Exchanging more than Complete Data

Exchanging more than Complete Data

Marcelo Arenas
PUC Chile

Joint work with Jorge P´rez (U. de Chile) and Juan Reutter (U. Edinburgh)
e

M. Arenas – Exchanging more than Complete Data - RR2011 1 / 68

Outline: First part

◮ The data exchange problem
◮ Some fundamental results in relational data exchange

◮ The need for a more general data exchange framework
◮ Two important scenarios: Incomplete databases (open-world
databases: RDF) and knowledge bases


Outline: First part




The problem of data exchange

Given: A source schema S, a target schema T and a specification
Σ of the relationship between these schemas

Data exchange: Problem of materializing an instance of T given
an instance of S
◮ Target instance should reflect the source data as accurately as
possible, given the constraints imposed by Σ and T
◮ It should be efficiently computable
◮ It should allow one to evaluate queries on the target in a way
that is semantically consistent with the source data


Data exchange in a picture

Query Q

Σ

Schema S Schema T


Data exchange in a picture

Σ

Schema S Schema T


Data exchange: Some fundamental questions

What are the challenges in the area?
◮ What is a good language for specifying the relationship
between source and target data?
◮ Expressiveness versus complexity

◮ What is a good instance to materialize?
◮ What does it mean to answer a query over target data?
◮ How do we answer queries over target data? Can we do this
eﬃciently?


Exchanging relational data

The data exchange problem has been extensively studied in the
relational world.
◮ It has also been commercially implemented: IBM Clio

Relational data exchange setting:
◮ Source and target schemas: Relational schemas
◮ Relationship between source and target schemas:
Source-to-target tuple-generating dependencies (st-tgds)

Semantics of data exchange has been precisely deﬁned.
◮ Eﬃcient algorithms for materializing target instances and for
answering queries over the target schema have been developed


Schema mapping: The key component in relational data
exchange

Schema mapping: M = (S, T, Σ)
◮ S and T are disjoint relational schemas
◮ Σ is a ﬁnite set of st-tgds:

∀¯∀¯ (ϕ(¯ , y ) → ∃¯ ψ(¯ , z ))
x y x ¯ z x ¯

ϕ(¯, y ): conjunction of relational atomic formulas over S
x ¯
ψ(¯, z ): conjunction of relational atomic formulas over T
x ¯


Relational schema mappings: An example

Example
◮ S: Employee(name)

◮ T: Dept(name, number)

◮ Σ:

∀x Employee(x) → ∃y Dept(x, y )


Relational schema mappings: An example

Example


◮ Σ:

∀x Employee(x) → ∃y Dept(x, y )

Note
We omit universal quantiﬁers in st-tgds:
Employee(x) → ∃y Dept(x, y )


Relational data exchange problem

Fixed: M = (S, T, Σ)

Problem: Given instance I of S, find an instance J of T such that
(I , J) satisfies Σ
◮ (I , J) satisfies ϕ(¯ , y ) → ∃¯ ψ(¯ , z ) if whenever I satisfies
x ¯ z x ¯
a ¯
ϕ(¯, b), there is a tuple c such that J satisfies ψ(¯, c )
¯ a ¯


Relational data exchange problem

Fixed: M = (S, T, Σ)

Problem: Given instance I of S, find an instance J of T such that
(I , J) satisfies Σ
◮ (I , J) satisfies ϕ(¯ , y ) → ∃¯ ψ(¯ , z ) if whenever I satisfies
x ¯ z x ¯
a ¯
ϕ(¯, b), there is a tuple c such that J satisfies ψ(¯, c )
¯ a ¯

Notation
J is a solution for I under M
◮ SolM (I ): Set of solutions for I under M


The notion of solution: Example

Example
◮ Σ: Employee(x) → ∃y Dept(x, y )

Solutions for I = {Employee(Peter)}:



