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WSML Presentation

WSML Presentation. The F-Logic Approach for Description Languages Uwe Keller based on a paper by Mira Balaban published in „Annals of Mathematics and Artificial Intelligence“ (1995). Overview …. Motivation for the work (DFL) Embedding DL into F-Logic Using F-Logic to extend DLs

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  1. WSML Presentation The F-Logic Approach for Description Languages Uwe Keller based on a paper by Mira Balaban published in „Annals of Mathematics and Artificial Intelligence“ (1995)

  2. Overview … • Motivation for the work (DFL) • Embedding DL into F-Logic • Using F-Logic to extend DLs • Integrating terminologies in F-Logic knowledge bases

  3. Why a Description F-Logic (DFL)? • Description Logics … • Representation of terminological knowledge • Various DLs, always with emphasis on • Terminological operators (constructors) • Direct semantics • Inference algorithms (wrt. the def. semantics) • Trend: Strenghtening of DLs to meet more faithfully the user requirements • Non-taxonomic relations (part-whole, n-ary ..) • More flexible definitions • Uniform proof theory • Integration with LP, OO-Systems, DB-Technology

  4. Why a Description F-Logic (II)? • F-Logics … • Reasoning about OO-domains • Class-Hierarchies and Membership • Class == Object • Infos about objects by attributes and methods • Allows for reasoning about inheritance of types and methods • Allows for (syntactically) higher-order constructs (i.e. quantification over (named) methods) • Proof and sound proof theory

  5. Why a Description F-Logic (III)? • Advantages • Results in a full-fledged logic integrating Descriptions, reasoning, OO and LP • Can serve as a unifying formalism for various DLs • In particular • DLP has a few term constructors • Other constructors are axiomatized (Flexibility) • Uniform proof theory (for several definable DLs) • Integration of DL inference componts possible Idea: Develop a rich DL (called DLP) by starting from F-Logic !

  6. Description Logic Model … • Model for a familiy of DLs (Lp) • Symbols: • Primitive concept/name [role/rolename] symbols, object symbols, T, , and a finite set P of concept/role forming operators • Terms: • Concept terms vs. Role terms, built from primitive symbols and concept/role operators • Formulae: c = c-term; r = r-term; c1=>c2, r1=>r2, o:c, (o1,o2):r • Semantics: Formulas are intereted set-theoretically • Interpretation I = (D, ) • (c)  D, (r)  DxD, (T) = D, () = ø,(o)  D • Extension of  to complex terms by fixing the meaning of each operator in P • Satisfaction of formulae by I is defined as set equality for „=“, set inclusion for „=>“ and membership over D (resp. DxD) for „:“

  7. From DL to FDL … • Straightforward observation ... • F-Logic Ontologies can represent DL Ontologies • Now, we have to look at the … • standard elements of DL • which constructs have analogons • which are missing in F-Logic • terminological constructors • Set-theoretic semantic vs. OO semantics • Result: Nothing from DLs is lost! • F-Logic is adequate as a basis for a general DL

  8. Standard Elements of DLs … • DL-Concept = F-Logic object • Concept subsumption = partial order U(subclass) • Concept-Membership = binary relation U (element-of) • Roles = Methods • Expressions in DL vs Expressions in F-Logic • Meaning of term. constructors is not directly represented in F-Logic • Declaration of object-pairs in roles = Method value definition • Relationships between methods are not directly representable in F-Logic and thus have to be unfolded • Approach for terminological Constructors: • Axiomatize them on the basis of a few primary operators • That means: The semantics of the operators is not built in to the logic • This way, the logic DLP is likely to remain stable when adding new Ops • Advantage of F-Logic over First-Order Logic in this respect • No DL elements has a direct meaning / represenation in FOL

  9. Terminological Operators … • [And] • Semantics • Obj.-Constructor and denotes the glb-operator on (U, U) • Axiomatization

  10. Terminological Operators … • [exists] • Semantics • Selects all objects on which a (set-valued) role is defined • Axiomatization: Several flavours possible The usual way Objects on which a role R is defined Objects on which a role R is applicable

