Nominal Schema DL 2011

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Talk at DL 2011 on Description Logics with Nominal Schema

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Nominal Schema DL 2011

  1. 1. Nominal Schemas for Integrating Rules and Description LogicsMarkus Krötzsch1 Frederick Maier2 Adila Krisnadhi2 Pascal Hitzler2 1 University of Oxford 2 Kno.e.sis Center – Wright State University The 24th Description Logic Workshop – Barcelona, 2011
  2. 2. Nominal Schema as An Extension of Standard DLs• Sufficient expressivity for standard DLs to incorporate (Datalog) rule-based modeling.• Seamless integration with standard DLs. • Syntax is not hybrid, i.e., completely DL-based. • Easy to incorporate in current OWL standard.• Semantics is not complicated.• Decidable. • A large tractable fragment is identified. Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 2
  3. 3. What is Nominal Schema?• Nominal-like DL syntax. • {v} where v is a variable.• Semantically behaves like DL-safe variable (without hybrid combination with rule syntax) • Treated as a “macro” representing named individuals in the knowledge base.• Results in the paper: • N2ExpTime-completeness of SROIQV = SROIQ + nominal schema, i.e., no harder than SROIQ. (V is to indicate nominal schema extension). • Tractable fragment of DLs SROELV n (n ≥ 0). • Expressing DL-safe datalog with nominal schema Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 3
  4. 4. Example• Consider the (hybrid) knowledge base K : hasParent(x, y) ∧ hasParent(x, z) ∧ married(y, z) → C (x) hasParent(mary, john) (∃hasParent.∃married.{john})(mary) Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 4
  5. 5. Example• Consider the (hybrid) knowledge base K : hasParent(x, y) ∧ hasParent(x, z) ∧ married(y, z) → C (x) hasParent(mary, john) (∃hasParent.∃married.{john})(mary)• FOL semantics of SWRL: K |= C (mary) • Undecidable in general Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 4
  6. 6. Example• Consider the (hybrid) knowledge base K : hasParent(x, y) ∧ hasParent(x, z) ∧ married(y, z) → C (x) hasParent(mary, john) (∃hasParent.∃married.{john})(mary)• FOL semantics of SWRL: K |= C (mary) • Undecidable in general• DL-safe rules semantics: K |= C (mary), • John’s spouse is not named by any constant. • K |= C (mary) if z is DL-safe while x and y can be arbitrary values Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 4
  7. 7. Example• Consider the (hybrid) knowledge base K : hasParent(x, y) ∧ hasParent(x, z) ∧ married(y, z) → C (x) hasParent(mary, john) (∃hasParent.∃married.{john})(mary)• FOL semantics of SWRL: K |= C (mary) • Undecidable in general• DL-safe rules semantics: K |= C (mary), • John’s spouse is not named by any constant. • K |= C (mary) if z is DL-safe while x and y can be arbitrary values• With nominal schemas, the rule can be expressed as: ∃hasParent.{z} ∃hasParent.∃married.{z} C Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 4
  8. 8. SROIQV = SROIQ + Nominal Schema (Syntax)• Signature of SROIQV: NI , NC , Nr , NV .• Concept expression: C ::= | ⊥ | NC | {NI } | {NV } | ¬C | C C|C C| ∃R.C | ∀R.C | ∃S.Self | k S.C | k S.C k ∈ N, R (S) is a (simple) SROIQ role, incl. the universal role U .• Knowledge base axioms are as usual as in SROIQ with regular RBoxes Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 5
  9. 9. SROIQV = SROIQ + Nominal Schema (Semantics)• Standard semantics for concept and roles with variables interpreted as placeholder of named individuals: • embed a restricted form of variable assignment to the interpretation; • equivalent to replacing nominal schemas with finitely many nominals they can represent and applying the standard SROIQ semantics to the result.• ground(α): the set of all axioms obtained by uniformly replacing nominal schemas in α with nominals in NI ground(KB) := α∈KB ground(α)• I |= α iff I |= ground(α) and I |= KB iff I |= ground(KB).• Satisfaction and entailment are defined as usual.• Grounding does not affect restrictions on simplicity and regularity. Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 6
  10. 10. Complexity of reasoning in SROIQV• Grounding provides direct (and naive) approach for reasoning in SROIQV: • yields exponentially larger SROIQ KB after grounding; • (not tight) complexity upper bound is exponentially larger than that of SROIQ, i.e., N3ExpTime.• N2ExpTime-hardness follows from SROIQ.• Retaining N2ExpTime upper bound by extending the N2ExpTime upper bound proof for SROIQ from [Kazakov, KR08]: exponential reduction to satisfiability of theories of C 2 (two-variable fragment of FOL with counting quantifiers) that is in NExpTime • transform axioms (except complex RIAs) into normal form • eliminate RIAs using automata construction • express remaining axioms as C 2 theories Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 7
