Annotated RDF

Octavian UdreaDiego Reforgiato RecuperoV.S. Subrahmanian

University of Maryland

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

Foundations of aRDF

Annotations are partial orders (A,≤) Afuzzy , Atime , Atime-intervals , Apedigree

Cartesian products can generate others Such as Afuzz-time = Afuzzy X Atime

Builds on annotated logic (Kifer et al. 1992)

aRDF syntax

Student

PhD-Student

Organiza-tionUnit

Organization

Person

AffiliatedPerson

hasPartsubClassOf

affiliatedWith

Employee

EducationalEmployee subClassOf

subClassOf

subClassOf

subClassOf

I

rdf:typehasPart,

(0.95,2001)

rdf:type, (0.8, 2003)

rdf:type, (0.6, 2004)

Professor

subClassOf

rdf:type

rdf:type, (0.85, 1999)

AcademicResearcher

subClassOf

hasAdvisor, (0.9, 2004)

(hasAdvisor,rdfs:subpropertyOf, hasSupervisor)Triples not explicitly annotated are assumed annotated with (1, now).

affiliatedWith, (0.7, 2003)

hasSupervisor,

(0.95,2003)

hasAdvisor, (0.7, 2003)

Mary

William

Adam

ACME University

ACME Foundation

Max

Set of annotated triples

(r,p:a,v)

aRDF syntax

Student

PhD-Student

Organiza-tionUnit

Organization

Person

AffiliatedPerson

hasPartsubClassOf

affiliatedWith

Employee

EducationalEmployee subClassOf

subClassOf

subClassOf

subClassOf

I

rdf:typehasPart,

(0.95,2001)

rdf:type, (0.8, 2003)

rdf:type, (0.6, 2004)

Professor

subClassOf

rdf:type

rdf:type, (0.85, 1999)

AcademicResearcher

subClassOf

hasAdvisor, (0.9, 2004)

(hasAdvisor,rdfs:subpropertyOf, hasSupervisor)Triples not explicitly annotated are assumed annotated with (1, now).

affiliatedWith, (0.7, 2003)

hasSupervisor,

(0.95,2003)

hasAdvisor, (0.7, 2003)

Mary

William

Adam

ACME University

ACME Foundation

Max

We’re .9 sure that Max had Adam as an advisor until

2004

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

Satisfying interpretation example

PhD-Student

rdf:type, (0.6, 2004)rdf:type, (0.85, 1999)

AcademicResearcher

hasAdvisor, (0.9, 2004)

(hasAdvisor,rdfs:subpropertyOf, hasSupervisor)Triples not explicitly annotated are assumed annotated with (1, now).

hasSupervisor, (0.95,2003)

hasAdvisor, (0.7, 2003)

Mary

William Adam Max

Satisfying interpretation example

PhD-Student

rdf:type, (0.6, 2004)rdf:type, (0.85, 1999)

AcademicResearcher

hasAdvisor, (0.9, 2004)

(hasAdvisor,rdfs:subpropertyOf, hasSupervisor)Triples not explicitly annotated are assumed annotated with (1, now).

hasSupervisor, (0.95,2003)

hasAdvisor, (0.7, 2003)

Mary

William Adam Max

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

Satisfying interpretation example

hasSupervisor, Personal_Webpage

hasSupervisor, Graduate_School

hasSupervisor, Faculty_List

hasSupervisor, Dept_Webpage

hasAdvisor, Faculty_ListSteve William

Mary

Max

Personal_Webpage ≤ Dept_WebpageFaculty_List ≤ Graduate_School

Satisfying interpretation example

hasSupervisor, Personal_Webpage

hasSupervisor, Graduate_School

hasSupervisor, Faculty_List

hasSupervisor, Dept_Webpage

hasAdvisor, Faculty_ListSteve William

Mary

Max

Personal_Webpage ≤ Dept_WebpageFaculty_List ≤ Graduate_School

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

aRDF consistency

The existence of a satisfying I: For (r,p:ai,v), the set {ai} has an upper

bound Let Ak(r,p,v) be the set of annotations

on the kth p-path from r to v (for transitive p)

The set B = {LB(Ak)} has an upper bound

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 *(n3 * e + n*a2))

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

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

aRDF atomic query examples

PhD-Student

rdf:type, (0.6, 2004)rdf:type, (0.85, 1999)

AcademicResearcher

hasAdvisor, (0.9, 2004)

(hasAdvisor,rdfs:subpropertyOf, hasSupervisor)Triples not explicitly annotated are assumed annotated with (1, now).

hasSupervisor, (0.95,2003)

hasAdvisor, (0.7, 2003)

Mary

William Adam Max

Query: (Max,?p:(0.8,2002), William)

Answer: {(Max, hasSupervisor:(0.9,2003), William)}

aRDF atomic query examples

PhD-Student

rdf:type, (0.6, 2004)rdf:type, (0.85, 1999)

AcademicResearcher

hasAdvisor, (0.9, 2004)

(hasAdvisor,rdfs:subpropertyOf, hasSupervisor)Triples not explicitly annotated are assumed annotated with (1, now).

hasSupervisor, (0.95,2003)

hasAdvisor, (0.7, 2003)

Mary

William Adam Max

Query: (Mary,hasSupervisor:(0.7,2002), ?v)

Answer: {(Mary, hasAdvisor:(0.7,2003), William)}

aRDF theory closure

At each step, add to O one of: For (r,p:a1,v), (r,p’:a2,v), p’ is a subProperty of p

(or p = p’), add (r,p:a,v), where a is a minimal upper bound for a1,a2

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)

Naïve query answer algorithm

1. Compute closure lfp(O)2. Choose semi-unifiable triples with

annotations “above” the query’s3. Eliminate any triples entailed by

subsets

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

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

atomicAnswerX complexity

atomicAnswerV (and R) are running in time O(n2 * e + n * e * a2) O(n2 * e + n * e * a2) when annotation is a

complete lattice atomicAnswerP is has the same worst-

case complexity atomicAnswerA is O(n * e * a2) Complexity results given for finite partial

orders For lattices, the “a” factors dissapear

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.

Consistency running time

atomicAnswer running time

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

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

Questions and comments