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Reducing OWL Entailment to Description Logic Satisfiability

Reducing OWL Entailment to Description Logic Satisfiability. Ian Horrocks and Patel Schneider Presented by: Muhammed Al-Muhammed. The problem. Applying DL reasoning techniques to OWL Specifically: ontology entailment This cannot be done directly because:

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Reducing OWL Entailment to Description Logic Satisfiability

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

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

  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)

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

  7. Translate every axiom and fact in OWL DL to one or more axioms in SHOIN+(D) Easy translation for OWL DL axioms OWL DL to SHOIN+(D) A  (C1) U … U (Cn) , (C1) U … U (Cn)  A Class (A partial C1…Cn)

  8. 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) OWL DL to SHOIN+(D)

  9. OWL DL to SHOIN+(D)

  10. 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

  11. 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: UzV f.{z} Π  g.{z} Plus one fresh data value for each datatype in K

  12. 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 K1 and K2 are two SHOIN+(D): K1 |=K2 <=> (K1) U {g(A)} is unsatisfiable for all A in K2 c  d a: C Trans(r) r  s f  g

  13. 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

  14. Transforming OWL Lite to SHIF+(D) OWL Lite facts F Translation F’(F) Individual(x1…xn) F’(a: x1),…,F’(a: xn) a: type(C) a: (C) a: value(R, x) <a, b>: R, F’(b: x) a: value(U v) <a, v>: U a: o a = o

  15. Transforming SHIF+(D) Entailment to SHIF(D) Unsatisfiability SHIF+(D) axiom A g’(A) a: C a:  C <a, b>: R b: B, a: R. B <a, v>: 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

  16. 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)?

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