Query Answering based onStandard and Extended
Modal Logic
Evgeny ZolinThe University of Manchester
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)?
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!