  11. 11. N2ExpTime upper bound• Given a SROIQV KB, depth of ground(KB) = depth of KB.• ground(KB) is in SROIQ, hence the exponential reduction applies.• The resulting C 2 theories T is equisatisfiable to ground(KB) that is equisatisfiable with KB• The size of T is bounded by a function that is linear in the size of ground(KB) and exponential in the depth of KB.• The size of ground(KB) is exponential in the size of KB.• Thus, the size of T is bounded exponentially in the size and depth of KB• Satisfiability of T is in NExpTime, hence satisfiability of KB is in N2ExpTime Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 8
  12. 12. A tractable fragment: consideration• Consider adding nominal schemas to the tractable SROEL• Normalization destroys complex dependencies between nominal schemas. • Exponential blow up due to grounding nominal schemas is unavoidable.• Limiting the occurrences of all nominal schemas (with global constant from the language) trivially yields to tractability.• Non-trivial tractability result: only limit (globally) the occurrences of problematic nominal schemas • no limit on occurrences of safe nominal schemas. • inspired from tree-shapedness notion from the rule language ELP [Krötzsch,et.al., ISWC08] Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 9
  13. 13. Safe Environment of Nominal Schema x, y, z ∈ NV ∃reviews.({x} ∃hasAuthor.{y} ∃atVenue.{z}) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{x} v x u y z Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 10
  14. 14. Safe Environment of Nominal Schema p ∈ NI y, z ∈ NV ∃reviews.({p} ∃hasAuthor.{y} ∃atVenue.{z}) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{p} v p p u y z Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 10
  15. 15. Safe Environment of Nominal Schema p ∈ NI y, z ∈ NV ∃reviews.({p} ∃hasAuthor.{y} {p} ∃atVenue.{z}) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{p} v p p u y z Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 10
  16. 16. Safe Environment• An occurrence of a nominal schema {x} in a concept C is safe if • C has a sub-concept of the form {a} ∃R.D for some a ∈ NI • D contains the occurrence of {x} but no other occurrence of any nominal schema.• {a} ∃R.D is a safe environment S(a, x) for this occurrence of {x}.• A nominal schema {x} is safe for a SROIQV TBox axiom C D if • {x} does not occur in D, and • at most one occurrence of {x} in C is not safe. Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 11
  17. 17. Safe Environment• An occurrence of a nominal schema {x} in a concept C is safe if • C has a sub-concept of the form {a} ∃R.D for some a ∈ NI • D contains the occurrence of {x} but no other occurrence of any nominal schema.• {a} ∃R.D is a safe environment S(a, x) for this occurrence of {x}.• A nominal schema {x} is safe for a SROIQV TBox axiom C D if • {x} does not occur in D, and • at most one occurrence of {x} in C is not safe.• The DL SROELV n : • concept/roles: SROIQV concepts/roles using only , ⊥, , ∃, Self, U , {a}, nominal schemas • no , ¬, ∀, number restrictions, inverse roles • TBox axioms: SROIQV TBox axioms with SROELV n concept/roles, and at most n nominal schemas are not safe for each axiom. • KB axioms: uses only SROELV n axioms Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 11
  18. 18. Transformation Example (1)• Assume KB has only p, a, b as named individuals. Consider an axiom α: ∃reviews.(({p} ∃hasAuthor.{y}) ({p} ∃atVenue.{z})) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{p} Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 12
  19. 19. Transformation Example (1)• Assume KB has only p, a, b as named individuals. Consider an axiom α: ∃reviews.(({p} ∃hasAuthor.{y}) ({p} ∃atVenue.{z})) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{p}• For each nominal schema {x} safe for α with safe environments Si (ai , x), i = 1, . . . , , introduce a fresh concept name Ox,α • Oy,α , Oz,α Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 12
  20. 20. Transformation Example (1)• Assume KB has only p, a, b as named individuals. Consider an axiom α: ∃reviews.(({p} ∃hasAuthor.{y}) ({p} ∃atVenue.{z})) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{p}• For each nominal schema {x} safe for α with safe environments Si (ai , x), i = 1, . . . , , introduce a fresh concept name Ox,α• For each individual c in KB, add polynomially many ground axioms ( for empty conjunction): ∃U .Si (ai , c) ∃U .({c} Ox,α ) i=1 Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 12
