Representation and a Proposal to WebOnt
by Peter F. Patel-Schneider

Abstract

A formal view of representation languages has a number of useful consequences. These consequences are discussed in the context of a proposal for a web ontology language (OWL-1).

What is a Representation Formalism?

• a formal system
  • syntax and semantics
  • form and meaning
• designed to represent information about the world
  • has a commitment to how the world works
  • determines what is important and what is not
• examples
  • relational data bases, propositional logic, first-order logic, modal logics, Montague logic, description logics

Why are formal semantics needed?

• People cope without them don't they?
  • well, sort of
  • people's repair mechanisms are very complex
• Lack of well-defined meaning leads to misunderstanding between communicants.

Horror stories

• range and domain in RDFS
  • meaning not specified in RDFS
  • two very different interpretations
• lists, bags, and alternatives in RDF
  • meaning not specified in RDF
  • valid syntax not even well-defined
  • notions of identity and equivalence difficult
• reification in RDF
• QUA in KL-ONE
  • two different interpretations for same syntax
• C++ (vs ML or FORTRAN)
  • many different versions of C++

Determining Meaning

• Data Models
• Proof Theory
• Model Theory
• Axiomatizations
• Operational Meaning

Determining Meaning - Data models

• single data structure that carries meaning of syntax
• fine for relational data bases
• not so bad for RDF and RDFS
• don't work for more-expressive formalisms

Determining Meaning - Proof Theory

• specifies what syntactic structure follow from (are derivable from) others
• generally in the form of a collection of axioms and rule of inference
• very (too) flexible
• can (often) be directly mechanized
• examples of rules of inference
  • modes ponens
  • resolution

Determining Meaning - Model Theory

• start with interpretations
  • how the world can be
• specifies which interpreations are compatible with which syntactic structures
  • called satisfaction or models
  • gives a check for intuitions
• define entailment as a relation between two syntactic structures
  • one kb entails another if all models of the first are also models of the second
• not usually directly mechanizable

Determining Meaning - Axiomatization

• provide a mapping into another formalism (often first-order logic)
• parasitic on the semantics of that other formalism

Determining Meaning - Operational

• very flexible
• generally impossible to determine exactly what is going on

Determining Meaning - Ideal Situation

• best situation is both model theory and proof theory
  • that have the same result
• model theory gives intuitions
• proof theory provides mechanization

Pitfalls in formal systems

• conflict with intuitions
  • prescriptive vs descriptive in RDFS
  • DAML+OIL entailment
• inconsistency/paradoxes
  • set theory
  • Liar's paradox

Issues in the design of a Web Ontology Language

• what syntax to use
  • choices appear to be XML and RDF
• how to specify semantics
  • model theory or proof theory or ....
• how much expressive power is needed
  • frames
  • disjunction
  • ...
  • what computational properties result
• what computational properties are wanted
• relationship with XML and RDF(S)

Relationships between (Representation) Formalisms

1. semantic extension (only)
   • RDF and RDFS
2. syntactic and semantic extension
   • propositional logic and predicate logic
3. syntactic embedding
   • XML and RDF/XML

A simple Web Ontology Language (OWL-2)

• syntax basics
  • datatypes
  • XQuery 1.0 Data model
  • statements about basic objects
    • like RDF statements
  • definitions of classes and properties
    • not like RDF
  • descriptions
    • not like RDF
• model theory, interpretations, and entailment
• type 2 extension of RDFS

Proposed Approaches

• RDF syntax
• type 1 extension of RDF
• problem - entailment not correct
• example - AND [Person Student] vs AND [Student Person]
• possible solution - require all lists (and other syntactic structures) in all interpretations
• problem - semantic paradoxes

Proposed Approaches

• XML syntax, relatively compatible with RDF
  • type 2 extension of RDFS
• semantic extension of RDF, includes RDF meta theory
• problem - complex constructs - e.g., conditional transitivity
• possible solution - forbid such conditional constructs
• problem - unprincipled

Proposed Approaches

• XML syntax, object stuff in RDF
• semantics different from RDF
  • not an extension of RDF
• no semantic problems

Problems with Proposed Approaches

• DAML+OIL, compatible with RDF
  • need to add entailment to DAML+OIL model theory
  • semantic paradoxes
• OWL-1
  • conditional meta-constructs are computationally ugly
  • interpretations are very complex, so possibility of paradoxes
• OWL-2
  • diverges from RDF

Why is Entailment in DAML+OIL broken?

• don't get the appropriate entailments
  • lists don't exist in all interpretations
  • classes don't exist in all interpretations
• not a problem for RDF(S) because RDF(S) has no class constructors
• possible fixes
  • abandon entailment
  • diverge from RDF(S)
    • some RDF(S) entailments will not be DAML+OIL entailments
  • place more restrictions on DAML+OIL interpretations
    • add in missing stuff

Why is Fixed Entailment in DAML+OIL problematic?

• need all possible syntax in all interpretations
• lists (including circular, etc.)
• classes (including self-referential)
• some of this (self-referential) syntax cannot be given meaning

Why is Entailment in OWL-1 difficult?

• quasi-logical predicates (like swol:TransitiveProperty) can show up in conditional constructs
• all of John's friends are transitive properties, John has at least one friend which is oneof friend or enemy