Example

J1 : {Dept(Peter,1)}



Example

J2 : {Dept(Peter,1), Dept(Peter,2)}



Example

J3 : {Dept(Peter,1), Dept(John,1)}



Example

J4 : {Dept(Peter,n1 )}



Example

J4 : {Dept(Peter,n1 )}
J5 : {Dept(Peter,n1 ), Dept(Peter,n2 )}


Canonical universal solution

Algorithm (Chase)
Input : M = (S, T, Σ) and an instance I of S
Output : Canonical universal solution J ⋆ for I under M

let J ⋆ := empty instance of T
for every ϕ(¯ , y ) → ∃¯ ψ(¯ , z ) in Σ do
x ¯ z x ¯
a ¯ a ¯
for every ¯, b such that I satisﬁes ϕ(¯, b) do
create a fresh tuple n of pairwise distinct null values
¯
insert ψ(¯, n) into J
a ¯ ⋆


Canonical universal solution: Example

Example
Consider mapping M speciﬁed by dependency:

Canonical universal solution for
I = {Employee(Peter), Employee(John)}:
◮ For a = Peter do
◮ Create a fresh null value n1
◮ Insert Dept(Peter, n1 ) into J ⋆
◮ For a = John do
◮ Create a fresh null value n2
◮ Insert Dept(John, n2 ) into J ⋆

Result: J ⋆ = {Dept(Peter, n1 ), Dept(John, n2 )}


Query answering in data exchange

Given: Mapping M, source instance I and query Q over the target
schema
◮ What does it mean to answer Q?


Query answering in data exchange

Given: Mapping M, source instance I and query Q over the target
schema
◮ What does it mean to answer Q?

Deﬁnition (Certain answers)
certainM (Q, I ) = Q(J)


Certain answers: Example

Example
Consider mapping M speciﬁed by:

Given instance I = {Employee(Peter)}:

certainM (∃y Dept(x, y ), I ) = {Peter}
certainM (Dept(x, y ), I ) = ∅


Query rewriting: An approach for answering queries

How can we compute certain answers?
◮ Na¨ algorithm does not work: inﬁnitely many solutions
ıve


Query rewriting: An approach for answering queries

How can we compute certain answers?
◮ Na¨ algorithm does not work: inﬁnitely many solutions
ıve

Approach proposed in [FKMP03]: Query Rewriting
Given a mapping M and a target query Q, compute a query
Q ⋆ such that for every source instance I with canonical
universal solution J ⋆ :

certainM (Q, I ) = Q ⋆ (J ⋆ )


Query rewriting over the canonical universal solution

Theorem (FKMP03)
Given a mapping M speciﬁed by st-tgds and a union of
conjunctive queries Q, there exists a query Q ⋆ such that for every
source instance I with canonical universal solution J ⋆ :



Query rewriting over the canonical universal solution

Theorem (FKMP03)
Given a mapping M speciﬁed by st-tgds and a union of
conjunctive queries Q, there exists a query Q ⋆ such that for every
source instance I with canonical universal solution J ⋆ :


Proof idea: Assume that C(a) holds whenever a is a constant.

Then:
Q ⋆ (x1 , . . . , xm ) = C(x1 ) ∧ · · · ∧ C(xm ) ∧ Q(x1 , . . . , xm )


Computing certain answers: Complexity

Data complexity: Data exchange setting and query are considered
to be ﬁxed.

Corollary (FKMP03)
For mappings given by st-tgds, certain answers for UCQ can be
computed in polynomial time (data complexity)


Relational data exchange: Some lessons learned

Key steps in the development of the area:
◮ Definition of schema mappings: Precise syntax and semantics
◮ Definition of the notion of solution

◮ Identification of good solutions
◮ Polynomial time algorithms for materializing good solutions
◮ Definition of target queries: Precise semantics
◮ Polynomial time algorithms for computing certain answers for
UCQ


Relational data exchange: Some lessons learned

Key steps in the development of the area:
◮ Definition of schema mappings: Precise syntax and semantics
◮ Definition of the notion of solution

◮ Identification of good solutions
◮ Polynomial time algorithms for materializing good solutions
◮ Definition of target queries: Precise semantics
◮ Polynomial time algorithms for computing certain answers for
UCQ

Creating schema mappings is a time consuming and expensive
process
◮ Manual or semi-automatic process in general


Outline: First part




Ongoing project: Reusing schema mappings

S T U
ΣST ΣTU

ΣSU



S T U
ΣST ΣTU

ΣSU ?



S T U
ΣST ΣTU

ΣSU ?

We need some operators for schema mappings



S T U
ΣST ΣTU

ΣSU = ΣST ◦ ΣTU

We need some operators for schema mappings
◮ Composition in the above case


Metadata management

Contributions mentioned in the previous slides are just a ﬁrst step
towards the development of a general framework for data exchange.

