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Query Answering Based on the Modal Correspondence Theory

This presentation explores the intersection of modal logic and query answering within the context of description logics and knowledge bases. Key topics include foundational concepts of description logics, answering conjunctive queries, and the practical implications of modal correspondence theory, particularly the application of Kracht’s Theorem and extensions beyond its original fragment. We examine the complexity of query answering, instance retrieval, and the transformations between modal logic and query mechanisms. The talk concludes with insights into future directions in this area of research.

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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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