Computability and Complexity Issues of Extended RDF

1
Computability and Complexity Issues of Extended
RDF

• Anastasia Analyti
• Institute of Computer Science, FORTH-ICS, Greece
• Grigoris Antoniou
• Dept. of Computer Science, Univ. of Crete, Greece
• Carlos Viegas Damásio
• CENTRIA, Depart. De Informatica, Univ. Nova de
  Lisboa, Portugal
• Gerd Wagner
• Inst. Of Informatics, Brandenburg Univ. of
  Technology at Cottbus, Germany

Presenter Grigoris Antoniou
2
Presentation Overview

• Define ERDF Ontologies
• provide an example
• Define ERDF stable model semantics for ERDF
  Ontologies
• it extends RDFS
• it is undecidable
• Propose ERDF n-stable model semantics
• a slight modification of ERDF stable model
  semantics
• it extends RDFS
• it is decidable
• Future work

3
ERDF Framework

• The ERDF framework extends the semantic web
  language RDFS with
• weak negation (),
• strong negation (?),
• derivation rules.
• ERDR distinguishes between
• partial properties p
• p(x,y) is possibly neither true nor false
  (interpretation level)
• partial classes c
• rdftype(x,c) is possibly neither true nor false
  (interpretation level)
• total properties p
• p(x,y) is either true or false (interpretation
  level)
• total classes c
• rdftype(x,c) is either true or false
  (interpretation level)

4
Open-World Closed-World Reasoning

• ERDF enables the combination of
• closed-world reasoning through the default
  closure rules
• ? p(?x, ?y) ? p(?x, ?y). OR
• p(?x, ?y) ? ? p(?x, ?y).
• ? rdftype(?x, c) ? rdftype(?x, c). OR
• rdftype(?x, c) ? ? rdftype(?x, c).
• Example ? rdftype(?x, exChild) ?
  rdftype(?x, exChild).
• open-world reasoning
• through the metaclasses erdfTotalProperty
  and erdfTotalClass
• (on total properties and total classes, and
  ? coincide)
• Example rdftype(exAdult, erdfTotalClass).

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ERDF Ontologies

• An ERDF ontology OltG,Pgt is the combination of
• an ERDF graph G containing (implicitly
  existentially quantified)positive and negative
  information,
• thus, G may contain variables (blank nodes)
• Example Grdftype(?x, Guest), ?
  rdftype(?x, Adult).
• an ERDF program P containing derivation rules,
  with possibly
• all connectives , ?, ?, ?, ?, ? in the
  body of a rule, and
• strong negation ?, false in the head of a
  rule.
• thus, P may contain constraints
• Example false ? rdftype(?x, exChild),
  rdftype(?y, exWine), exserve(?x, ?y).
  

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Example Drink Selection Problem

• We want to select drinks for a dinner such that
• for each adult guest that we (know that) likes
  wine, there on the table exactly one wine that
  he/she likes,
• guests who are not adults and not children should
  be served Coca-Cola,
• adult guests for whom we do not know if they
  like wine,
• they should also be served Coca Cola.
• In contrast to a child, we cannot decide if guest
  is an adult or not.
• i.e. on exAdult an OWA applies
• on exChild a CWA applies

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Example Drink Selection problem (cont.)

• We define O ltG,Pgt, where
• G rdftype(Carlos, Guest), rdftype(Gerd,
  Guest), rdftype(Anne, Guest),
  rdftype(Riesling, Wine), rdftype(Retsina,
  Wine), likes(Gerd, Riesling), likes(Gerd,
  Retsina), likes(Carlos, Retsina), rdftype(Gerd,
  Adult), rdftype(Carlos, Adult),
  rdftype(?x, Guest), ? rdftype(?x, Adult).
• P id(?x,?x). rdftype(Adult,
  erdfTotalClass).
• ? rdftype(?x, Child) ? rdftype(?x,
  Child).
• rdftype(?y, SelectedWine) ? rdftype(?x,
  Guest), rdftype(?x, Adult),
• 
  rdftype(?y,Wine), likes(?x,?y),
• ??z (rdftype(?z,
  SelectedWine), id(?x,?y) ? likes(?x, ?z)).
• serveSoftDrink(?x, Coca-Cola) ?
  rdftype(?x, Guest), ? rdftype(?x, Adult).
  ?
  rdftype(?x, Child).
• serveSoftDrink(?x, Coca-Cola) ?
  rdftype(?x, Guest), rdftype(?x, Adult),
• 
  ??z (rdftype(?y, Wine) ? likes(?x,
  ?y)).

