A Tale of Two Layerings
by Peter F. Patel-Schneider
for presentation at W3C Technical Plenary, March 2003

Abstract

Semantic layering is a current problem in the Semantic Web. This is strange, as there are many examples of semantic layering in logical formalisms that have been around for decades. The problem is that semantic layering in the Semantic Web is a much stronger notion than what one would think it should be. Some effects of this very strong notion and how to get around them are presented.

The Two Layerings

How to Work in the Semantic Web Layering

1. Design a logical language
• Devise a syntax and semantics
• Make it compatible with lower level(s)
2. Encode the syntax in RDF triples
   • Lot's of work here
3. Devise a set of semantic conditions on RDF interpretations that make the syntactic encoding work
   • Even more work here

Syntactic Problems in RDF Triple Encoding

Complex syntactic constructs have to be split into pieces

1. _:r rdf:type owl:Restriction .
_:r owl:onProperty ex:friend .
_:r owl:allValuesFrom ex:Person .
2. _:or rdf:type ll:or .
_:or ll:arg1 raining .
_:or ll:arg2 Tuesday .
3. _:l rdf:first ex:one .
_:l rdf:rest rdf:nil .

Possibility of missing or duplicate pieces

Syntactic Problems in RDF Triple Encoding (cont'd)

Everything is top-level

1. ex:John ll:believes _:pa .
_:pa rdf:subject ex:ClarkKent .
_:pa rdf:predicate owl:sameIndividualAs .
_:pa rdf:object ex:SuperMan .

Can't treat components differently

Too few syntactic categories

1. _:a rdf:type ll:forall .
_:a ll:variable ex:x .
_:a ll:formula _:f .
_:f rdf:subject ex:x .
...

Can't treat components differently

Semantic Problems in RDF Triple Encoding

Have to handle malformed syntactic constructs

• e.g., missing parts of syntactic constructs
• e.g., duplicated parts of syntactic constructs
• e.g., extra information associated with syntactic constructs

All aspects of syntax contribute to meaning

• Ordering is significant in syntax, e.g. OWL intersections
• Structural names are significant, e.g., rdf:_1 vs rdf:_2

Semantic Problems in RDF Triple Encoding (cont'd)

All triples have semantic impact

• Syntactic encodings exist in the domain of discourse
• Some syntactic constructs need to be represented as data
  • e.g., disjunctions
• Need semantic theory of syntax
• e.g., comprehension principles for OWL descriptions
• Can easily lead to semantic paradoxes
• Handled in OWL by weakening comprehension principles

All names denote the same way

• How to handle logical variables and modal constructs?

Semantic Layering - weakest version

Logics:

A logic is a formal language equipped with a model theory that has an entailment relationship between top-level syntax in a logic defined in the more-or-less usual way.

Semantic Layering: L_b under L_t

Mapping M from top-level syntax of L_b to top-level syntax of L_t such that S₁ entails S₂ in L_b if and only if M(S₁) entails M(S₂) in L_t.

Example - Propositional Logic and First-Order Logic:

Different syntax - P ∨ Q vs P(a) ∨ Q(b); different semantics - mapping from letters to truth values vs mapping from predicates to sets of tuples; but there is an entailment-preserving mapping

Semantic Layering - strengthenings

Weak semantic correspondence

Require a ``nice'' mapping between interpretations - as in PL and FOL

Medium semantic correspondence

Require mapping be an extension - not as in DL and FOL

Strong semantic correspondence

Require that interpretations be the same - as in RDF and RDFS

Weak syntactic correspondence

Require that the languages be in a superset relationship - as in FOL and HOL

Strong syntactic correspondence

Require that the languages be the same - as in RDF and RDFS

Problems introduced by Stronger Semantic Layerings

Medium semantic correspondence

Can't reinterpret the semantics to make it work in the more-general setting

Strong semantic correspondence

Can't even extend the semantics to make it work in the more-general setting

Strong syntactic correspondence

Can't introduce new basic syntactic constructs

Semantic Problems - A Theory of Classes

A Desirable Inference:

Premises:

John is an instance of Student.
John is an instance of Employee.

Conclusion:

John is an instance of the intersection of Student and Employee.

• If intersections have to be encoded as RDF triples, this requires that the intersection automatically exists whenever Student and Employee exists (comprehension principles).
• Many more of these examples can be devised.

Semantic Problems - A Theory of Classes

A Unfortunate Inference from too-strong comprehension principles:

Premise:

(empty)

Conclusion 1:

rdf:type is an instance of the restriction whose instances do not have an rdf:type link to the restriction itself.

Conclusion 2:

rdf:type is not an instance of the restriction whose instances do not have an rdf:type link to the restriction itself.

Removing the RDF Straightjacket

• Treat RDF (or XML) as a language for providing facts.
• Still use XML for syntax.
• Use different syntaxes for the other Semantic Web languages.
  • Allow RDF syntax, but also allow extensions.
  • E.g., syntax like OIL's for Ontologies.
  • E.g., more-normal syntax for logic languages.
• Use the RDF model theory as the basis for the model theory of the other Semantic Web Languages.
  • Other Semantic Web languages extend the RDF semantics.
• Not the most-general layering principle but removes most of the limitations