Integrity checking for combined databases Davide Martinenghi - - PowerPoint PPT Presentation

Davide Martinenghi

CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003 Computer Science, building 42.1 Roskilde University Universitetsvej 1 P.O. Box 260 DK-4000 Roskilde Denmark Phone: +45 4674 2000 Fax: +45 4674 3072 www.dat.ruc.dk

Integrity checking for combined databases

Davide Martinenghi 2 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Content

• Description of the problem
• A simplification framework
• GaV and LaV mappings
• Application to data integration
• Examples

Davide Martinenghi 3 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Description of the problem

• ICs are properties of the DB that must

always hold

• The integrity must be checked wrt the ICs

after every update (typically tested in an ad hoc way at the application level)

• In a data integration system, it’s the same
• Idea: generate specialized versions of the

ICs to be automatically executed

• For expected kinds of updates
• Assuming the integrity before the update
• Generalize this technique to data integration

systems

Davide Martinenghi 4 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

A simplification framework

• 1. Produce a weakest precondition

Ex: ϕ=←p(x) U=p(a) AfterU(ϕ)=←(p(x)∨x=a)

• 2. Use the fact that ϕ was known to hold

before the update (Cond. Weak. Prec.).

• 3. Take the weakest CWP.

DEF: SimpU(ϕ) =Weakenϕ(AfterU(ϕ))

A condition about the updated state that can be checked in the present state

Davide Martinenghi 5 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

A simplification framework - Example

ϕ = ← m(x, y) ∧ m(x, z)∧ y ≠ z checked by posing it as a query against the DB and expecting an empty answer U = m(Bob, Alice) simpU(ϕ) = ← m(Bob, y) ∧ y ≠ Alice

Davide Martinenghi 6 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Mappings

• Mapping = a way to associate n local DBs to

a global DB

• GaV mapping = the global DB is expressed

as a set of views over the local sources.

• LaV mapping = the local DBs are expressed

as a set of views over the global DB.

• We assume:
• sound mappings (the views produce only but not

necessarily all correct information)

• no existential quantifier in LaV mappings
• ⇒ LaV mappings can be rewritten as GaV

mappings without skolemization

Davide Martinenghi 7 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Mappings - example

LaV mapping L = { m1(x, y) → m(x, y) ∧ n(x, it), m2(x, y) → m(x, y) ∧ n(x, dk) } GaV mapping ML = { m(x, y) ← m1(x, y), m(x, y) ← m2(x, y), n(x, y) ← m1(x, z) ∧ y=it, n(x, y) ← m2(x, z) ∧ y=dk }

Davide Martinenghi 8 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Application to data integration

• AfterM(ϕ) is a weakest precondition

M is a GaV mapping

• SimpO∆(ϕ) =Weaken∆(AfterO(ϕ))

A condition about the global DB that can be checked on the local DBs

Conditions to check globally Conditions known to hold locally

Davide Martinenghi 9 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Example 1

ϕ = ← m(x, y) ∧ m(x, z)∧ y ≠ z ϕ1 = ← m1(x, y) ∧ m1 (x, z)∧ y ≠ z ϕ2 = ← m2(x, y) ∧ m2 (x, z)∧ y ≠ z Check: { ← m1(x, y) ∧ m1 (x, z)∧ y ≠ z, ← m1(x, y) ∧ m2 (x, z)∧ y ≠ z, ← m2(x, y) ∧ m1 (x, z)∧ y ≠ z, ← m2(x, y) ∧ m2 (x, z)∧ y ≠ z } …

Davide Martinenghi 10 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Example 1

ϕ = ← m(x, y) ∧ m(x, z)∧ y ≠ z ϕ1 = ← m1(x, y) ∧ m1 (x, z)∧ y ≠ z ϕ2 = ← m2(x, y) ∧ m2 (x, z)∧ y ≠ z Check: { ← m1(x, y) ∧ m1 (x, z)∧ y ≠ z, ← m1(x, y) ∧ m2 (x, z)∧ y ≠ z, ← m2(x, y) ∧ m1 (x, z)∧ y ≠ z, ← m2(x, y) ∧ m2 (x, z)∧ y ≠ z } = SimpM