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Vocabulary of an ERDF Ontology

• Vocabulary of an ERDF ontology OltG,Pgt
• VOVsk(G) ? VP ? VRDF ? VRDFS ? VERDF,
  where
• Vsk(G) terms appearing in the skolemized
  version of ERDF graph G.
• VP terms appearing in ERDF program P.
• VRDF RDF vocabulary terms.
• VRDFS RDFS vocabulary terms.
• VERDFerdfTotalProperty, erdfTotalClass.

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ERDF Stable Models

• Each stable model M ? Mst(O)
• interprets the terms in VO,
• assigns intended truth and falsity extensions to
  the classes and properties in VO
• satisfies all semantic conditions of an RDFS
  interpretation on VO, as well as new semantic
  conditions, particular to ERDF,
• starting of an intended interpretation for sk(G),
  a stratified sequence of rule applications is
  produced, where all applied rules remain
  applicable throughout the generation of M
• definition is based on Parial Logic Herrer,
  Jaspars, Wagner 1999), which extends Answer
  Set Programming.

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Example Drink Selection problem (cont.)

• We define O ltG,Pgt, where
• G rdftype(Carlos, Guest), rdftype(Gerd,
  Guest), rdftype(Anne, Guest),
  rdftype(Riesling, Wine), rdftype(Retsina,
  Wine), likes(Gerd, Riesling), likes(Gerd,
  Retsina), likes(Carlos, Retsina), rdftype(Gerd,
  Adult), rdftype(Carlos, Adult),
  rdftype(?x, Guest), ? rdftype(?x, Adult).
• P id(?x,?x). rdftype(Adult,
  erdfTotalClass).
• ? rdftype(?x, Child) ? rdftype(?x,
  Child).
• rdftype(?y, SelectedWine) ? rdftype(?x,
  Guest), rdftype(?x, Adult),
• 
  rdftype(?y,Wine), likes(?x,?y),
• ??z (rdftype(?z,
  SelectedWine), id(?x,?y) ? likes(?x, ?z)).
• serveSoftDrink(?x, Coca-Cola) ?
  rdftype(?x, Guest), ? rdftype(?x, Adult).
  ?
  rdftype(?x, Child).
• serveSoftDrink(?x, Coca-Cola) ?
  rdftype(?x, Guest), rdftype(?x, Adult),
• 
  ??z (rdftype(?y, Wine) ? likes(?x,
  ?y)).

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Properties of ERDF Stable Model Semantics

• Syntactically restricted ERDF ontologies
• simple ERDF ontologies contain only , ? , ?,
• objective ERDF ontologies contain only ? , ?.
• ERDF stable model semantics
• extends RDFS,
• is undecidable even for simple ERDF ontologies
  without the terms erdfTotalClass and
  erdfTotalProperty,
• entailment of an ERDF graph is co-NP-complete for
  objectiveERDF ontologies,
• entailment of a general ERDF formula is
  undecidable for objective ERDF ontologies.

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ERDF n-Stable Model Semantics

• VO VOn rdf_i i gtn.
• The ERDF n-stable model semantics of an ERDF
  ontology O is defined similarly to the ERDF
  stable model semantics of O, but now only the
  interpretation of the terms in VOn is
  considered.
• ERDF n-stable model semantics
• extends RDFS,
• is decidable,
• entailment of an ERDF graph is co-NP-complete for
  objective and simple ERDF ontologies,
• is equivalent to ERDF stable model semantics for
  objective ERDF ontologies.

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Future Work

• Identify the complexity classes of entailment
  under ERDF n-stable model semantics for
• general ERDF ontologies,
• syntactically restricted ERDF ontologies (other
  than objective and simple).
• Extend ERDF with
• handling of contradiction,
• terms from the OWL vocabulary and datatypes,
• modularity constructs.