1. Reducing OWL Entailment to Description Logic Satisfiability Ian Horrocks and Patel Schneider Presented by: Muhammed Al-Muhammed

2. 2 The problem Applying DL reasoning techniques to OWL  Specifically: ontology entailment This cannot be done directly because:  The syntax of OWL does not directly correspond to DL axioms Careful: OWL allows for anonymous individuals in axioms  OWL inference defined in terms of ontology entailment not ontology satisfiability Careful: the role negation not supported by any implemented DL reasoner

3. 3 Why This Reduction? OWL entailment is costly Reduction of OWL (DL, Lite) Ontology entailment to knowledge base satisfiability in SHOIN(D) and SHIF(D) provides two advantages  OWL entailment has the same complexity as knowledge base satisfiability  Use of well-known DL reasoners (e.g. RACER)

4. 4 Reduction from OWL Entailment to DL Unsatisfiability OWL DL Ontology Entailment O1 |= O2 SHOIN + (D) Knowledge Base Entailment K1 |= K2 SHOIN(D) Knowledge Base Unsatisfiability K1 |= A for all A in K2 OWL Lite Ontology Entailment O1 |= O2 SHIF + (D) Knowledge Base Entailment K1 |= K2 SHIF(D) Knowledge Base Unsatisfiability K1 |= A for all A in K2

5. 5 OWL DL and OWL Lite Subsets of OWL full OWL DL restricts OWL Full  No cycles  Classes, properties, and individuals are disjoint OWL Lite is a subset of OWL DL  Eliminates some of the constructs (e.g. oneOf)

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7. 7 SHOIN(D), SHOIN + (D) SHIF(D), SHIF + (D)

8. 8 OWL DL to SHOIN + (D) Translate every axiom and fact in OWL DL to one or more axioms in SHOIN + (D) Easy translation for OWL DL axioms OWL DL axiomSHOIN + (D) axiom Class (A complete C 1 …C n ) A  (C 1 ) Π … Π (C n ) (C 1 ) Π … Π (C n )  A DisjointClass(C 1 …C n ) (C i )   (C j )  i  j Class (A partial C 1 …C n ) A  (C 1 ) U … U (C n ), (C 1 ) U … U (C n )  A

9. 9 OWL DL to SHOIN + (D) Translation of OWL facts to SHOIN + (D) axioms is more complex  Facts can be stated with respect to anonymous individuals  E.g. Individual(type(C) value(R Individual(type(D)))) Solution: use the existence axiom  The above fact can be translated:  (C Π  R.D)

10. 10 OWL DL to SHOIN + (D) OWL DL SHOIN + (D) Individual(x 1 …x n )  ( F(x 1 )…F(x n )) Type(C) (C) Value(R x)  R. F(x) Value(U v)  U. {v} o{o}

11. 11 OWL DL to SHOIN + (D) The translation preserves the equivalence The size of the resulted KB is  (OWL DL ontology) 2 The translation can be done in linear time in the size of KB

12. 12 SHOIN + (D) KB Entailment to SHOIN(D) KB Unsatisfiability Axiom A Transformation g(A) c  d  c Trans(r) r  s f  g x: c Π  d T   c x:  r.  r.{y} Π   r.{y} x:  r.{y} Π   s.{y} x: U z  V  f.{z} Π   g.{z} Plus one fresh data value for each datatype in K

13. 13 SHOIN + (D) KB Entailment to SHOIN(D) KB Unsatisfiability K—SHOIN + (D)  (K)—SHOIN(D) c  d  c Trans(r) r  s f  g Given the transformation, if K 1 and K 2 are two SHOIN + (D): K 1 |= K 2  (K1) U {g(A)} is unsatisfiable for all A in K 2 c  d a: C Trans(r) r  s f  g

14. 14 Consequences Polynomial time translation from OWL DL to SHOIN(D) Polynomial number of knowledge base satisfiability problem However, most of the satisfiability problems in SHOIN(D) are in NExpTime No optimized inference algorithm

15. 15 Transforming OWL Lite to SHIF + (D) OWL Lite facts F Translation F’(F) Individual(x 1 …x n ) F’(a: x 1 ),…,F’(a: x n ) a: type(C) a: (C) a: value(R, x) : R, F’(b: x) a: value(U v) : U a: o a = o

16. 16 Transforming SHIF + (D) Entailment to SHIF(D) Unsatisfiability SHIF + (D) axiom A g’(A) a: C a:  C : R b: B, a:  R.  B : U a:  U. v K |= r  s iff  (K) U {x: B Π  r(  s --.  B )} is unsatisfiable K |= r transitive iff  (K) U {x:  r(  r. T)} is unsatisfiable

17. 17 Analysis The paper provides proof of correctness, completeness through theorems The paper lacks any empirical experiments Transforming to SHOIN(D) that has no practical reasoning algorithm—is that weakness for the paper? (Yes/No)?