Exchanging More than Complete Data

Exchanging More than Complete Data

Jorge PerezUniversidad de Chile

joint work with Marcelo Arenas (PUC-Chile) and Juan Reutter (Univ. Edinburgh)

Father

Andy BobBob DannyDanny Eddie

Mother

Carrie Bob

Father(x , y) → Parent(x , y)Mother(x , y) → Parent(x , y)

Parent(x , y) ∧ Parent(y , z) → Gr-Parent(x , z)

Father


Mother

Carrie Bob



SourceFather(x , y) → Pateras(x , y)

Gr-Parent(x , y) → Pappoi(x , y)Target

Father


Mother

Carrie Bob





Pateras


Pappoi

Andy DannyCarrie DannyBob Eddie

Father


Mother

Carrie Bob





Pateras(x , y) ∧ Pateras(y , z) → Pappoi(x , z)

Father


Mother

Carrie Bob





Pateras


Pappoi

Carrie Danny

Pateras(x , y) ∧ Pateras(y , z) → Pappoi(x , z)

We can exchange more than complete data

◮ In data exchange we begin with a database instancewe begin with complete data

◮ What if we have a representation of a set ofpossible instances?

◮ We propose a new general formalism to exchangerepresentations of possible instancesand apply it to incomplete instances and knowledge bases

Outline

Formalism for exchanging representations systems

Applications to incomplete instances

Applications to knowledge basesWhat is the complexity of testing kb-solutions?How can we compute a good kb-solution?

Concluding remarks

Representation systems

A representation system is composed of:

◮ a set W of representatives

◮ a function rep from W to sets of instances

Vrep−→ {I1, I2, I3, . . .} = rep(V)

Representation systems

A representation system is composed of:

◮ a set W of representatives

◮ a function rep from W to sets of instances

Vrep−→ {I1, I2, I3, . . .} = rep(V)

◮ Incomplete instances and knowledge bases arerepresentation systems

In classical data exchange

we consider only complete data

M = (S,T,Σ) a schema mapping from S to T

I ∈ Inst(S), J ∈ Inst(T)

J is a solution for I under M iff (I , J) |= Σ

J ∈ SolM(I )

In classical data exchange

we consider only complete data

M = (S,T,Σ) a schema mapping from S to T

I ∈ Inst(S), J ∈ Inst(T)

J is a solution for I under M iff (I , J) |= Σ

J ∈ SolM(I )

We can extend this to set of instances:given a set X ⊆ Inst(S)

SolM(X ) =⋃

I∈X

SolM(I )

Extending the definition to representation systems

Given:

◮ a mapping M = (S,T,Σ)

◮ a representation system R = (W, rep)

◮ U ,V ∈ W


Given:



◮ U ,V ∈ W

Definition

U is an R-solution of V if

rep(U) ⊆ SolM(rep(V))


Given:



◮ U ,V ∈ W

Definition

U is an R-solution of V if

rep(U) ⊆ SolM(rep(V))

or equivalently,

∀J ∈ rep(U), ∃I ∈ rep(V) such that J ∈ SolM(I )

Universal solutions and strong representation systems

Definition

U is a universal R-solution of V if

rep(U) = SolM(rep(V))


Definition



Let C be a class of mappings,

Definition

R = (W, rep) is a strong representation system for C iffor every M ∈ C, and V ∈ W, there exists a U ∈ W such that:



Definition



Let C be a class of mappings,

Definition

R = (W, rep) is a strong representation system for C iffor every M ∈ C, and V ∈ W, there exists a U ∈ W such that:


Strong representation systems ⇒ universal solutions forrepresentatives in R can always be represented in the same system.