In fact, as pointed in [B03],
many information system problems involve not only the design
and integration of complex application artifacts, but also their
subsequent manipulation.


Metadata management

This has motivated the need for the development of a general
infrastructure for managing schema mappings.

The problem of managing schema mappings is called metadata
management.

High-level algebraic operators, such as compose, are used to
manipulate mappings.
◮ What other operators are needed?


An inverse operator is also needed

S T U
ΣST ΣTU

ΣVS =ΣVS ?
Σ−1
SV ΣSV
Σ−1 ◦ ΣST
VS (Σ−1 ◦ ΣST ) ◦ ΣTU
VS

V



S T U
ΣST ΣTU

ΣVS =ΣVS ?
ΣSV
−1
ΣSV
Σ−1 ◦ ΣST
VS (ΣVS ◦ ΣST ) ◦ ΣTU
−1

V



S T U
ΣST ΣTU
ΣVS = Σ−1
SV

ΣVS ? ΣSV
Σ−1 ◦ ΣST
VS

V



S T U
ΣST ΣTU

ΣVS = Σ−1
SV ΣSV
Σ−1 ◦ ΣST
VS

V



S T U
ΣST ΣTU

ΣVS = Σ−1
SV ΣSV
Σ−1 ◦ ΣST
SV (Σ−1 ◦ ΣST ) ◦ ΣTU
VS

V

Composition and inverse operators have to be combined



S T U
ΣST ΣTU

ΣVS = Σ−1
SV ΣSV
Σ−1 ◦ ΣST
SV (Σ−1 ◦ ΣST ) ◦ ΣTU
SV

V

Composition and inverse operators have to be combined


Metadata management: A more general data exchange
framework is needed

Composition and inverse operators have been extensively studied in the
relational world.
◮ Semantics, computation, . . .

Combining these operators is an open issue.


framework is needed

relational world.

◮ Key observation: A target instance of a mapping can be the source
instance of another mapping


framework is needed

relational world.

◮ Sources instances may contain null values


framework is needed

relational world.

◮ Sources instances may contain null values

There is a need for a data exchange framework that can handle databases
with incomplete information.


Data exchange in the RDF world

There is an increasing interest in publishing relational data as RDF
◮ Resulted in the creation of the W3C RDB2RDF Working Group

The problem of translating relational data into RDF can be seen as a
data exchange problem
◮ Schema mappings can be used to describe how the relational data is
to be mapped into RDF



There is an increasing interest in publishing relational data as RDF
◮ Resulted in the creation of the W3C RDB2RDF Working Group

The problem of translating relational data into RDF can be seen as a
data exchange problem
◮ Schema mappings can be used to describe how the relational data is
to be mapped into RDF

But there is a mismatch here: A relational database under a closed-world
semantics is to be translated into an RDF graph under an open-world
semantics
◮ There is a need for a data exchange framework that can handle
both databases with complete and incomplete information



An issue discussed at the W3C RDB2RDF Working Group: Is a
mapping information preserving?
◮ In particular: For the default mapping deﬁned by this group

How can we address this issue?
◮ Metadata management can help us





Question to answer: Is a mapping invertible?





◮ This time an RDF graph is to be translated into a relational
database!





◮ This time an RDF graph is to be translated into a relational
database!
◮ We want to have a unifying framework for all these cases


But these are not the only reasons . . .

Nowadays several applications use knowledge bases to represent data.
◮ A knowledge base has not only data but also rules that allows to
infer new data
◮ In the Semantics Web: RDFS and OWL ontologies



infer new data

In a data exchange application over the Semantics Web:
The input is a mapping and a source speciﬁcation including data
and rules, and the output is a target speciﬁcation also including
data and rules



infer new data

In a data exchange application over the Semantics Web:
The input is a mapping and a source speciﬁcation including data
and rules, and the output is a target speciﬁcation also including
data and rules

There is a need for a data exchange framework that can handle
knowledge bases.