ϕ1∧ ϕ2(ϕ)

…

Davide Martinenghi 11 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Example 1

ϕ = ← m(x, y) ∧ m(x, z)∧ y ≠ z ϕ1 = ← m1(x, y) ∧ m1 (x, z)∧ y ≠ z ϕ2 = ← m2(x, y) ∧ m2 (x, z)∧ y ≠ z Check: { ← m1(x, y) ∧ m1 (x, z)∧ y ≠ z, ← m1(x, y) ∧ m2 (x, z)∧ y ≠ z, ← m2(x, y) ∧ m1 (x, z)∧ y ≠ z, ← m2(x, y) ∧ m2 (x, z)∧ y ≠ z } = SimpM

ϕ1 ∧ ϕ2 ∧ ϕ1,2(ϕ)

If we knew ϕ1,2 = ← m1(x, y) ∧ m2 (x, z)

Davide Martinenghi 12 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Example 2

M={ f(i,t,r)←m(i,t,y)∧r(i,r)} ϕ1 ={← m(i,t1,y1) ∧ m(i,t2,y2) ∧ t1 ≠ t2, ← m(i,t1,y1) ∧ m(i,t2,y2) ∧ y1 ≠ y2} ϕ1,2 = ← r(i, r) ∧ ¬m(i,t,y) ϕ ={ ← f(i,t1,r1) ∧ f(i,t2,r2) ∧ t1 ≠ t2, ← f(i,t1,r1) ∧ f(i,t2,r2) ∧ r1 ≠ r2} Global check: {

← m(i,t1,y1) ∧ r(i, r1) ∧ m(i,t2,y2) ∧ r(i, r2) ∧ t1 ≠ t2, ← m(i,t1,y1) ∧ r(i, r1) ∧ m(i,t2,y2) ∧ r(i, r2) ∧ r1 ≠ r2}

…

Davide Martinenghi 13 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Example 2

M={ f(i,t,r)←m(i,t,y)∧r(i,r)} ϕ1 ={← m(i,t1,y1) ∧ m(i,t2,y2) ∧ t1 ≠ t2, ← m(i,t1,y1) ∧ m(i,t2,y2) ∧ y1 ≠ y2} ϕ1,2 = ← r(i, r) ∧ ¬m(i,t,y) ϕ ={ ← f(i,t1,r1) ∧ f(i,t2,r2) ∧ t1 ≠ t2, ← f(i,t1,r1) ∧ f(i,t2,r2) ∧ r1 ≠ r2} Global check: {

← m(i,t1,y1) ∧ r(i, r1) ∧ m(i,t2,y2) ∧ r(i, r2) ∧ t1 ≠ t2, ← m(i,t1,y1) ∧ r(i, r1) ∧ m(i,t2,y2) ∧ r(i, r2) ∧ r1 ≠ r2}

= SimpM

ϕ1∧ ϕ1,2(ϕ)

Davide Martinenghi 14 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

Summary

• Express the data integration in terms of a

GaV-mapping

• Reformulate the condition to check in terms
• f the sources by calculating a weakest

precondition wrt the mapping

• Remove from it all conditions known to hold

locally (plus possible cross-conditions)

Davide Martinenghi 15 CoLogNET Workshop on Logic-Based Methods for Information Integration - 23 August 2003

References

• 1. H. Christiansen, D. Martinenghi

Simplification of integrity constraints for data integration Submitted to FoIKS '04

• 2. H. Christiansen, D. Martinenghi

Simplification of database integrity constraints revisited: A transformational approach Accepted for presentation at LOPSTR '03 http://www.dat.ruc.dk/~dm/publications