  21. 21. Transformation Example (1)• Assume KB has only p, a, b as named individuals. Consider an axiom α: ∃reviews.(({p} ∃hasAuthor.{y}) ({p} ∃atVenue.{z})) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{p}• For each nominal schema {x} safe for α with safe environments Si (ai , x), i = 1, . . . , , introduce a fresh concept name Ox,α• For each individual c in KB, add polynomially many ground axioms ( for empty conjunction): ∃U .Si (ai , c) ∃U .({c} Ox,α ) i=1 ∃U .({p} ∃hasAuthor.{p}) ∃U .({p} Oy,α ) ∃U .({p} ∃hasAuthor.{a}) ∃U .({a} Oy,α ) ∃U .({p} ∃hasAuthor.{b}) ∃U .({b} Oy,α ) ∃U .({p} ∃atVenue.{p}) ∃U .({p} Oz,α ) ∃U .({p} ∃atVenue.{a}) ∃U .({a} Oz,α ) ∃U .({p} ∃atVenue.{b}) ∃U .({b} Oz,α ) Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 12
  22. 22. Transformation Example (2) ∃reviews.(({p} ∃hasAuthor.{y}) ({p} ∃atVenue.{z})) ∃submits.(∃hasAuthor.{y} ∃atVenue.{z}) ∃hasConflictingAssignedPaper.{p}• Let C be the LHS of α. For each nominal schema {x} safe for α: • replace all safe occurrences S(a, x) in C by {a} Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 13
  23. 23. Transformation Example (2) ∃reviews.{p} ∃submits.(∃hasAuthor.Oy,α ∃atVenue.Oz,α ) ∃hasConflictingAssignedPaper.{p}• Let C be the LHS of α. For each nominal schema {x} safe for α: • replace all safe occurrences S(a, x) in C by {a} • replace the non-safe occurrence (if any) of {x} in C by Ox,α Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 13
  24. 24. Transformation Example (2) ∃reviews.{p} ∃submits.(∃hasAuthor.Oy,α ∃atVenue.Oz,α ) ∃U .Oy,α ∃U .Oz,α ∃hasConflictingAssignedPaper.{p}• Let C be the LHS of α. For each nominal schema {x} safe for α: • replace all safe occurrences S(a, x) in C by {a} • replace the non-safe occurrence (if any) of {x} in C by Ox,α • add a new conjunct ∃U .Ox,α to C . Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 13
  25. 25. Transformation Example (2) ∃reviews.{p} ∃submits.(∃hasAuthor.Oy,α ∃atVenue.Oz,α ) ∃U .Oy,α ∃U .Oz,α ∃hasConflictingAssignedPaper.{p}• Let C be the LHS of α. For each nominal schema {x} safe for α: • replace all safe occurrences S(a, x) in C by {a} • replace the non-safe occurrence (if any) of {x} in C by Ox,α • add a new conjunct ∃U .Ox,α to C .• After the above steps, C has only nominal schemas (if any) that are not safe for α. We thus ground α. Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 13
  26. 26. Tractability Result• Given a SROELV n knowledge base KB, the size of KB after unfolding is exponential in n and polynomial in the size of KB before unfolding.• KB before transformation is equisatisfiable with KB after transformation.• A knowledge base is unsatisfiable iff it entails {a} ⊥ for arbitrary a ∈ NI .• KB after transformation is in SROEL(×).• Instance retrieval for SROEL(×) is polynomial [Krötzsch, IJCAI11].• If n is constant, satisfiability of KB is P-complete. Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 14
  27. 27. Expressing DL-safe Rules with Nominal Schemas• dl(C (t)) := ∃U .({t} C)• dl(R(s, t)) := ∃U .({s} ∃R.{t})• dl(p(t1 , . . . , tk )) := ∃U .(∃p1 .{t1 } ··· ∃pk .{tk })• dl(B → H ) := F∈B dl(F ) dl(H )• For a set of DL-safe rules RB, dl(RB) := B→H ∈RB dl(B → H ).• RB is semantically equivalent dl(RB)• If RB is a set of n-variable rules with n constant, then satisfiability of RB is P-complete. Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 15
  28. 28. Summary• Conclusion: • Nominal schemas provides sufficient expressivity to incorporate rule-based modeling. • DL-safe datalog with arbitrary arity is covered. • SROIQV is no harder than SROIQ • Tractable reasoning is possible with SROELV n .• Outlook: • Concrete serialization formats (currently proposed). • Deferred grounding for inference algorithms (ongoing work). Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 16
  29. 29. Thank you!
  30. 30. Recall: N2ExpTime upper bound proof from SROIQ1 SROIQ axioms can be transformed in linear time into normal form where R(i) , S1 , S2 ∈ Nr , S1 , S2 simple: A ∀R.B Ai Bj S1 S2 A n S.B A ≡ {a} R1 R− A n S.B A ≡ ∃S.Self R1 ◦ · · · ◦ Rn R2 For each non-simple role R, construct an NFA AR that accepts all role chains (viewed as strings) that imply R • The number of states of AR is bounded exponentially in the depth of KB: maximum cardinality of any strict linear order of role names that witnesses regularity of KB. Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 18
  31. 31. Recall: N2ExpTime upper bound proof from SROIQ (cont.)3 Drop complex RIAs and replace each A ∀R.B with axioms (with NFA AR = (QR , NR , ∆R , q0R , FR )): S {A Xq0R } ∪ {Xq B | q ∈ FR } ∪ {Xq ∀S.Xq | q → q ∈ ∆R } For each axiom A ∀R.B: • The number of new axioms is linearly bounded by |QR | + |∆R | • |∆R | is linearly bounded by |QR | and |Nr | • |QR | is exponentially bounded by the depth of KB The overall size of KB after adding new axioms is now bounded by a function that is linear in the size of original KB and exponential in the depth of original KB4 Rewrite KB into C 2 (not increase the size of KB) Krötzsch, Maier, Krisnadhi, and Hitzler. Nominal Schemas for Integrating Rules and DLs — DL2011 19

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