Knowledge exchange: A more general data exchange
framework is needed

Example
Assume given the following source knowledge base:

Data:
Father Mother
Andy Bob Carrie Bob
Bob Danny
Danny Eddie

Rules:
Father(x, y ) → Parent(x, y )
Mother(x, y ) → Parent(x, y )
Parent(x, y ) ∧ Parent(y , z) → Grandparent(x, z)


framework is needed

Example (cont’d)
Given a mapping:
Father(x, y ) → F(x, y )
Grandparent(x, y ) → G(x, y )

What is a good translation of the initial knowledge base?


framework is needed

Example (cont’d)
Given a mapping:
Father(x, y ) → F(x, y )
Grandparent(x, y ) → G(x, y )

What is a good translation of the initial knowledge base?

Data:
F G
Andy Bob Andy Danny
Bob Danny Carrie Danny
Danny Eddie Bob Eddie

Rules: ∅


framework is needed

Example (cont’d)
Our ﬁrst alternative does not include any translation of the source rules:
Father(x, y ) → Parent(x, y )


framework is needed

Example (cont’d)



framework is needed

Example (cont’d)

F(x, y ) ∧ F(y , z) → G(x, z)


framework is needed

Example (cont’d)

F(x, y ) ∧ F(y , z) → G(x, z)

What data should we materialize?


framework is needed

Example (cont’d)

F(x, y ) ∧ F(y , z) → G(x, z)

F G
Andy Bob Andy Danny
Danny Eddie Bob Eddie


framework is needed

Example (cont’d)

F(x, y ) ∧ F(y , z) → G(x, z)

F G
Andy Bob
Danny Eddie


framework is needed

Example (cont’d)

F(x, y ) ∧ F(y , z) → G(x, z)

F G
Andy Bob
Danny Eddie

Is this a good translation? Why?


One can exchange more than complete data

◮ In data exchange one starts with a database instance (with
complete information).

◮ What if we have an initial object that has several
interpretations?
◮ A representation of a set of possible instances

◮ We propose a new general formalism to exchange
representations of possible instances
◮ We apply it to the problems of exchanging instances with
incomplete information and exchanging knowledge bases


Outline: Second part

◮ Formalism for exchanging representations systems

◮ Applications to incomplete instances

◮ Applications to knowledge bases

◮ Concluding remarks


Representation systems

A representation system R = (W, rep) consists of:
◮ a set W of representatives
◮ a function rep that assigns a set of instances to every element
in W
rep(V) = {I1 , I2 , I3 , . . .} for every V ∈ W

Uniformity assumption: For every V ∈ W, there exists a relational
schema S (the type of V) such that rep(V) ⊆ Inst(S)


Representation systems

A representation system R = (W, rep) consists of:
◮ a set W of representatives
◮ a function rep that assigns a set of instances to every element
in W
rep(V) = {I1 , I2 , I3 , . . .} for every V ∈ W

Uniformity assumption: For every V ∈ W, there exists a relational
schema S (the type of V) such that rep(V) ⊆ Inst(S)

Incomplete instances and knowledge bases are representation
systems


In classical data exchange we consider only complete data

Recall that given M = (S, T, Σ), I ∈ Inst(S) and J ∈ Inst(T): J is
a solution for I under M if (I , J) |= Σ

J ∈ SolM (I )


In classical data exchange we consider only complete data

Recall that given M = (S, T, Σ), I ∈ Inst(S) and J ∈ Inst(T): J is
a solution for I under M if (I , J) |= Σ

J ∈ SolM (I )

This can be extended to set of instances. Given X ⊆ Inst(S):

SolM (X ) = SolM (I )
I ∈X


Extending the deﬁnition to representation systems

Given:
◮ a mapping M = (S, T, Σ)
◮ a representation system R = (W, rep)
◮ U, V ∈ W of types S and T, respectively



Given:

Deﬁnition (APR11)
V is an R-solution of U under M if
rep(V) ⊆ SolM (rep(U))



Given:

Deﬁnition (APR11)
V is an R-solution of U under M if
rep(V) ⊆ SolM (rep(U))

Or equivalently: V is an R-solution of U if for every J ∈ rep(V),
there exists I ∈ rep(U) such that J ∈ SolM (I ).


Universal solutions

What is a good solution in this framework?


Universal solutions

What is a good solution in this framework?

Deﬁnition (APR11)
V is an universal R-solution of U under M if
rep(V) = SolM (rep(U))


Strong representation systems

Let C be a class of mappings.




