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### An Ontology-Extended Relational Algebra

Piero Bonatti

Università di Napoli "Federico II" Yu Deng

V.S. Subrahmanian

University of Maryland College Park

Outline

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

Ontology Extended Algebra

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.

Ontology Extended Algebra

Approach

- Associate ontologies to data sources.
- Ontology interoperation.
- Extend relational data model and relational algebra.

Ontology Extended 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)

Ontology Extended Algebra

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.

Ontology Extended Algebra

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 Algebra

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 Extended Algebra

Ontology

- Suppose ∑ is some finite set of strings and S is some set. An ontologyw.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 Extended Algebra

arson

Air Gasket

Spark Plug

Wheel

burglary

mugging

Hubcap

Valve

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

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

Ontology Extended Algebra

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.

Ontology Extended Algebra

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 ≤iyφi(x) ≤ φi(y).
- (x Hi)(y Hj) (x:iopy:j) IC φi(x)opφj(y).

φ1

H1

φ2

H2

H

.

φn

.

.

Hn

Ontology Extended Algebra

arson

burglary

mugging

fire

auto-accident

theft

arson

burglary

mugging

robbery

auto-accident

fire

burglary

Example of Hierarchy Integrationisa hierarchy with Claims1 relation

Integrated isa hierarchy for Claims1 and Claims2

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

isa hierarchy with Claims2 relation

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.

Ontology Extended Algebra

arson

theft

robbery

fire

auto-accident

burglary

mugging

arson

burglary

mugging

robbery

auto-accident

fire

burglary

Example of Canonical Hierarchyisa hierarchy with Claims1 relation

Canonical Hierarchy with Claims1 and Claims2

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

isa hierarchy with Claims2 relation

Ontology Extended Algebra

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.

Ontology Extended Algebra

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 Algebra

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:.

Ontology Extended Algebra

Ontology Extended Relational Algebra (2)

- Operations in simple select conditions:
- XopY, 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) ≤HYt, and Xt dom(Yt).
- X subtype_of Y: Xt T , Yt T, Xt≤HYt.
- 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 YX subtype_of Y.
- X above Y: Y below X.
- The operators instance_of, subtype_of, below and above are applicable to arbitrary hierarchies.

Ontology Extended Algebra

Ontology Extended Relational Algebra (3)

- 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 = {tR’ (R’, S, F), t|= c}.

Ontology Extended Algebra

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.

Ontology Extended Algebra

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.

Ontology Extended Algebra

Ontology Extended Relational Algebra (4)

- 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)opS22S(R2).
- If E = (S)E’, where S is a schema and [E’]F = (R, S’, F), then [E]F = (S’2S(R), S, F).

Ontology Extended Algebra

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.

Ontology Extended Algebra

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

Ontology Extended Algebra

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.

Ontology Extended Algebra

Experimental Results (1)

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

Experimental Results (2)

Performance of HOME for join queries based on GNIS data sets

Experimental Results (3)

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

Experimental Results (4)

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.

Ontology Extended Algebra

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.

Ontology Extended Algebra

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.

Ontology Extended Algebra

Finally

Thank you!

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