An Ontology-Extended Relational Algebra

1
An Ontology-Extended Relational Algebra

• Piero Bonatti
• Università di Napoli "Federico II" Yu Deng
• V.S. Subrahmanian
• University of Maryland College Park

2
Outline

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

3
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 users assumptions
  of semantic meaning about the terms being used.

4
Approach

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

5
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)

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Parts1 and Parts2 Relations
Parts1 relation
Parts2 relation
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
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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.

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Claims1 and Claims2 Relation
Claims1 relation
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
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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.

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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.

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Ontology

• Suppose ? is some finite set of strings and S is
  some set. An ontology w.r.t. ? is a partial
  mapping T 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.

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Ontology Example
Ontology associated with Claims1 relation (?
isa)
Ontology associated with Parts2 relation (?
part_of)
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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.

14
Definition of Hierarchy Integration

• Suppose (Hi, i), 1i 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), 1i n iff
  there are n injective mappings f1,,fn from
  H1,,Hn respectively to H such that
• (?i ? 1,,n)x i y ? fi(x) fi(y).
• (?x ? Hi)(?y ? Hj) (xi op yj) ? IC ? fi(x) op
  fj(y).

f1
H1
f2
H2
H
.
fn
.
.
Hn
15
Example of Hierarchy Integration
isa hierarchy with Claims1 relation
Integrated isa hierarchy for Claims1 and Claims2
IC theft1 robbery2, arson1 fire2
isa hierarchy with Claims2 relation
With the integrated hierarchy, system can
recognize that burglaries, muggings and robberies
count as thefts.
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Canonical Hierarchy

• Suppose (Hi, i), 1i n are n different
  hierarchies and suppose IC is a finite set of
  interoperation constraints. The canonical
  hierarchy (H, ) of (Hi, i), 1i n is defined
  as follows.
• H is the set of all strongly connected
  components of the graph associated with (Hi, i),
  1i n.
• If x, y ? H, then x y iff either x
  y or there exists a directed path from xi to
  yj (for some xi ? x and yj ? y ) in the
  hierarchy graph associated with (Hi, i), 1i n.

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Example of Canonical Hierarchy
isa hierarchy with Claims1 relation
Canonical Hierarchy with Claims1 and Claims2
IC theft1 robbery2, arson1 fire2
isa hierarchy with Claims2 relation
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Theorems about Hierarchy Integrability

• Let (Hi, i), 1i n be a family of hierarchies
  and suppose (H, ) is its canonical hierarchy.
  Suppose (H, ), f1,,fn is any arbitrary witness
  to the integration of (Hi, i), 1i n. Then
  xi yj ? fi(x) fj(y).
• A set (Hi, i), 1i n of hierarchies is
  integrable if and only if the canonical witness
  of (Hi, i), 1i n is a witness to the
  integrability of (Hi, i), 1i n.

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

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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(?)

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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 ?i2?j 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?.

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Ontology Extended Relational Algebra (2)

• Operations in simple select conditions
• X op Y, op ? , ltgt, lt, ?, gt, ? 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.

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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, EF (R, S, ?F), where R
  trF(Ri), S (A1trF(?1), , An trF(?n)).
• If E is ?Ai1,, Aik(E) (1 ? ij ? n, 1 ? j ? k)
  and if EF (R, (A1?1, , An?n), ?F), then
  EF (R, S, ?F), where R ?Ai1,, Aik(R) and
  S (Ai1?i1, , Aik?ik).
• If E is E1 x E2 and EiF (Ri, Si, ?F), (i 1,
  2), then EF (R, S, ?F), where R R1 x R2, S
  S1S2.
• If E is ?c(E), EF (R, S, ?F), then EF
  (R, S, ?F), where R t ? R ? (R, S, ?F), t
  c.

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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.

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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.

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Ontology Extended Relational Algebra (4)

• If E E1 op E2 where op ??, ?, ?, and EiF
  (Ri, Si, ?F), (i1,2), and S1, S2 have a least
  common super schema S, then EF (R, S, ?F),
  where R S12S(R1) op S22S(R2).
• If E (S)E, where S is a schema and EF (R,
  S, ?F), then EF (S2S(R), S, ?F).

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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.

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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

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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.

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Experimental Results (1)
Performance of HOME for conjunctive selection
queries based on GNIS data sets
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Experimental Results (2)
Performance of HOME for join queries based on
GNIS data sets
32
Experimental Results (3)
Join queries with varying selectivity and number
of tuples based on GNIS data sets
33
Experimental Results (4)
Performance of ontology integration algorithms
34
Contributions

• Theory about ontologies and ontology integration.
• Theory about ontology extended relational
  algebra.
• HOME a platform for ontology-based data
  integration.

35
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.

36
Future Work

• Integrate non-relational data sources, such as
  semi-structured sources, textual sources, etc.
• More effort on Semantic Web, DAMLOIL, RDF,
  metadata, etc.
• Extension to richer ontology structures.
• Indexing for ontology based data retrieval.
• Scaling ontology integration.

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Finally

• Thank you!