Three Theses of Representation

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• Three Theses of Representation
• in the Semantic Web

Ian Horrocks University of Manchester Manchester,
UK horrocks_at_cs.man.ac.uk
Peter F. Patel-Schneider Bell Labs
Research Murray Hill, NJ, USA pfps_at_research.bell-l
abs.com
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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

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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

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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, DAMLOIL, 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

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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

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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)

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(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

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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 DAMLOIL, C subClassOf D equivalent to
  8 x.C(x) ! D(x)
• But have to axiomatise as holds2(subClass, C, D)

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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)

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(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
  rdftype being in domain of discourse
• Still no direct semantic interoperability with
  RDFS
• Computational pathway only via (performance-damagi
  ng) FOL mapping

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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

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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

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(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-damagi
  ng) FOL mapping

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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)

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