Query Answering based onStandard and Extended

Modal Logic

Evgeny ZolinThe University of Manchester

zolin@cs.man.ac.uk

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

3/12

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) = 9y1…yk (term1 &…& termn ), where each termi is z:C or zRu, z and u are among {x, y1 , … , yk }

– 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)

4/12

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

5/12

Deploying Modal Logic for Q. Answering

• q(x) = xRx (reflexivity) ! p ! § p

KB ² aRa , KB ² a:(:P t 9R.P )

• q(x) = 9y (xRy & xSy) ! ¤1 p ! §2 p

) the concept is: :8R.P t 9 S.P

q(x) = 9y (xRy & xSy & y:C )

) the concept is: :8R.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

xR

yS C

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)?

7/12

The harvest: Queries answered by concepts

x

yx

xq(x)

q(x)

q(x)

8/12

Introducing Variable Modalities• The language is extended in two ways:

• Modal formulas:

• The dual variable modalities are defined as:

propositional variables: p0 , p1 , …

constant modalities: ¤1 ,…, ¤mpropositional constants: A1 ,…,An

variable modalities: ¡0, ¡1, …

9/12

Semantics for Variable Modalities• Frame: F =hW ; V1 ,…,Vn; R1,…,Rm i, Vi µ W, Ri µ W£W

• Model: M =hF ,; S0,S1 ,…i, (pi) µ W ; Si 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 pi and variable modalities ¡i

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?

11/12

Mary Likes All CatsTask: 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: :8Likes.P t 8S.(:Cat t P )

12/12

Conclusions and outlookRelationship 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!