1. Annotated RDF Octavian Udrea Diego Reforgiato Recupero V.S. Subrahmanian University of Maryland

2. Motivation  Many RDF extensions for specific scenarios: Temporal (Gutierrez et. al 2005) Uncertainty (Dubois et. al 2005, Straccia et al. 2005) Provenance (Carroll et. al 2005)  Can we construct a common syntax and semantics for RDF extensions? Together with efficient query mechanism

3. Foundations of aRDF  Annotations are partial orders (A,≤) A fuzzy, A time, A time-intervals, A pedigree  Cartesian products can generate others Such as A fuzz-time = A fuzzy X A time  Builds on annotated logic (Kifer et al. 1992)

4. aRDF syntax Set of annotated triples (r,p:a,v)

5. aRDF syntax We’re.9 sure that Max had Adam as an advisor until 2004

6. aRDF satisfying interpretation  We consider transitive properties as a simple inference capability  A mapping I from the universe of possible triples (r,p,v) to A  A satisfying interpretation I for O has: For all (r,p:a,v) in O, a ≤ I(r,p,v) For all paths on transitive properties, the lower bounds of the set of annotations is less than I(r,p,v)  Entailment defined in the usual way

7. Satisfying interpretation example

8. (0.9,2003) ≤ I(Max,hasSupervisor,William)

9. Satisfying interpretation example

10. No matter what we assign to I(Mary,hasSupervisor,William), I will not satisfy O

11. aRDF consistency  The existence of a satisfying I: For (r,p:a i,v), the set {a i } has an upper bound Let A k (r,p,v) be the set of annotations on the k th p-path from r to v (for transitive p) The set B = {LB(A k )} has an upper bound

12. aRDF consistency results  All RDF instances annotated with partial orders with top elements are consistent  For general partial orders, consistency verification runs in O(p *(n 3 * e + n*a 2 ))

13. aRDF atomic queries  (R,P:A,V) where at most one is variable  Examples: (Max, ?p:(0.8,2002), William) (Mary, hasSupervisor:(0.7,2002),?v)  (r,p:a,v) and (r’,p’:a’,v’) are semi- unifiable if there is a substitution θ: r θ = r’ θ, p θ = p’ θ, v θ = v’ θ

14. aRDF atomic query answers  The answer to (R,P:A,V) is the set of (r,p:a,v) such that: (r,p:a,v) is semi-unifiable with (R,P:A,V) and A ≤ a (where applicable) (r,p:a,v) is entailed by the aRDF ontology (r,p:a,v) is not entailed by a subset of the answer  The minimal (w.r.t. entailment) set of triples entailed by the theory that semi- unifies with the query

15. aRDF atomic query examples Query: (Max,?p:(0.8,2002), William) Answer: {(Max, hasSupervisor:(0.9,2003), William)}

16. aRDF atomic query examples Query: (Mary,hasSupervisor:(0.7,2002), ?v) Answer: {(Mary, hasAdvisor:(0.7,2003), William)}

17. aRDF theory closure  At each step, add to O one of: For (r,p:a 1,v), (r,p’:a 2,v), p’ is a subProperty of p (or p = p’), add (r,p:a,v), where a is a minimal upper bound for a 1,a 2 Add (r,p:a,v) for (r,p’:a1,r’), (r’,p’’:a2,v), where  p’,p’’ are subProperty* of p  For all a’, (a’ ≤ a1) and (a’ ≤ a2) => (a’ ≤ a)  Monotonic operator => there exists a fixpoint lfp(O)

18. Naïve query answer algorithm 1. Compute closure lfp(O) 2. Choose semi-unifiable triples with annotations “above” the query’s 3. Eliminate any triples entailed by subsets

19. atomicAnswerX algorithms  lfp(O) can be exponential But the minimal set we look for in the answer is not  atomicAnswerV computes the answer to (R,P:A,V?) queries  atomicAnswerP computes the answer to (R,P?:A,V) queries  conjunctAnswer answers conjunctions of atomic queries

20. atomicAnswerX algorithms  atomicAnswerV: For the maximal transitive p-paths starting at r, compute: The lower bound(s) on the sets of annotations The least upper bound(s) of the previous set  atomicAnswerP: Similar approach for the maximal paths between r and v

21. atomicAnswerX complexity  atomicAnswerV (and R) are running in time O(n 2 * e + n * e * a 2 ) O(n 2 * e + n * e * a 2 ) when annotation is a complete lattice  atomicAnswerP is has the same worst- case complexity  atomicAnswerA is O(n * e * a 2 )  Complexity results given for finite partial orders For lattices, the “a” factors dissapear

22. Experimental results  Existing RDF ontologies with randomly generated annotations  Synthetically generated data up to 100,000 nodes Also varied number of properties, node degree, number of transitive properties, etc.

23. Consistency running time

24. atomicAnswer running time

25. Applications  We have started using aRDF on the STORY project http://om.umiacs.umd.edu  An online aRDF system will be released in August 2006 Features such as graphical editing and annotation, custom annotations, view maintenance

26. Conclusions  We have presented a general framework for extending RDF Based on annotated logic  Simple syntax and semantics  Query algorithms are very efficient in practice

27. Questions and comments