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 (L p ) 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) C2C2 C 1 ≤  R.C 2  R.C 2 ≤ 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)