View-based query answering: on the relationship
between rewriting, answering, and losslessness
Diego Calvanese
Faculty of Computer Science
Free University of Bolzano/Bozen
Giuseppe De Giacomo, Maurizio Lenzerini
Dipartimento di Informatica e Sistemistica
Università di Roma “La Sapienza”
Moshe Y. Vardi
Rice University, Houston
ICDT 2005 – Edinburgh, U.K., January 2005
View based query processing
Computing the answer to a query by relying solely on a set of
views
Relevant problem in data integration, data warehousing, query
optimization, authorization, etc.
Two different approaches:
• view based query answering
• view based query rewriting
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View based query answering
certain
answers
certQ,V
we are
interested in
answers
to Q
Q
View definition V
V1 V2 … Vn
Database schema
R1 R2 … Rm
…
View extension E
…
Database B
Open world assumption (sound views): E ⊆ V(B)
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View based query rewriting
certain
answers
certQ,V
answers
to RmaxQ,V
we are
interested in
answers
to Q
RmaxQ,V
Q
View definition V
V1 V2 … Vn
Database schema
R1 R2 … Rm
…
View extension E
…
Database B
Open world assumption (sound views): E ⊆ V(B)
max
RQ,V
expressed in the “same” language as Q (but on V -symbols)
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Answering vs rewriting
• Answering and rewriting coincide in some interesting cases
(notably, in the case of conjunctive queries and views – see
later)
• However, they do not coincide in general, and therefore, it
makes sense to compare the query, the rewriting and the
certain answers
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The main focus of this work
Principles and tools for comparing:
• query Q
max
• maximal rewriting RQ,V
of Q wrt views V
(a maximal rewriting of Q wrt V is a maximal query R over V such
that ∀B ∀E ⊆ V(B) : we have R(E) ⊆ Q(B))
• function (i.e., query) cert Q,V that computes the certain answers to
Q wrt views V , given V -extension E
(i.e., ~t ∈ cert Q,V (E) iff ∀B : E ⊆ V(B) we have ~t ∈ Q(B))
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
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Outline
1. Framework
2. Rewriting vs answering
3. Exactness
4. Perfectness
5. Losslessness
6. Conclusions
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Framework
Two settings:
• Relational dbs
– Conjunctive queries and views
• Semistructured data: edge labeled graph, with set Σ of basic
binary relations (edge labels) on nodes
– Queries and views: variants of regular path queries
(RPQs)
∗ an RPQ is a regular expression Q over the edge labels
∗ it returns the set of pairs of nodes connected by a path
in L(Q)
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Two-way regular path queries (2RPQs)
Expressed as regular expression over Σ± = Σ ∪ {p− | p ∈ Σ}
(p− denotes the inverse of the binary relation p), e.g.,
Q = r·(p− + q)·p·p− ·q ∗
Answer over DB: set of pairs of nodes connected by a semipath in
DB conforming to the regular expression
d1
d3
r
DB:
p
q
d5
p
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q
p
r
r
Q returns
d1 d5
d1 d3
.. ..
. .
Rewriting, Answering, and Losslessness
via rqpp−
via rp− pp− q
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Rewriting vs answering for conjunctive queries
v1 (T )
= { (T ) | movie(T, Y, D) ∧ european(D) }
v2 (T, Z) = { (T, Z) | movie(T, Y, D) ∧ review(T, Z) }
The certain answers to Q are computed by evaluating the goal Q
wrt this nonrecursive logic program [Abiteboul&Duschka 1998]:
movie(T, f1 (T ), f2 (T )) ← v1 (T )
european(f2 (T )) ← v1 (T )
movie(T, f4 (T, Z), f5 (T, Z)) ← v2 (T, Z)
review(T, Z)) ← v2 (T, Z)
The goal and the logic program can be equivalently transformed
into a finite union of conjunctive queries over the view symbols,
which is the maximal rewriting of Q wrt V
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Rewriting vs answering for 2RPQs
Views V :
V1 = d
Query Q:
df + eg
max
=∅
RQ,V
V2 = e
V3 = f + g
cert Q,V = { (x, y) | ∃z . x V1 z ∧ x V2 z ∧ z V3 y }
x
V1 (d)
z
V3 (f + g)
y
V2 (e)
Furthermore, computing the certain answers is coNP-complete in
data complexity, while evaluating the maximal rewriting can be
done in polynomial time
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max
Exactness: comparing RQ,V
and Q
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
max
The maximal rewriting RQ,V
of Q wrt views V is exact if for every
max
(V(B))
database B we have that Q(B) = RQ,V
Exactness means losslessness of rewriting wrt the query (note
that exactness = perfectness + losslessness)
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Exactness in the case of conjunctive queries
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
max
• RQ,V
is a union of conjunctive queries over the V -symbols
• To check whether such union is equivalent to Q modulo V , it
suffices to check whether there is a disjunct in the unfolding
max
of RQ,V
that is equivalent to Q
• Checking whether there exists an exact rewriting of a
conjunctive query is NP-complete [Halevy&al 1995]
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Exactness in the case of 2RPQs
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
From [Calvanese&al 2000]:
max
• RQ,V
can be constructed in 2EXPTIME, via an
automata-theoretic approach
• To check exactness, we check whether Q is contained in the
max
unfolding of RQ,V