Deﬁnition (APR11)
R = (W, rep) is a strong representation system for C if for every
M∈C and for every U ∈ W , there exists a
V ∈W :





Deﬁnition (APR11)
M ∈ C from S to T, and for every U ∈ W , there exists a
V ∈W :





Deﬁnition (APR11)
M ∈ C from S to T, and for every U ∈ W of type S, there exists a
V ∈W :





Deﬁnition (APR11)
V ∈ W of type T:





Deﬁnition (APR11)
V ∈ W of type T:


If R = (W, rep) is a strong representation system, then the
universal solutions for the representatives in W can be represented
in the same system.


Motivating questions

What is a strong representation system for the class of mappings
speciﬁed by st-tgds?
◮ Are instances including nulls enough?

Can the fundamental data exchange problems be solved in
polynomial time in this setting?
◮ Computing (universal) solutions
◮ Computing certain answers


Naive instances

We have already considered naive instances: Instances with null values
◮ Example: Canonical universal solution

A naive instance I has labeled nulls:
R(1, n1 )
R(n1 , 2)
R(1, n2 )


Naive instances

We have already considered naive instances: Instances with null values
◮ Example: Canonical universal solution

A naive instance I has labeled nulls:
R(1, n1 )
R(n1 , 2)
R(1, n2 )

The interpretations of I are constructed by replacing nulls by constants:

rep(I) = {K | µ(I) ⊆ K for some valuation µ}


Are naive instances expressive enough?

Naive instances have been extensively used in data exchange:

Proposition (FKMP03)
Let M = (S, T, Σ), where Σ is a set of st-tgds. Then for every
instance I of S, there exists a naive instance J of T such that:

rep(J ) = SolM (I )

In fact, the canonical universal solution satisﬁes the property
mentioned above.



But naive instances are not expressive enough to deal with
incomplete information in the source instances:

Proposition (APR11)
Naive instances are not a strong representation system for the class
of mappings speciﬁed by st-tgds



Example
Consider a mapping M speciﬁed by:
Manager(x, y ) → Reports(x, y )
Manager(x, x) → SelfManager(x)

The canonical universal solution for I = {Manager(n, Peter)} under M:
J = {Reports(n, Peter)}

But J is not a good solution for I.
◮ It cannot represent the fact that if n is given value Peter, then
SelfManager(Peter) should hold in the target.


Conditional instances

What should be added to naive instances to obtain a strong
representation system?



◮ Answer from database theory: Conditions on the nulls



◮ Answer from database theory: Conditions on the nulls

Conditional instances: Naive instances plus tuple conditions

A tuple condition is a positive Boolean combinations of:
◮ equalities and inequalities between nulls, and between nulls
and constants



Example
R(1, n1 ) n1 = n2
R(n1 , n2 ) n1 = n2 ∨ n2 = 2



Example
R(1, n1 ) n1 = n2
R(n1 , n2 ) n1 = n2 ∨ n2 = 2

Semantics:



Example
R(1, n1 ) n1 = n2
R(n1 , n2 ) n1 = n2 ∨ n2 = 2

Semantics:
µ(n1 ) = µ(n2 ) = 2 µ(n1 ) = µ(n2 ) = 3 µ(n1 ) = 2, µ(n2 ) = 3



Example
R(1, n1 ) n1 = n2
R(n1 , n2 ) n1 = n2 ∨ n2 = 2

Semantics:
µ(n1 ) = µ(n2 ) = 2 µ(n1 ) = µ(n2 ) = 3 µ(n1 ) = 2, µ(n2 ) = 3
R(1, 2)
R(2, 2)



Example
R(1, n1 ) n1 = n2
R(n1 , n2 ) n1 = n2 ∨ n2 = 2

Semantics:
µ(n1 ) = µ(n2 ) = 2 µ(n1 ) = µ(n2 ) = 3 µ(n1 ) = 2, µ(n2 ) = 3
R(1, 2) R(1, 3)
R(2, 2)



Example
R(1, n1 ) n1 = n2
R(n1 , n2 ) n1 = n2 ∨ n2 = 2

Semantics:
µ(n1 ) = µ(n2 ) = 2 µ(n1 ) = µ(n2 ) = 3 µ(n1 ) = 2, µ(n2 ) = 3
R(1, 2) R(1, 3)
R(2, 2) R(2, 3)



