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WSML Presentation

The F-Logic Approach for Description Languages

Uwe Keller

based on a paper by Mira Balabanpublished in „Annals of Mathematics and Artificial Intelligence“ (1995)

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Overview … Motivation for the work (DFL)

Embedding DL into F-Logic

Using F-Logic to extend DLs

Integrating terminologies in F-Logic knowledge bases

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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

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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

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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 !

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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 „:“

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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

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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

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Terminological Operators … [And]

Semantics Obj.-Constructor and denotes the glb-operator on (U, U)

Axiomatization

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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

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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)

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Terminological Operators … [inverse roles]

Semantics Denotes the inverse relation R- of relation R

Axiomatization

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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

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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 !

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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

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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

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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}]

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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!

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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

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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!

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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)