• Checking whether there exists an exact rewriting of a 2RPQ
is 2EXPSPACE-complete
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max
Perfectness: comparing RQ,V
and cert Q,V
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
max
of Q wrt views V is
New notion: The maximal rewriting RQ,V
perfect, if for every database B and every view extension E with
max
(E)
E ⊆ V(B) we have that cert Q,V (E) = RQ,V
Perfectness means that the maximal rewriting is powerful enough
to compute the certain answers
max
over
Perfectness allows us to compute cert Q,V by evaluating RQ,V
the view extension
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Perfectness in the case of conjunctive queries
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
max
is always equivalent to cert Q,V
• RQ,V
max
is always perfect
• RQ,V
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Perfectness in the case of 2RPQs
Perfectness means
max
∀B ∀E ⊆ V(B) : cert Q,V (E) ⊆ RQ,V
(E)
that is a form of view-based query containment
max
Q ⊆V RQ,V
From [Calvanese&al 2003], this can be checked in NEXPTIME,
max
is
resulting in N3EXPTIME for perfectness, since the size of RQ,V
doubly exponential in Q
New result: checking perfectness can be done in N2EXPTIME,
via characterization of view-based query answering through CSP
(lower bound open)
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Losslessness: comparing cert Q,V and Q
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
A set of views V is lossless wrt a query Q, if for every database B
we have that Q(B) = cert Q,V (V(B))
Losslessness means that the views are powerful enough to
precisely answer the query
In the case where we have access to B , losslessness allows us to
compute cert Q,V by evaluating Q over the database
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Losslessness in the case of conjunctive queries
RmaxQ,V
perfectness
certQ,V
losslessness
Q
exactness
max
is always equivalent to cert Q,V
• RQ,V
• Losslessness and exactness coincide, i.e., checking
losslessness is NP-complete
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Losslessness in the case of 2RPQs: example
Views V :
Query Q:
V1 = 0 + 1
V2 = 01
V3 = 10
V4 = 000
V5 = 111
010 + 101 + 000 + 111
max
RQ,V
= V4 + V5
cert Q,V is equivalent to Q
V3
database B’
V3
database B
x1
0
V1
x2
V2
1
V1
x3
x1
0
V1
a
V1
x2
b
V1
V2
x4
x1
0
V1
x2
x3
0
V1
x4
1
V1
x4
V3
b
V1
x3
V2
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Loslessness in the case of 2RPQs
• In [Calvanese&al 2003] we showed that losslessness is
EXPSPACE-complete for RPQs
• We show in the paper that perfectness is
EXPSPACE-complete also for 2RPQs
• To this end, we introduce the notion of linear approximation to
the certain answers
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Losslessness in the case of 2RPQs
The linear fragment of certain answers clin Q,V for a 2RPQ Q wrt a
set V of views is the maximal two-way path query Q0 over Σ such
that ∀B : Q0 (B) ⊆ cert Q,V (V(B))
New results:
• We have a method for constructing clin Q,V (always a 2RPQ)
• We show that losslessness means ∀B Q(B) ⊆ clin Q,V (B)
• Checking losslessness is EXPSPACE-complete
perfectness
RmaxQ,V
clinQ,V
certQ,V
Q
losslessness
exactness
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The case of lossiness (with perfectness)
• In case of losslessness, cert Q,V computes exactly Q (which is
equivalent to clin Q,V ), and Q explains cert Q,V at best
• In case of lossiness, still, we would like to express cert Q,V in
the language of the user (2RPQ over the database)
max
max
– If RQ,V
is perfect, then RQ,V
is equivalent to clin Q,V , and
max
explains cert Q,V at best
the unfolding of RQ,V
perfectness
Rmax
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Q,V
clinQ,V
Rewriting, Answering, and Losslessness
certQ,V
losslessness
Q
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The case of lossiness (without perfectness)
max
• In case of lossiness and if RQ,V
is not perfect, still, we would
like to express cert Q,V as a 2RPQ over the database
– If cert Q,V is equivalent to clin Q,V , then clin Q,V explains
cert Q,V at best (and, if we have access to B , cert Q,V can be
computed by evaluating clin Q,V over the database)
perfectness
Rmax
Q,V
clinQ,V
certQ,V
losslessness
Q
New result: checking whether cert Q,V is equivalent to clin Q,V
can be done in N3EXPTIME (lower bound open)
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Conclusions
• Answering and rewriting are different notions
• Exactness, perfectness and losslessness are different notions
• We introduced the concept of “good” approximation of cert Q,V
for 2RPQs, i.e., the linear fragment
• In our past work, we addressed
– answering, via the relationship with CSP
– rewriting, via an automata-theoretic approach
max
The paper also proposes a technique for building RQ,V
that
reconciles the two approaches (see the proceedings)
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Future work
• A few lower bounds open
• In the case cert Q,V is not equivalent to clin Q,V , we would like
to exhibit a nonlinear counterexample database explaining
lossiness to the user
• Many open problems with exact views:
– perfectness and losslessness in the case of conjunctive
max
and cert Q,V do not coincide)
queries and views (RQ,V
– perfectness and losslessness in the case of 2RPQs
• More expressive language, i.e., C2RPQs
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