Example
R(1, n1 ) n1 = n2
R(n1 , n2 ) n1 = n2 ∨ n2 = 2

Semantics:
µ(n1 ) = µ(n2 ) = 2 µ(n1 ) = µ(n2 ) = 3 µ(n1 ) = 2, µ(n2 ) = 3
R(1, 2) R(1, 3)
R(2, 2) R(2, 3)

Interpretations of a conditional instance I:

rep(I) = {K | µ(I) ⊆ K for some valuation µ}


Positive conditional instances

Many problems are intractable over conditional instances.
◮ We also consider a restricted class of conditional instances

Positive conditional instances: Conditional instances without
inequalities


(Positive) conditional instances are enough

Theorem (APR11)
Both conditional instances and positive conditional instances are strong
representation systems for the class of mappings speciﬁed by st-tgds.

Example
Consider again the mapping M speciﬁed by:
Manager(x, y ) → Reports(x, y )
Manager(x, x) → SelfManager(x)

The following is a universal solution for I = {Manager(n, Peter)}
Reports(n, Peter) true
SelfManager(Peter) n = Peter


Positive conditional instances are exactly the needed
representation system

Positive conditional instances are minimal:

Theorem (APR11)
All the following are needed to obtain a strong representation
system for the class of mappings speciﬁed by st-tgds:
◮ equalities between nulls
◮ equalities between constant and nulls
◮ conjunctions and disjunctions

Conditional instances are enough but not minimal.


Positive conditional instance can be used in practice!

Let M = (S, T, Σ), where Σ is a set of st-tgds.




Theorem (APR11)
There exists a polynomial time algorithm that, given a positive
conditional instance I over S, computes a positive conditional instance
J over T that is a universal solution for I under M.




Theorem (APR11)
conditional instance I over S, computes a positive conditional instance
J over T that is a universal solution for I under M.

Let Q be a union of conjunctive queries over T.

Q(J ) = Q(J)
J∈rep(J )

certainM (Q, I) = Q(J )



Theorem (APR11)
conditional instance I over S, computes certainM (Q, I).



Theorem (APR11)
conditional instance I over S, computes certainM (Q, I).

The same result holds for the class of unions of conjunctive queries with
at most one inequality per disjunct.
◮ The other important class of queries in the data exchange area for
which certain answers can be computed in polynomial time


The semantics of knowledge bases is given by sets of
instances

Knowledge base over S: (I , Γ) such that
◮ I ∈ Inst(S)
◮ Γ a set of rules over S

Semantics: ﬁnite models

Mod(I , Γ) = {K ∈ Inst(S) | I ⊆ K and K |= Γ}


We can apply our formalism to knowledge bases

(I2 , Γ2 ) is a KB-solution for (I1 , Γ1 ) under M if:

Mod(I2 , Γ2 ) ⊆ SolM (Mod(I1 , Γ1 ))

(I2 , Γ2 ) is a universal KB-solution for (I1 , Γ1 ) under M if:

Mod(I2 , Γ2 ) = SolM (Mod(I1 , Γ1 ))


Motivating questions

Same as for the case of instances with incomplete information.
◮ Constructing universal KB-solutions
◮ Answering target queries

New fundamental problem: Construct solutions including as much
implicit knowledge as possible.


What are good knowledge-base solutions?

First alternative: universal KB-solutions

But there exist some other KB-solutions desirable to materialize
◮ Minimality comes into play


What are good knowledge-base solutions?

First alternative: universal KB-solutions

But there exist some other KB-solutions desirable to materialize
◮ Minimality comes into play

Given sets X , Y of instances:
◮ X ≡min Y if X and Y coincide in the minimal instances under ⊆

Deﬁnition
(I2 , Γ2 ) is a minimal KB-solution of (I1 , Γ1 ) under M if:

Mod(I2 , Γ2 ) ≡min SolM (Mod(I1 , Γ1 ))


Two requirements to construct minimal knowledge-base
solutions

Given (I1 , Γ1 ) and M, when constructing a minimal KB-solution
(I2 , Γ2 ) we would like:


solutions


1. Γ2 to only depend on Γ1 and M:

Γ2 is safe for Γ1 and M


solutions


1. Γ2 to only depend on Γ1 and M:

Γ2 is safe for Γ1 and M

Deﬁnition
Γ2 is safe for Γ1 and M, if for every I1 there exists I2 :

(I2 , Γ2 ) is a minimal KB-solution of (I1 , Γ1 ) under M


solutions

2. Γ2 to be as informative as possible (thus minimizing the size
of I2 ):


solutions

2. Γ2 to be as informative as possible (thus minimizing the size
of I2 ):

Deﬁnition
Γ2 is optimal-safe if for every other safe set Γ′ :

Γ2 |= Γ′


Computing minimal KB-solutions

To obtain algorithms for computing minimal KB-solutions, we need
to specify the language used in knowledge bases.
◮ Full st-tgd:

∀¯∀¯ (ϕ(¯ , y ) → ψ(¯ ))
x y x ¯ x



To obtain algorithms for computing minimal KB-solutions, we need
to specify the language used in knowledge bases.
◮ Full st-tgd:

∀¯∀¯ (ϕ(¯ , y ) → ψ(¯ ))
x y x ¯ x

Theorem (APR11)
There exists a polynomial-time algorithm that, given
M = (S, T, Σ), where Σ is a set of full st-tgds, and given a set Γ1
of full tgds over S, computes a set Γ2 of second-order logic
sentences over T that is optimal-safe for Γ1 and M.



Unfortunately, ﬁrst-order logic is no expressive enough.

Theorem (APR11)
There exist M = (S, T, Σ), where Σ is a set of full st-tgds, and a
set Γ1 of full tgds over S such that:

no FO-sentence is optimal-safe for Γ1 and M.




Theorem (APR11)


How can we deal with these problems in practice?




Theorem (APR11)


How can we deal with these problems in practice?
◮ We need to restrict the language used to specify knowledge
bases: Description logics [ABC11]


We can exchange more than complete data

We propose a general formalism to exchange representation
systems

Next step: Apply our general setting to the Semantic Web
◮ Semantic Web data has nulls (blank nodes)
◮ Semantic Web speciﬁcations have rules (RDFS, OWL)

Lots of interesting problems to solve if knowledge bases are
speciﬁed by means of description logics.
◮ Better results can be obtained


Thank you!


Bibliography

[ABC11] M. Arenas, E. Botoeva, D. Calvanese. Knowledge Base Exchange.
DL 2011.

[APR11] M. Arenas, J. P´rez, J. Reutter. Data Exchange beyond Complete
e
Data. PODS 2011.

[B03] P. A. Bernstein. Applying Model Management to Classical Meta
Data Problems. CIDR 2003.

[FKMP03] R. Fagin, P. G. Kolaitis, R. J. Miller, L. Popa. Data Exchange:
Semantics and Query Answering. ICDT 2003.


Bonus track: Computation of solutions and its associated
decision problem

Decision problem: Check-KB-Sol
Input: M = (S, T, Σ), where Σ is a set of st-tgds
(I1 , Γ1 ) KB over S with Γ1 a set of tgds
(I2 , Γ2 ) KB over T with Γ2 a set of tgds

Output: Is (I2 , Γ2 ) a KB-solution of (I1 , Γ1 ) under M?


decision problem

Decision problem: Check-KB-Sol
Input: M = (S, T, Σ), where Σ is a set of st-tgds
(I1 , Γ1 ) KB over S with Γ1 a set of tgds
(I2 , Γ2 ) KB over T with Γ2 a set of tgds

Output: Is (I2 , Γ2 ) a KB-solution of (I1 , Γ1 ) under M?

Theorem (APR11)
Check-KB-Sol is undecidable (even for a ﬁxed M).


decision problem

Undecidability is a consequence of using ∃ in knowledge bases.
◮ We need to restrict the input

Check-Full-KB-Sol: Γ1 , Γ2 are assumed to be sets of full tgds


decision problem

Theorem (APR11)
Check-Full-KB-Sol is EXPTIME-complete.


decision problem

Theorem (APR11)
Check-Full-KB-Sol is EXPTIME-complete.

Theorem (APR11)
If M = (S, T, Σ) is ﬁxed:

Check-Full-KB-Sol is ∆P [O(log n)]-complete.
2

∆P [O(log n)]:
2 P NP with a logarithmic number of
calls to the NP oracle.


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