an ontology-extended relational algebra piero bonatti università di napoli "federico ii"...

An Ontology-Extended Relational Algebra

Piero BonattiUniversità di Napoli "Federico II"

Yu DengV.S. Subrahmanian

University of Maryland College Park

10/28/2003 Ontology Extended Algebra 2

Outline Problem statement Approach Motivating example Ontology-extended relational algebra HOME system Contributions Related work Future work


Problem Statement Integrating heterogeneous data sources is

an important problem. There are many projects in this area, but at syntactic level.

Our goal: Integrate data sources with diverse structures

and assumptions at the semantic level. Answer queries correctly under user’s

assumptions of semantic meaning about the terms being used.


Approach Associate ontologies to data

sources. Ontology interoperation. Extend relational data model

and relational algebra.


Motivating Example Two parts relations:

Relation Parts1 with the schema (Name, Cost, Shipping)

Relation Parts2 with the schema (Item, Price, ShipCost)

Two insurance claim relations: Relation Claims1 with the schema (ClaimId,

Type, Cost) Relation Claims2 with the schema

(ClaimNumber, Type, Value)

10/28/2003 Ontology Extended Algebra

Parts1 and Parts2 Relations

Name Cost Shipping

Tire 54.19 20.05

Gasket 3.05 1.55

Valve 3.35 1.55

Brake pads

78.50 8.50

Evaporator

305.00

11.50

Item Price ShipCost

Wheel 50.05 18.00

Air Gasket

3.00 1.70

Valve 3.35 1.55

Hubcap 11.50 6.00

Spark Plug

20.00 8.50

Parts1 relation Parts2 relation


Problems (1) When users specify a query spanning these

two relations, they may wonder: Do the fields Cost and Price mean the same thing? Is wheel a part of tire? Is air gasket a gasket?

Furthermore, does the field Cost use the unit US dollar? Does the field Price use the unit Euro?

Users may be at a loss to determine these by looking at the fields.


Claims1 and Claims2 Relation

ClaimId Type Cost

1 burglary 2000

2 theft 150

3 mugging 860

4 arson 1800

ClaimNumber

Type Value

1 robbery 400

2 fire 550

3 auto accident

500

4 burglary

250

Claims1 relation Claims2 relation


Problem (2) Users may have a query such as “Find

all the thefts that involved a cost of over $1000 dollars”. The system should automatically recognize that burglaries, muggings and robberies count as thefts.

In addition, conversions between units are needed if costs are represented in different units in above query.


Ontology Extended Relation (OER)

We use ontology to convey semantics about terms in a domain and associate ontologies with relations.

Intuitively, an Ontology extended relation is an ordinary relation as well as an associated ontology.


Ontology Suppose ∑ is some finite set of strings

and S is some set. An ontology w.r.t. ∑ is a partial mapping Θ from ∑ to hierarchies for S.

For example, ∑ = {isa, part_of, affects} A hierarchy can be regarded as a Hasse

diagram associated with a partial ordering. We provide formal definition in our paper.


Ontology Example

theft

mugging

arson

burglary

Ontology associated with Claims1 relation (∑ = {isa})

Wheel

Valve

Air Gasket

Hubcap

Spark Plug

Ontology associated with Parts2 relation (∑ = {part_of})


Ontology Integration Example query: Find all the thefts that

involved a cost of over $1000 dollars. Ontology integration is needed to answer

this query when performing binary operations between two ontology extended relations.

Interoperation constraints are needed to specify the connections between ontologies. We consider: x = y, x ≤ y, x ≠ y, x !≤ y, suppose x and y are from two different hierarchies.


Definition of Hierarchy Integration

Suppose (Hi, ≤i), 1≤i ≤n are n different hierarchies and suppose IC is a finite set of interoperation constraints. A hierarchy (H, ≤) is said to be an integration of (Hi, ≤i), 1≤i ≤n iff there are n injective mappings φ1,…,φn from H1,…,Hn respectively to H such that:

(i {1,…,n})x ≤i y φi(x) ≤ φi(y).

