Query Answering Based on the Modal Correspondence Theory

Evgeny ZolinUniversity of Manchester

Manchester, UKzolin@cs.man.ac.uk

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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

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Description Logics• A family of knowledge representation formalisms

• Vocabulary:

– concept names A, 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] ALC is a notational variant of the multi-

modal logic K(m): replace Ri and Ri with ◊i and □i

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Description Logics (continued)• A knowledge base KB = T , A consists of:

– T : TBox (“terminology”) contains axioms: C D

– 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 }

Q – num.restr.: ( ≥n R.C )

H – role hierarchy: R S

S – transitive roles: Trans(R)

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Query answering• A conjunctive query q(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

– SHIQ is ExpTime-complete [S.Tobies,2001]. Query answering:

• 3coNExpTime upper bound, if KB has no transitive roles;

• 4coNExpTime in general case [Calvanese et al., DL2005].

– SHOIQ is NExpTime-complete, but the decidability of the query answering problem has only recently been established

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A closer look at instance retrieval• Consider KB a : C, where the concept C 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 )

all individuals will be retrieved

no individuals will be retrieved

{ a | KB aRa }

{ a | KB y (aRy aSy) }

KB a : X

KB a : (X X )

KB a : (X R.X )

KB a : (R.X S.X )

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Query answered by a conceptDefinition. 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 query q(x) xRx

R.X S.X answers the query q(x) y(xRy xSy)

From modal logic:From modal logic: F ||– p ◊p R is reflexive: x xRx

F,e ||– p ◊p R is reflexive at e: eRe

F,e ||– □R p ◊S p y (eRy eSy) holds in F

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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 valuation v(pi) W

• A formula is true at a point e of a model M: M,e

• Local validity: F,e ||– iff M,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).

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“From modal logic to query answering”

Given , denote by C the corresponding ALC-concept

(with variables pi replaced by fresh concept names Xi ).

Theorem (Reduction) Suppose that

• q(x) is a relational query (with one free variable);

is a modal formula.

Then:

if q(x) locally corresponds to

then q(x) is answered by the

ALC-concept C

(over any KB)?

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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,j x Ri yj x Rt x

k,l yk Rl x

x : C s ys: Ds )

x

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Queries within Kracht’s fragmentxRx 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 ))

x R

xR

yS C

xR

yS C

x

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Beyond Kracht’s fragment

Parallel-serial queries (with two poles)Parallel-serial queries (with two poles)x y

x y x yq1(x)

q2(x)

serial connection (q1 o q2)

x y

x y

parallel connection (q1 || q2)

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 o q2):=R(q1) o

R(q2)

Then q(x) is answered by the concept R(q).T

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Beyond Kracht’s fragment (continued)Family of queries Z : For any query of the following

shape, there exists a concept answering it. If q(x) is relational, then the concept belongs to ALC.

yx

yx

Reversed tree with the root y, whose all leaves merged in x

A parallel-serial query, where only atomic q2 are allowed in (q1 o q2)

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Adding role inverses

Theorem (Family of queries Y )

• For any connected query q(x) without cycles consisting of bound variables 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

x

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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, F q(e) F,e ||– .

Recently: we can replace “” with “” in the above Lemma for finitely branching frames F.

Definition A frame F is finitely branching if, for any its point e and a relation R, the set { d | eRd } is finite.

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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 )

a

c

b

d

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Conclusions and outlookRelationship 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!