Download presentation

Presentation is loading. Please wait.

Published byCorinne Egle Modified over 4 years ago

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!

Similar presentations

Presentation is loading. Please wait....

OK

An Introduction to Description Logics

An Introduction to Description Logics

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google