(x Hi)(y Hj) (x:i op y:j) IC φi(x) op φj(y).

H1

H2

Hn

.

.

.

H

φ1

φ2

φn


Example of Hierarchy Integration

theft

mugging

arson

burglary

isa hierarchy with Claims1 relation

robbery fireauto-accident

burglary


theft

mugging arsonburglary

fire auto-accident

Integrated isa hierarchy for Claims1 and Claims2

IC = {theft:1 = robbery:2, arson:1 ≤ fire:2}

With the integrated hierarchy, system can recognize that burglaries, muggings and robberies count as thefts.


Canonical Hierarchy Suppose (Hi, ≤i), 1≤i ≤n are n different

hierarchies and suppose IC is a finite set of interoperation constraints. The canonical hierarchy (H*, ≤*) of (Hi, ≤i), 1≤i ≤n is defined as follows. H* is the set of all strongly connected components

of the graph associated with (Hi, ≤i), 1≤i ≤n. If x*, y* H*, then x* ≤ * y* iff either x* = y* or there

exists a directed path from x:i to y:j (for some x:i x* and y:j y* ) in the hierarchy graph associated with (Hi, ≤i), 1≤i ≤n.


Example of Canonical Hierarchy

theft

mugging

arson

burglary


robbery fireauto-accident

burglary


Canonical Hierarchy with Claims1 and Claims2

IC = {theft:1 = robbery:2, arson:1 ≤ fire:2}

theftrobbery

burglary mugging

fire

arson

auto-accident


Theorems about Hierarchy Integrability

Let (Hi, ≤i), 1≤i ≤n be a family of hierarchies and suppose (H*, ≤*) is its canonical hierarchy. Suppose (H, ≤), φ1,…,φn is any arbitrary witness to the integration of (Hi, ≤i), 1≤i ≤n. Then: [x:i] ≤* [y:j] φi(x) ≤ φj(y).

A set (Hi, ≤i), 1≤i ≤n of hierarchies is integrable if and only if the canonical witness of (Hi, ≤i), 1≤i ≤n is a witness to the integrability of (Hi, ≤i), 1≤i ≤n.

This shows how to integrate hierarchies very efficiently: compute canonical hierarchy and check integrability.


Definition of Ontology Integrability

Suppose is some finite set of strings, S is some set, and 1,…,n are ontologies w.r.t. , S. Suppose IC is a finite set of interoperation constraints. The ontologies 1,…,n are integrable iff for every x , 1(x),…, n(x) are integrable.


Definition of OER An ontology extended relation is a

triple (R, S, Hisa), where S is a schema (A1:1, …,An:n), Hisa is an isa hierarchy and the following constraints are satisfied: 1,…,n Tisa

R belowHisa(1) x … x belowHisa

(n)

BelowH() = {’|’≤} dom()


Ontology Extended Relational Algebra (1)

Example query: Find the car parts from Parts1 relation which are more expensive than Wheel in Parts2 relation. Conversion function is needed to answer this query.

Conversion Function: for each pair of types i and j, we assume there exists at most one conversion function i2j : dom(i) dom(j)

Given a term X, Xt is defined as: t.Ai, if X = Ai, where t is a tuple of relation R. , if X = . v, if X = v:.



Operations in simple select conditions: X op Y, op { =, <>, <, , >, }: Let be the least

common supertype of X and Y, then (type(X)2)(Xt) op (type(Y)2)(Yt) is true.

X instance_of Y: Yt T, type(X) ≤H Yt, and Xt dom(Yt). X subtype_of Y: Xt T , Yt T, Xt ≤H Yt.

If c1, c2 are select conditions, c1 c2, c1 c2, and c1 are select conditions.

Complex operations in select conditions: X below Y: X instance_of Y X subtype_of Y. X above Y: Y below X.

The operators instance_of, subtype_of, below and above are applicable to arbitrary hierarchies.



