Imposing Restrictions Over Temporal Properties in OWL: A Rule Based Approach

1. Imposing Restrictions Over Temporal Properties in OWL: A Rule Based Approach Sotiris Batsakis Euripides G.M. Petrakis Technical university of crete Intelligent systems laboratory

2. Introduction • Temporal Properties are not binary • Representation in OWL involves additional objects • Cardinality restrictions over temporal properties cannot apply directly • A rule based approach is proposed • Two different interpretations of restrictions over temporal properties Technical University of Crete

3. Motivation • OWL property semantics • Domains, Ranges, Subproperty, Equivalence, Symmetric, Assymetric, Functional, Inverse Functional, Reflexive, Irreflexive, Disjoint, Transitive • OWL property restrictions • All values from, Some Values From, Intersection , Union, Min Cardinallity, Max Cardinality, Exact Cardinality • Representation of temporal properties affects their semantics and restrictions Technical University of Crete

4. Temporal Representation (N-ary) Professor Course teaches Course Professor Teaching Interval Technical University of Crete

5. Temporal Representation (4D-fluents) Professor Course teaches Professor Course timesliceOf timesliceOf teaches Professor TimeSlice Course Timeslice Interval interval interval Technical University of Crete

6. Property Restrictions & Semantics • Domains-Ranges are adjusted • Domain timesliceOf Professor • Range timesliceOf Course • Property Semantics Retained • Symmetric, Equivalent, Reflexive, Subproperty Course Professor Interval Professor Timeslice Course TimeSlice teaches Technical University of Crete

7. Property Restrictions Problems • Cardinality Restrictions (min, max, exact) • Imposing cardinality on “new” property affects meaning (many timeslices, perhaps for overlapping intervals exist) • Imposing restriction on property chains is not supported because it leads to undecidability (Horrocks et.al. “Practical Reasoning for Expressive Description Logics” , 1999). Professor Course Interval Professor TimeSLice Course TimeSlice Technical University of Crete

8. Imposing Cardinality Restrictions • SWRL DL safe rules are applied • Decidability is retained, supported by reasoners (e.g. Pellet) • Rules apply only on named individuals (ABox) and not class descriptions (TBox) into the ontology • Open world assumption is adopted, thus min cardinality restrictions cannot be directly applied. • Restrictions have two different interpretations • On the entire existence of the object • On every specific temporal interval Technical University of Crete

9. First Interpretation-entire existence • A professor can’t teach more than n different courses in his career: Professor(x) ⋀ (timesliceOf(x1, x) ⋀ … ⋀timesliceOf(xn+1,x) ⋀ teaches(x1, y1) ⋀ teaches(xn+1, yn+1) ⋀ timesliceOf(y1 ,z1)… ⋀timesliceOf(yn+1, zn+1) ⋀ Alldifferent(z1, z2,…, zn+1) ⋀ Course(z1)… error(x, z1) • Rule directly detects inconsistencies for max cardinality • For min cardinality a similar rule asserts which individuals are related with more than n objects, and a SPARQL query detects individuals without the assertion. Technical University of Crete

10. Second Interpretation-every interval • A professor can’t teach more than n different courses simultaneously : Professor(x) ⋀ (timesliceOf(x1, x) ⋀ … ⋀timesliceOf(xn+1,x) ⋀ teaches(x1, y1) ⋀ teaches(xn+1, yn+1) ⋀ timesliceOf(y1 ,z1)… ⋀hasinterval(x1,w1)… ⋀ hasinterval(xn+1,wn+1) ⋀ timesliceOf(yn+1, zn+1) ⋀ Alldifferent(z1, z2,…, zn+1) ⋀pairwiseoverlapping(w1, …wn+1) ⋀ Course(z1)… error(x, z1) • Rule directly detects inconsistencies for max cardinality • Detecting overlapping intervals is achieved using temporal reasoning rules (S. Batsakis and E.G.M. Petrakis. “SOWL: A Framework for Handling Spatio-Temporal Information in OWL 2.0”, RuleML 2011) Technical University of Crete

11. Temporal Reasoning • Implemented in SWRL • Applies on interval Allen’s relations (e.g., before, after, overlaps) • Based on Path Consistency • Intersects and composes existing relations until no rules apply or inconsistency is detected • Example Composition • During(x,y) ⋀ Meets(y,z) Before(x,z) • Example Intersection • (Before(x,y) OR Meets(x,y)) ⋀ Meets(x,y)Meets(x,y) • Tractable Sound and Complete for specific sets of temporal relations Technical University of Crete

12. Additional Property Semantics • Functional and Inverse functional are handled as at most one cardinality restrictions • Asymmetric: This is handled as a cardinality restriction, where the same property cannot hold for interchanged subjects and objects for timeslices with overlapping intervals. • Irreflexive: This is handled as a cardinality restriction; two timeslices of an object cannot be related with the property. • Transitive: Fluent properties are declared transitive since related timeslices must have equal intervals (by the definition of the 4D-fluent model) and for these intervals transitivity is applied. Technical University of Crete

13. Contributions and limitations • Contributions • Offer support for property restrictions and semantics over temporal representations in OWL • Rule based approach that retains decidability • Compliance with existing standards and tools (OWL, SWRL, Pellet) • Limitations • Applies only on named individuals • Exponential to the number of the cardinality restriction at hand (e.g. at most n rule is exponential to n) Technical University of Crete

14. Future Work • Detecting the maximal decidable description logic that supports temporal cardinality restrictions • Optimize SWRL implementations of OWL reasoners • Optimize the rules Technical University of Crete

15. Thank You Questions?