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
ECAI-2008

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 is decidable
Future work
ECAI-2008

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
rdf:type(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
rdf:type(x,c) is either true or false (interpretation level)
ECAI-2008

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).
 rdf:type(?x, c)  ~ rdf:type(?x, c). OR
rdf:type(?x, c)  ~  rdf:type(?x, c).
Example:  rdf:type(?x, ex:Child)  ~ rdf:type(?x, ex:Child).
open-world reasoning
through the metaclasses erdf:TotalProperty and erdf:TotalClass
(on total properties and total classes, ~ and  coincide)
Example: rdf:type(ex:Adult, erdf:TotalClass).
ECAI-2008

5. ERDF Ontologies An ERDF ontology O=<G,P> is the combination of:
an ERDF graph G containing (implicitly existentially quantified) positive and negative information,
thus, G may contain variables (blank nodes)
Example: G={rdf:type(?x, Guest),  rdf:type(?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  rdf:type(?x, ex:Child), rdf:type(?y, ex:Wine), ex:serve(?x, ?y).
ECAI-2008

6. 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 ex:Adult an OWA applies
on ex:Child a CWA applies
ECAI-2008

7. Example: Drink Selection problem (cont.)
We define O= <G,P>, where:
G = {rdf:type(Carlos, Guest), rdf:type(Gerd, Guest), rdf:type(Anne, Guest), rdf:type(Riesling, Wine), rdf:type(Retsina, Wine), likes(Gerd, Riesling), likes(Gerd, Retsina), likes(Carlos, Retsina), rdf:type(Gerd, Adult), rdf:type(Carlos, Adult), rdf:type(?x, Guest),  rdf:type(?x, Adult)}.
P = {id(?x,?x). rdf:type(Adult, erdf:TotalClass).
 rdf:type(?x, Child)  ~ rdf:type(?x, Child).
rdf:type(?y, SelectedWine)  rdf:type(?x, Guest), rdf:type(?x, Adult),
rdf:type(?y,Wine), likes(?x,?y),
?z (rdf:type(?z, SelectedWine), ~id(?x,?y)  ~likes(?x, ?z)).
serveSoftDrink(?x, Coca-Cola)  rdf:type(?x, Guest),  rdf:type(?x, Adult)  rdf:type(?x, Child).
serveSoftDrink(?x, Coca-Cola)  rdf:type(?x, Guest), rdf:type(?x, Adult),
?z (rdf:type(?y, Wine)  ~likes(?x, ?y)).}
blank nodes are handled by skolemization
ECAI-2008

8. Vocabulary of an ERDF Ontology
Vocabulary of an ERDF ontology O=<G,P>:
VO=Vsk(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.
VERDF={erdf:TotalProperty, erdf:TotalClass}.
ECAI-2008

9. 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.
ECAI-2008

10. Example: Drink Selection problem (cont.)
We define O= <G,P>, where:
G = {rdf:type(Carlos, Guest), rdf:type(Gerd, Guest), rdf:type(Anne, Guest), rdf:type(Riesling, Wine), rdf:type(Retsina, Wine), likes(Gerd, Riesling), likes(Gerd, Retsina), likes(Carlos, Retsina), rdf:type(Gerd, Adult), rdf:type(Carlos, Adult), rdf:type(?x, Guest),  rdf:type(?x, Adult)}.
P = {id(?x,?x). rdf:type(Adult, erdf:TotalClass).
 rdf:type(?x, Child)  ~ rdf:type(?x, Child).
rdf:type(?y, SelectedWine)  rdf:type(?x, Guest), rdf:type(?x, Adult),
rdf:type(?y,Wine), likes(?x,?y),
?z (rdf:type(?z, SelectedWine), ~id(?x,?y)  ~likes(?x, ?z)).
serveSoftDrink(?x, Coca-Cola)  rdf:type(?x, Guest),  rdf:type(?x, Adult)  rdf:type(?x, Child).
serveSoftDrink(?x, Coca-Cola)  rdf:type(?x, Guest), rdf:type(?x, Adult),
?z (rdf:type(?y, Wine)  ~likes(?x, ?y)).}
O has two two stable models M1 and M2.
M1 |=  rdf:type(Anne, Adult) and M2 |= rdf:type(Anne, Adult).
M1, M2 |= serveSoftDrink(Anne, Coca-Cola).
M1, M2 |= serveSoftDrink(skG(?x), Coca-Cola).
M1, M2 |= rdf:type(Retsina, SelectedWine)  ~ rdf:type(Riesling, SelectedWine).
ECAI-2008

11. 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 erdf:TotalClass and erdf:TotalProperty,
entailment of an ERDF graph is co-NP-complete for objective ERDF ontologies,
entailment of a general ERDF formula is undecidable for objective ERDF ontologies.
ECAI-2008

12. ERDF n#-Stable Model Semantics
VO= VO#n – {rdf:_i | i >n}.
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 VO#n 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.
ECAI-2008

13. 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.
ECAI-2008