Suppose (R1, S1, H1),…,(RZ, SZ, HZ) are ontology extended relations, F is a fusion of H1,…,HZ via witness trF.

If E is a relation Ri, [E]F = (R, S, F), where R = trF(Ri), S = (A1:trF(1), …, An: trF(n)).

If E is Ai1,…, Aik(E’) (1 ij n, 1 j k) and if [E’]F = (R’,

(A1:1, …, An:n), F), then [E]F = (R, S, F), where R = Ai1,

…, Aik(R’) and S = (Ai1

:i1, …, Aik

:ik).

If E is E1 x E2 and [Ei]F = (Ri, Si, F), (i = 1, 2), then [E]F = (R, S, F), where R = R1 x R2, S = S1S2.

If E is c(E’), [E’]F = (R’, S, F), then [E]F = (R, S, F), where R = {t R’ (R’, S, F), t |= c}.


Example of Selection Example query: Find all the items

from Parts1 relation which are parts of Tire.

To answer this query: Ontology of Parts1 including part_of

hierarchy. Retrieve the set of subtypes of Tire with

regard to part_of relationship. Transform the query based on the set of

subtypes.


Example of Join Example query: Find the items from

Claims2 relation which are a kind of theft and cost more than the item theft in Claims1 relation.

To answer this query: Integrated ontology of Claims1 and Claims2

including isa hierarchy. Conversion function between the

corresponding units. Transform the query with regard to the

ontology and conversion function.



If E = E1 op E2 where op {, , }, and [Ei]F = (Ri, Si, F), (i=1,2), and S1, S2 have a least common super schema S, then [E]F = (R, S, F), where R = S12S(R1) op S22S(R2).

If E = (S)E’, where S is a schema and [E’]F = (R, S’, F), then [E]F = (S’2S(R), S, F).


Example of Union Example query: Find all the items from

Claims1 and Claims2 that are a kind of theft and involve a cost of over $1000 dollars.

To answer this query: Integrated ontology including isa hierarchy which

contains not only values, but also field names, such as Cost and Value.

Conversion function between corresponding units. Compute least common super schema of Claims1

and Claims2. Convert the selected records to the least common

super schema and compute the union of them.


HOME We built the HOME (Heterogeneous

Ontology Management Engine) system to prove the proposed concepts and implement the algorithms.

The main components in HOME: GUI Ontology maker

Rule maker Ontology inference

Query Executor


Current Status of HOME HOME is implemented in Java. Briefly, HOME has the following major

functionalities: Learn ontology from relational and XML data sources. Modify ontology with a rule maker. Browse ontology with zoomable interface. Import ontology from XML files and write ontology

back to XML files. Ontology integration. Ontology extended query processing for relational

data sources and XML sources.


Experimental Results (1)

Performance of HOME for conjunctive selection queries based on GNIS data sets



Performance of HOME for join queries based on GNIS data sets



Join queries with varying selectivity and number of tuples based on GNIS data sets



Performance of ontology integration algorithms


Contributions Theory about ontologies and

ontology integration. Theory about ontology extended

relational algebra. HOME: a platform for ontology-

based data integration.


Related Work Integrate heterogeneous data sources:

TSIMMIS from Stanford HERMES from UMD SIMS from USC DISCO from INRIA and UMD

Ontology algebra Scalable Knowledge Composition Project from

Stanford Focused on computing union, intersection, and

difference of ontologies, instead of answering queries with ontologies.

Did not consider embedding ontologies into existing data models.


Future Work Integrate non-relational data sources,

such as semi-structured sources, textual sources, etc.

More effort on Semantic Web, DAML+OIL, RDF, metadata, etc.

Extension to richer ontology structures. Indexing for ontology based data

retrieval. Scaling ontology integration.


Finally

Thank you!

an ontology-extended relational algebra piero bonatti università di napoli "federico ii"...

Documents

associated ontology

ontology interoperation

relation claims1

arson burglary ontology

relation parts1

value slide

ordinary relation

cost relation claims2