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Query Answering Based on the Modal Correspondence Theory

Query Answering Based on the Modal Correspondence Theory. Evgeny Zolin University of Manchester Manchester, UK zolin@cs.man.ac.uk. Talk Outline. Description Logics, knowledge bases Answering conjunctive queries Modal correspondence theory “From modal logic to query answering”

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Query Answering Based on the Modal Correspondence Theory

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  1. Query Answering Based on the Modal Correspondence Theory Evgeny Zolin University of Manchester Manchester, UK zolin@cs.man.ac.uk

  2. Talk Outline • Description Logics, knowledge bases • Answering conjunctive queries • Modal correspondence theory • “From modal logic to query answering” • Applications: • Transferring Kracht’s Theorem • Beyond Kracht’s fragment • Adding inverse relations • “From query answering back to modal logic”? • Conclusions and outlook

  3. Description Logics • A family of knowledge representation formalisms • Vocabulary: • concept namesA, B, …; • role names R, S, … • individual names a, b, … • Syntax for the Description Logic ALC : • concepts are built up from concept names (A, B, …) using operations C, C D, CD, and R.C, R.C • [K.Schild,1991] ALCis a notational variant of the multi-modal logic K(m): replace Ri and Riwith ◊i and □i

  4. H–role hierarchy: RS Description Logics (continued) • A knowledge base KB =T, Aconsists of: • T : TBox (“terminology”) contains axioms: CD • A: ABox (“world description”) assertions: a:C, aRb • Extensions (indicated by adding letters to logic’s name): • Reasoning problems: • KB satisfiability: whether there is a model of a given KB • instance checking and instance retrieval: KB a :C I– inverse roles: R – O– nominals: { a } S–transitive roles: Trans(R) Q–num.restr.: ( ≥nR.C )

  5. Query answering • A conjunctive queryq(x) is an expression of the form: q(x)  (y) term1(x, y)  …  termk(x, y) where x,y are lists of variables, terms are either z :C or zRz’ (z,z’{x,y}) • The answer set of the query q(x) w.r.t. a KB: ans(q,KB) := { a IndNames: KB q(a) } • No tight complexity bounds for query answering known so far • SHIQis ExpTime-complete [S.Tobies,2001]. Query answering: • 3coNExpTime upper bound, if KB has no transitive roles; • 4coNExpTime in general case [Calvanese et al., DL2005]. • SHOIQis NExpTime-complete, but the decidability of the query answering problem has only recently been established

  6. KB a :X KB a : (XX) KB a : (X R.X) { a |KB aRa } KB a : (R.XS.X) { a |KB y (aRyaSy)} A closer look at instance retrieval • Consider KB a :C, where the conceptC contains fresh concept names (X, …) not occurring in the KB. • The concept X R.X “answers” the query q(x)  xRx • The concept R.XS.X “answers” the query q(x)  y (xRy  xSy ) no individuals will be retrieved all individuals will be retrieved

  7. Query answered by a concept Definition. A query q(x) is answered by a concept C if, for any KB and a constant a, KB q(a)  KB a :C • The conceptX R.X answers the queryq(x)  xRx • R.XS.X answers the query q(x)  y(xRy  xSy) From modal logic: • F ||–p  ◊p  R is reflexive: xxRx • F,e ||–p  ◊p  R is reflexive at e: eRe • F,e ||–□Rp  ◊Sp  y (eRy eSy) holds in F

  8. Modal correspondence theory • Modal logic K(m):  := pi |  |    | □i • (Kripke) semantics: • Frame:F = W, R1, …, Rm , where Ri W 2 • Model: M = F,v, where a valuationv(pi)  W • A formula  is true at a point e of a model M: M,e  • Local validity: F,e ||–  iffM,e  for any M = F,v Let (x) be a FO-formula over binary predicates {R1, …, Rm }. Definition. (x)locally corresponds to if, for any frame F and its point e, F,e ||–  F(e).

  9. ?  “From modal logic to query answering” Given ,denote byCthe correspondingALC-concept (with variablespireplaced by fresh concept namesXi ). Theorem (Reduction) Suppose that • q(x) is a relational query (with one free variable); •  is a modal formula. Then:

  10. Sahlqvist’s and Kracht’s theorems Modal formulas <~~~>First-order formulas [Sahlqvist,1975] {…  …} <~~~> {… (x) …} [Kracht,1993] Family of queries K :For any query of the following shape, there exists a concept that answers it. For a relational query q(x), the resulting concept is in ALC. q(x)  y (Tree(x,y)  i,jx Riyj  x Rt x  k,lyk Rlx x : C sys:Ds )

  11. x R S S y y x x R R x Queries within Kracht’s fragment xRx X R.X y(xRy  ySx) X R.S.X y(xRy  ySx  y:C) X R.(C  S.X) y(xRy  xSy) R.Y S.Y y(xRy  xSy  y:C) R.Y S.(C Y ) y(xR1y1  y1R2y2  y1R3y3  y1R2y2  y4R5y5  y4R6y6  xS1y1  xS4y6  y2S2x  y5S3x) ( S1.Y11  S4.Y46  X22  X53) R1. ( Y11 R2.S2.X22  R3.T  R4.( R6.Y46R5.S3.X53)) C C

  12. y x y x y x y x y x Beyond Kracht’s fragment Parallel-serial queries (with two poles) q(x)  y (xRy ) Fact: Any parallel-serial relational query q(x) is answered by some concept in ALC(,o): R(q):=R for atomic q(x) R(q1 || q2):=R(q1) R(q2) R(q1 oq2):=R(q1) o R(q2) Then q(x) is answered by the concept R(q).T q1(x) q2(x) parallel connection(q1 ||q2) serial connection(q1 oq2)

  13. y Beyond Kracht’s fragment (continued) Family of queries Z :For any query of the following shape, there exists aconcept answering it. If q(x) is relational, then the concept belongs to ALC. A parallel-serial query, where only atomic q2 are allowed in (q1 oq2) Reversed tree with the root y, whose all leaves merged in x

  14. x Adding role inverses Theorem (Family of queries Y) • For any connected queryq(x) without cycles consisting of boundvariables only, there is a concept answering it (and it can be built in linear time). • If q(x) is relational, then the resulting concept belongs to the Description Logic ALCI. • (KZ ) Y

  15. From query answering back to modal logic? Theorem (Reduction) q(x) loc. corresponds to  q(x) is answered by C Lemma If q(x) is answered by a concept C , then for any frame F and its point e, Fq(e)  F,e ||–  . Recently: we can replace “” with “” in the above Lemma for finitely branching frames F. DefinitionA frameFis finitely branchingif, for any its point e and a relationR, the set{ d | eRd } is finite.

  16. d c a b From query answering back to modal logic? • Validity of a modal formula ≈ closed world assumption Ex.: F = W,R , where W = {a,b,c,d}, R = {a,b, a,c, c,d }. • F, b||– ◊T (b has no R-successors) • F, c||– ◊p  □p (R is functional at the point c) • Entailment from a KB ≈ open world assumption KB=T, A , TBox T is empty, Abox A = { aRb, aRc, cRd } Then neither KB b:R.T, nor KB c : ( R.X R.X )

  17. Conclusions and outlook • Relationship between corr. theory and query answering • Two families of queries answered by ALC-concepts • A larger family of queries answered by ALCI-concepts • Questions and further directions: • Does the converse “” of the Reduction Theorem hold? • Characterisation of conj. queries answered by concepts? • More expressive queries? (disjunction, equality) • Adding number restrictions? ( ALCQ≈ Graded ML) • Relations of arbitrary arities? ( DLR≈ Polyadic ML) Thank you!

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