Outline




Concluding remarks

Incomplete Instances

We consider:

◮ Naive instances: instances with null values

R(1,N1)R(N1, 2)


We consider:


R(1,N1)R(N1, 2)

◮ Positive conditional instances: naive instances plusconditions for tuples

◮ equalities between nulls, and between nulls and constants◮ conjunctions and disjunctions

T (1,N1) trueR(1,N1,N2) N1 = N2 ∨ (N1 = 2 ∧ N2 = 3)


We consider:


R(1,N1)R(N1, 2)

◮ Positive conditional instances: naive instances plusconditions for tuples

◮ equalities between nulls, and between nulls and constants◮ conjunctions and disjunctions

T (1,N1) trueR(1,N1,N2) N1 = N2 ∨ (N1 = 2 ∧ N2 = 3)

Semantics given by valuations of nulls (+ open world):

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

Strong representation systems for st-tgds

Let M = (S,T,Σ) be specified by source-to-target tgds(st-tgd: φS(x) → ∃yψT(x , y), φS and ψT are CQs)

(FKMP03)

For every complete instance I there exists a naive instance J s.t.

rep(J ) = SolM(I )



(FKMP03)


rep(J ) = SolM(I )

Proposition

Naive instances are not a strong representation system for st-tgds

Idea

Mng(N ,Alice) Mng(x , y) → Reports(x , y)Mng(x , x) → Self-Mng(x)



(FKMP03)


rep(J ) = SolM(I )

Proposition

Naive instances are not a strong representation system for st-tgds

Idea

Mng(N ,Alice) Mng(x , y) → Reports(x , y) ?Mng(x , x) → Self-Mng(x)

Positive conditional instances are enough

Theorem

Positive conditional instances are astrong representation system for st-tgds.


Theorem


Mng(N, Alice) Mng(x , y) → Reports(x , y)Mng(x , x) → Self-Mng(x)


Theorem


Mng(N, Alice) Mng(x , y) → Reports(x , y) Reports(N ,Alice) true

Mng(x , x) → Self-Mng(x)


Theorem



Mng(x , x) → Self-Mng(x) Self-Mng(Alice)


Theorem



Mng(x , x) → Self-Mng(x) Self-Mng(Alice) N = Alice


Theorem



Mng(x , x) → Self-Mng(x) Self-Mng(Alice) N = Alice

Corollary

Positive conditional instances are astrong representation system for SO-tgds.

Positive conditional instances are

exactly the needed representation system

Theorem

Positive conditional instances are minimal:

◮ equalities between nulls◮ equalities between constant and nulls◮ conjunctions and disjunctions

are all needed for a strong representation system of st-tgds

Positive conditional instances

match the complexity bounds of classical data exchange

Theorem

For positive conditional instances and (fixed) st-tgds

◮ Universal solutions can be constructed in polynomial time.

◮ Certain answers of unions of conjunctive queriescan be computed in polynomial time.

Outline




Concluding remarks

The semantics of knowledge bases

is given by sets of instances

Knowledge base over S: (I ,Γ) s.t.

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 iff

Mod(I2,Γ2) ⊆ SolM(Mod(I1,Γ1))

Outline




Concluding remarks

We study the following decision problem

Check-KB-Sol:

Input: M = (S,T,Σ) with Σ st-tgds(I1,Γ1) knowledge base over S with Γ1 tgds(I2,Γ2) knowledge base over T with Γ2 tgds

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


Check-KB-Sol:



Theorem

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


Check-KB-Sol:



Theorem

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

Undecidability is a consequence of using ∃ in knowledge bases

Check-Full-KB-Sol: Γ1, Γ2 full-tgds (no ∃)

The complexity of Check-Full-KB-Sol

Theorem

Check-Full-KB-Sol is EXPTIME-complete.

The complexity of Check-Full-KB-Sol

Theorem

Check-Full-KB-Sol is EXPTIME-complete.

Theorem

If M = (S,T,Σ) is fixed

Check-Full-KB-Sol is ∆P2 [O(log n)]-complete.

∆P2 [O(log n)] ≡ PNP with only a logarithmic

number of calls to the NP oracle.

