1. Query Answering based on Standard and Extended Modal Logic Evgeny Zolin The University of Manchester zolin@cs.man.ac.uk

2. 2/12 Talk Outline Query Answering with standard Modal Logic: –How to generalise the rolling-up? –Deploying Correspondence Theory –The harvest: queries we are able to answer Modal Logic with variable modalities –semantics –expressivity –more queries Conclusions and further directions

3. 3/12 DLs and Query Answering Consider a DL: ALC or SHIQ or your favourite logic Given a knowledge base KB = hT, Ai that consists of: –a TBox T of axioms: C v D, R v S, Trans( R ), etc. –an ABox A of assertions: a : C, aRb Given a query q ( x ) that can be: –a conjunctive query: q ( x ) = 9 y 1 … y k ( term 1 &…& term n ), where each term i is z : C or zRu, z and u are among { x, y 1, …, y k } –or an arbitrary first-order formula with 1 (or 0) free variable x The task is: to find the answer to the query q ( x ), i.e., all individuals a that satisfy: KB ² q ( a )

4. 4/12 How to generalise the rolling-up? The rolling-up technique: a tree-like query q ( x ) into a concept C so that q ( x ) and C are equivalent, thus have the same instances: But equivalence of q ( x ) and C is not necessary for that: Take a query q ( x ) obtain a of a certain shape a concept C rolled up for any KB (in any DL) and any individual a : KB ² q ( a ), KB ² a : C

5. 5/12 Deploying Modal Logic for Q. Answering q ( x ) = xRx (reflexivity) ! p ! § p KB ² aRa, KB ² a :( : P t 9 R. P ) q ( x ) = 9 y ( xRy & xSy ) ! ¤ 1 p ! § 2 p ) the concept is: :8 R. P t 9 S. P q ( x ) = 9 y ( xRy & xSy & y : C ) ) the concept is: :8 R. P t 9 S.( P u C ) Definition. q ( x ) locally corresponds to  : if for any frame F and any point e, [H.Sahlqvist,1975] {…  … } ! {…  x  …} [M.Kracht,1993] x R x R y S C

6. 6/12 “From modal logic to query answering” Theorem (Reduction) If q(x) is relational, then: if q(x) locally corresponds to   then q(x) is answered by the ALC - concept C  (over any KB in any DL)  ?

7. 7/12 The harvest: Queries answered by concepts x y x x q(x)q(x) q(x)q(x) q(x)q(x)

8. 8/12 Introducing Variable Modalities The language is extended in two ways: Modal formulas: The dual variable modalities are defined as: propositional variables: p 0, p 1, … constant modalities: ¤ 1,…, ¤ m propositional constants: A 1,…, A n variable modalities: ¡ 0, ¡ 1, …

9. 9/12 Semantics for Variable Modalities Frame: F = h W ; V 1,…, V n ; R 1,…, R m i, V i µ W, R i µ W £ W Model: M = h F, ; S 0, S 1,… i, ( p i ) µ W ; S i µ W £ W A formula  is true at a point e of a model M : M, e ²  Validity of a formula  at a point e of a frame F : F, e ²  iff M, e ²  for any model M based on F In other words:  is true at e for any interpretation of propositional variables p i and variable modalities ¡ i

10. 10/12 Expressibility and advantages More properties of frames become expressible: All the above results are transferred: if a property q ( x ) is modally definable, then q ( x ) is answered by a concept. Correspondence Theory for the richer language?

11. 11/12 Mary Likes All Cats Task: KB ² “Mary Likes all Cats” Mary (individual), Likes (role), Cat (concept) Solution 1: KB ² Cat v 9 Likes —.{Mary} Need to introduce inverse roles and nominals… Solution 2: KB ² Mary: 8: Likes. : Cat Need to introduce role complement (ExpTime) Recall: Solution 3: KB ² Mary: :8 Likes. P t 8 S.( : Cat t P )

12. 12/12 Conclusions and outlook Relationship between corr. theory and query answering A family of conj. queries answered by ALC ( I )-concepts A modal language with variable modalities 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!