hasChild(MARY, PAUL) FatherΞ Man Π hasChild.Person MotherΞ Woman Π hasChild.Person MotherWithoutDaughterΞ Mother Π hasChild.¬Woman Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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 C I is nonempty: I |= C Subsumption: A concept C is subsumed by a concept D with respect to T if C I D I for every model I of T: C T D T |= ( C D) |Π Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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

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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 C I = D I for every model I of T: [ C Ξ T D and T |= C Ξ D ][ C D and D C ] Disjointness: Two concepts C and D are disjoint with respect to T if: C I D I = |Π Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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33 Description Logic Applications of DL Different extensions to AL-languages. ALU Union(C D) I AL ε Full existential quantification R.C ALN Number restrictions≥ n R and ≤ n R ALFL - Non-atomic negation AA AL in combination with U, ε, N, and FL - form 8 languages which are pairwise non-equivalent. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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34 Description Logic Applications of DL The special extension ALU ε, represented by ALC, describe a commonly used language with axioms for union and existential quantification. The ALC extension 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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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 Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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38 Description Logic Applications of DL OWL Example: “Daugther” (misspelling preserved) - Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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39 Description Logic Applications of DL OWL Example: “Daugther” (misspelling preserved) - DaugtherΞChild Π Woman Daugther Π SonΞ Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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. Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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) Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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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 Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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43 Description Logic References Enrico Franconi, DESCRIPTION LOGICS - Tutorial Course Information, Faculty of Computer Science, Free University of Bozen-Bolzano, Italy; 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; Bart Gajderowicz, Ryerson University, Dept. of Computer Science March-03-09

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44 Description Logic The End Thank you Q & A Bart Gajderowicz, Ryerson University, Dept. of Computer Science Bart Gajderowicz March-03-09

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