  11. Terminological Operators … • [all] • Semantics • Selects all objects for which the values of a (set-valued) role is restricted by a given class • Axiomatization Note: This is a finer definition than in DL! (Think about objects not involved R-tupels)

  12. Terminological Operators … • [inverse roles] • Semantics • Denotes the inverse relation R- of relation R • Axiomatization

  13. Example … • A student that drives sports cars of Italian makers, drives at least one and at most two such cars Assume addionally the following KB … Infer

  14. Replacing the Semantics … • What happens … • … if we change from the set-theoretic (DL) to the OO semantics (FL)? • … Nothing :- ) • Formally speaking: Logical Implications are preserved Uses a many- sorted F-Logic !

  15. Using DFL to extend DLs … • Question … • How and when can OO, intensional and higher-order nature of F-Logic be used to extend DLs? • Can we (by using DFL) integrate with general knowledge represenation systems? • Answer on Extensions … • DLF can account for desired features of DLs, that are problematic in the standard account of DLs • Higher-order roles and operator forming operators • Collective entities • n-ary relations • Roles as first class objects • Cycles and self-reference

  16. Higher-Order roles & Operator Forming Operators … • HO-Roles examples: Bin. Relation between Concepts • All-exists: AE(R) „Every person lives in some place“: (person,place) € AE(lives) • Subject-restriction: SR(R) „If X lives in an appartment the X is a person“: (person,appartmnt) € AE(lives) • Operator Forming Operators „A student who takes all math courses“: MA(take-course)(student, math-course) C2 C1≤ R.C2  R.C2 ≤C1 MA(R)(C1,C2) is not expressible in our original DL model

  17. Collective Entities … • Collective Entity = Set of other entities • Arise when sets have properties based on properties of their members: average(salary), min(bookPrice) • Examples • „John lead the Beatles“: John[lead ->> {beatles}] • „Every member of the Beatles sang Yellow-Submarine“beatles[distributive(sing) ->> {yellowSubmarine}] (uses HO-Roles!) • „Every member of the Beatles meet Brian within some subset of the Beatles “beatles[cumulative(meet) ->> {brian}]

  18. Integrating Terminologies in F-Logic Knowledge Bases … • Term. reasoning should be used in complex applications • Two common approaches: • Strenghten term. reasoners (i.e. in LOOM) • Integrate t.-reasoner in general KR systems (i.e. in CLASSIC) • Nonetheless, augmentation of term. reasoners is often needed • Regardless of the expressivity of DLs, an integration with other reasoners seems to be unavoidable • One major issue in designing the intergation scheme is how to avoid mismatches • F-Logic seems to be promising here as an underlying integration framework!

  19. Integrating Terminologies in F-Logic Knowledge Bases (II) … • Features of F-Logic as an integration platform: • Terminological definitions are possible • Standards DL algorithms are correct in F-Logic • Can support all typical deductive and OO-database reasoning • F-Logic is very expressive but computable (Why ???) • A Description F-Logic (DFL) reasoner can accomodate a seperate terminological component with independent processing methods

  20. Outcomes of the paper … • Proposes to use F-Logic … … as a unifying framework (DFL) to define description languages (resp. logics) • Adequate for capturing semantics of various DLs • Small set of modelling primitives allow to define many constructors of DLs (flexibility) • Integration of notions … • Description, Reasoning, Object-Orientation and Logic Programming • Allows for extending DLs in these directions • Main issue: Expressivity, Extendability, Uniformity • Not addressed: Decidability, Complexity issues!

  21. Relevance to WSMO/WSML … • Relevance for WSML-Full / WSML-DL • How to integrate DLs, DL-style modelling and inferencing cleanly and flexibly • No direct relevance for WSML-HL • we are interested in LP-style rule extensions of DLs • we are interested in understanding the tradeoff • Here the results of this paper can not be directly reused • Flexibility / Extendability of the Framework for DLs is interesting (F-Logic-based OWL?) • Integration of DLs with other KR-Formalisms via F-Logic might be interesting (in particular LP!) • Other papers by M. Balaban (1996)

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