Hardness is obtained by reducing

the Max-Odd-Clique problem

Given a graph G with n nodes:

S: E (·, ·), Succ(·, ·), Clique(·)




S: E (·, ·), Succ(·, ·), Clique(·)T: E ′(·, ·), Succ ′(·, ·), Clique′(·)





I1 and I2 are copies of G , plus a successor relation over [1, n]






Γ1: “E has a clique of even size x ≤ n” → Clique(x)






Γ1: “E has a clique of even size x ≤ n” → Clique(x)Γ2: “E ′ has a clique of odd size x ≤ n” → Clique′(x)







Σ: Clique(x) ∧ Succ(x , y) → Clique′(y)







Σ: Clique(x) ∧ Succ(x , y) → Clique′(y)

(I2,Γ2) is a kb-solution of (I1,Σ1) iffthe maximum size of a clique in G is odd.

Outline




Concluding remarks

Minimal knowledge bases as good kb-solutions

What are good knowledge-base solutions?

◮ first choice: universal kb-solutions, but

◮ there are some other kb-solutions desirable to materialize





X ≡min Y: X and Y coincide in the minimal instances





X ≡min Y: X and Y coincide in the minimal instances

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

Mod(I2,Γ2) ≡min SolM(Mod(I1,Γ1))

We consider minimal kb-solutions as the good solutions

Two requirements to construct

minimal knowledge-base solutions

Given (I1,Γ1) and Σ, when constructing aminimal-knowledge base solution (I2,Γ2) we would want:

1. Γ2 to only depend on Γ1 and Σ

Γ2 is safe for Γ1 and Σ




1. Γ2 to only depend on Γ1 and Σ


Definition

Γ2 is safe for Γ1 and Σ, if for every I1 there exists I2 s.t.

(I2,Γ2) is a minimal kb-solution of (I1,Γ1)




2. Γ2 to be as informative as possiblethus minimizing the size of I2




2. Γ2 to be as informative as possiblethus minimizing the size of I2

Γ2 is optimum-safe if for every other safe set Γ′

Γ2 |= Γ′

Safe sets: data independent target knowledge bases

Unfortunately, optimum-safe sets are not (always) FO-expressible

Theorem

There exist sets Γ1 (full & acyclic) and Σ (full) s.t.

no FO sentence is optimum-safe for Γ1 and Σ



Theorem



In our paper:

An algorithm to compute optimum-safe set in SO



Theorem



In our paper:

An algorithm to compute optimum-safe set in SO

Can we do better? not optimum but useful set?(non-trivial safe set in a good language)

We can compute a good safe set

for acyclic tgds knowledge bases

Input: Γ1 acyclic full-tgds, Σ full-tgds (from S to T)




1. Unfold Γ1 to get Γ+1





2. Construct Σ− (from T to S) as follows:

for every α(x) → R(x) in Γ+1







Rewrite α(x) in the target of Σ to obtain β(x)








Add β(x) → R(x) to Σ−









3. Let Γ2 = Σ− ◦ Σ









3. Let Γ2 = Σ− ◦ Σ

Theorem


(Γ2 is a set of full-tgds with 6=)

In the paper: a complete strategy for computing


Given Γ1 and Σ:

◮ We compute Γ2 safe for Γ1 and Σ

◮ We then compute minimal set Γ∗ (implied by Γ1) such that

for every I1



Given Γ1 and Σ:



for every I1

chaseΓ∗(I1)



Given Γ1 and Σ:



for every I1

chaseΣ( chaseΓ∗(I1))



Given Γ1 and Σ:



for every I1(

chaseΣ( chaseΓ∗(I1)),Γ2

)



Given Γ1 and Σ:



for every I1(

chaseΣ( chaseΓ∗(I1)),Γ2

)

is a minimal kb-solution for (I1,Γ1).

Outline




Concluding remarks


We propose a general formalismto exchange representation systems

◮ Applications to incomplete instances

◮ Applications to knowledge bases


We propose a general formalismto exchange representation systems

◮ Applications to incomplete instances

◮ Applications to knowledge bases

Next step: apply our general setting to the Semantic Web

◮ Semantic Web data have nulls (blank nodes)

◮ Semantic Web specifications have rules (RDFS, OWL)

Outline




Concluding remarks

Exchanging More than Complete Data

Technology

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