query answering based on the modal correspondence theory evgeny zolin university of manchester...
TRANSCRIPT
Query Answering Based on the Modal Correspondence Theory
Evgeny ZolinUniversity of Manchester
Manchester, [email protected]
2/17
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
3/17
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
4/17
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)
5/17
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
6/17
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 )
7/17
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
8/17
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).
9/17
“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)?
10/17
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
11/17
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
12/17
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
13/17
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)
14/17
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
15/17
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.
16/17
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
17/17
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!