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
Determining Meaning - Model Theory
- start with interpretations
- 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
- semantic extension (only)
- syntactic and semantic extension
- propositional logic and predicate logic
- syntactic embedding
A simple Web Ontology Language (OWL-2)
- syntax basics
- datatypes
- XQuery 1.0 Data model
- statements about basic objects
- definitions of classes and properties
- descriptions
- model theory, interpretations, and entailment
- type 2 extension of RDFS
Proposed Approaches
DAML+OIL
- 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
OWL-1 (OWL) (4 January 2002)
- XML syntax, relatively compatible with RDF
- 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
OWL-2 (OWL') (3 January 2002)
- XML syntax, object stuff in RDF
- semantics different from 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
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
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
so no models are possible
paradoxes
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