1. Three Theses of Representation in the Semantic Web
Ian Horrocks
University of Manchester
Manchester, UK
Peter F. Patel-Schneider
Bell Labs Research
Murray Hill, NJ, USA

2. Semantic Web Languages
SemWeb aims to make content accessible to automated processes
Add semantic markup (meta-data) describing content/function of resources
Need a common way of providing meta-data so that:
It can be understood and manipulated by automated processes (“agents”)
Agents can integrate meta-data from different sources
Proposed solution is famous language “layer cake”:

3. Language Architecture
Relationship between adjacent layers not clear
XML $ RDF relationship purely syntactic
RDF $ Ontology layer relationship should be something more?
RDF is proposed as base for SemWeb languages
Used to add metadata annotations to resources
Also used to define syntax and semantics of subsequent layers
Not clear that RDF is appropriate for all these functions
Limited set of syntax constructs (triples)
Not possible to extend syntax (as it is, e.g., when using XML)
Uniform semantic treatment of triple syntax
Non standard KR thesis and model theory
May facilitate development of SemWeb to use more standard KR thesis…

4. Ontology Language Layer
Ontologies set to play key role in SemWeb
source of shared and precisely defined terms for use in meta-data
RDF already extended to RDFS
Hierarchies of classes and properties
Domain and range constraints on properties
More expressive ontology languages clearly required
With logical connectives, quantifiers, transitive properties, etc.
E.g., OIL, DAML+OIL, and now OWL
Possible choices for language layering:
Base ontology language layer(s) on RDF(S)
Base ontology language layer(s) on “classical” FOL
Base ontology language layer(s) on SKIF/Lbase/CL languages

5. Semantics and Model Theories
Ontology/KR languages aim to model (part of) world
Constructs in language correspond to entities in world
Meaning given by mapping to some formal system
E.g., a logic such as FOL with its own well defined semantics
or a data model such as XQuery data model for XML
or (for more expressive languages) a Model Theory (MT)
MT defines relationship between syntax and interpretations
Can be many interpretations (models) of one piece of syntax
Models supposed to be analogue of (part of) world
E.g., elements of model correspond to objects in world
Formal relationship between syntax and models
Structure of models must reflect relationships specified in syntax
Inference (e.g., entailment) defined in terms of MT
E.g., A ² B iff every model of A is also a model of B

6. FOL Thesis Base SW languages on established FO hierarchy
Propositional logic
Decidable FOL subsets (e.g., DL, Horn)
Undecidable FOL subsets
Full FOL (and even HOL)
Higher layers extend syntax
Upwards compatibility, i.e., syntax retains same meaning in higher layers
Semantics via FOL mapping or standard FO model theory
Individual i ! element of domain (iI 2 D)
Class C ! sets of elements (CI µ D)
Property P ! binary rel on D (PI µ D £ D)

7. (Dis)advantages of FOL Thesis
Pros
Based on well known and extensively studied formalism
Wealth of theoretical knowledge and practical experience
Family of sub-languages with well known formal properties
E.g., decidability, complexity
Highly optimised reasoners for FOL and many sub-languages
E.g., DL reasoners, Horn (rule) reasoners, FOL provers
Mapping to FOL provides easy integration, e.g., of DL and Horn languages
FO subset of RDFS fits well in this framework
Cons
No classes as instances (unless extended to HOL)
Relatively poor fit with full RDFS
Can be axiomatised in FOL, but may damage semantic interoperability and computational properties

8. Axiomatisation
An Axiomatisation can be used to embed RDFS in FOL, e.g.:
Triple x P y translated as holds2(P,x,y)
Axioms capture semantics of language, e.g.:
Problems with axiomatisations include
May require large and complex set of axioms
Difficult to prove semantics have been correctly captured
Axiomatisation may greatly increase computational complexity
RDFS ! undecidable (subset of) FOL
No interoperability unless all languages similarly axiomatised
E.g., in DAML+OIL, C subClassOf D equivalent to x.C(x) ! D(x)
But have to axiomatise as holds2(subClass, C, D)

9. SKIF/Lbase/CL Thesis Base SW languages on SKIF/Lbase/CL
Similar to FOL thesis, but FOL replaced with CL
Higher layers extend syntax
Upwards compatibility, i.e., syntax retains same meaning in higher layers
Semantics via mapping into CL
CL provides model theory
Individual i ! element of domain (iV 2 D)
Class C ! element of domain (CV 2 D)
Property P ! element of domain (PV 2 D)
Second mapping (ext)
Class elt w ! set of elts (ext(w) µ D)
Prop elt k ! binary rel (ext(P) µ D £ D)

10. (Dis)advantages of CL Thesis
Pros
Classes as individuals without HOL extension
Can use as a basis for a family of sub-languages
Mapping to CL provides easy integration of sub-languages
Better fit with RDFS
Cons
Relatively new and untried
Little known about CL sub-languages
Confusion w.r.t. FOL compatibility
RDFS still requires axiomatisation due, e.g., to rdf:type being in domain of discourse
Still no direct semantic interoperability with RDFS
Computational pathway only via (performance-damaging) FOL mapping

11. Confusion w.r.t. FOL Compatibility
SKIF/Lbase/CL use same syntax as FOL
But allow variables to occur in predicate positions
Originally asserted that SKIF semantics coincide with FOL for well formed FOL sentences
Subsequently shown to be wrong for FOL with equality
E.g.,
Moral of the story
May confuse users more familiar with classical FOL
Easy to make mistakes with complex new formalisms
Risky to base future of SemWeb on such a new formalism

12. RDF Thesis All SW languages based on triples
Triple based syntax
Semantics compatible with semantics of triples as defined by RDF MT
Upwards & downwards compatibility
Syntax retains same meaning in higher layers
Higher layer syntax is valid in lower layers
Semantics via RDF model theory
Similar to CL, but only binary predicates
Language syntax also in domain of discourse
Higher layers impose additional constraints on models
Syntax must be encoded as triples
Awkward for complex constructs
Resulting triples also have meaning

13. (Dis)advantages of RDF Thesis
Pros
(Supposed) interoperability between language layers
RDF tools can be used to parse all SW languages into triples
Large ontologies/KBs can be stored in triple DBs
Cons
Achieving real (semantic) interoperability may be difficult or impossible
E.g., efforts to layer OWL on top of RDF(S)
Triple encoding of complex languages such as OWL is very clumsy
Triples introduced by encodings have semantic consequences
E.g., first-rest triples used in list syntax have same consequences as ground facts (even though ordering of list may be arbitrary)
Not clear if technique can be extended to more expressive languages
E.g., full FOL
Computational pathway only via (performance-damaging) FOL mapping

14. Summary Formal meaning of SW languages crucial to interoperability
Common semantic underpinning facilitates layered architecture
Widely assumed that RDF will provide this underpinning
But layering on top of RDF(S) may be difficult/impossible and does not lead to any direct computational pathway
Moreover, benefits are not clear
Alternative would be to use standard FOL as underpinning
Well established and well understood
Established family of languages capturing different trade-offs
Direct computational pathway for FOL and many sub-languages
FO subset of RDF(S) would fit well in this framework
Third approach is to use CL as underpinning
Relatively new and untested
May not solve problems with RDF(S)

15. Perhaps we should consider recalling the Semantic Web bandwagon in order to carry out a safety modification on the RDF component!