LLogics for DData and KKnowledgeRRepresentation

Semantic Matching

Outline Introduction

Matching Problems Graphs Matching

Syntactic vs. Semantic Semantic Relations

Semantic Matching in ClassL

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Introduction An important problem on graphs that we will apply to

richer representations like concept hierarchies, classifications, schemas and ontologies is “matching.”

This problem is also popular in semantic networks, where one may want to check whether a particular concept is present.

E.g.

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Date: a matching for marriage?

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Introduction (cont’) Some popular situations that can be modeled as a

matching problem are:

- Concept matching in semantic networks. - Schema matching in distributed databases. - Ontology matching (ontology “alignment”) in the Semantic Web. - ...

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The Semantic Matching Problem (Example)

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Outline Introduction

Matching Problems Graphs Matching

Syntactic vs. Semantic Semantic Relations

Classification matching L.O. and ClassL

Semantic Matching in ClassL

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Matching Problems There are two kinds of matching:

Syntactic: matching of nodes as objects or strings (so, as such, without meaning).

Semantic: matching of nodes as concepts.

A Matching Problem (syntactic or semantic) is a problem on graphs summarized as:Given two finite graphs, is there a matching between the (nodes of the) two graphs?

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Matching Problems A problem of matching can be decomposed in two

steps:

1. Extract the graphs from the conceptual models under consideration;

2. Match the resulting graphs.

Below we show some examples of step 1. (We follow [Giunchiglia & Shvaiko, 2007].)

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Relational DB Schemas Let us consider the following relational database

(RDB) model, say “BANK”:

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(Giunchiglia & Shvaiko, 2007)

Relational DB Schemas: Representation 1 We can represent the RDB model “BANK” as a graph

(a tree) with root “BANK”:

The RDB model is first partitioned into relations, then attributes and data instances.

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(Giunchiglia & Shvaiko, 2007)

Relational DB Schemas: Representation 2 We can represent the RDB model “BANK” as a graph

(a tree) with root “BANK”:

The model is partitioned into relations, then into tuples, attributes and data instances.

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(Giunchiglia & Shvaiko, 2007)

Relational DB Schemas: NOTEs Which of the two representations is more preferable

depends on the concrete task?

It is always possible to transform one representation into the other.

In contrast to the example of RDB “BANK”, DB schemas are seldom trees. More often, DB schemas are translated into Directed Acyclic Graphs (DAG’s).

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OODB Schemas Let us consider the RDB “BANK” in terms of an

object-oriented DB (OODB) schema:BRANCH (Street, City, Zip) PERSON (F_Name, L_Name) STAFF : PERSON (Position, Salary, Manager)

The resulting graph is:

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(Giunchiglia & Shvaiko, 2007)

OODB Schemas: NOTEs OODB schemas capture more semantics than the

relational DBs.

In particular, an OODB schema: explicitly expresses subsumption relations between

elements; admits special types of arcs for part/whole relationships

in terms of aggregation and composition.

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RDB vs. OODB

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Semi-structured Data Neither RDBs nor OODBs capture all the features of

semi-structured or unstructured data (Buneman, 1997): semi-structured data do not possess a regular structure

(schemaless); the “structure” of semi-structured data could be partial or

even implicit.

Typical examples are: HTML and XML.

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XML Schemas XML schemas can be represented as DAGs. The graph from the RDB “BANK” could also be

obtained from an XML schema.

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(Giunchiglia & Shvaiko, 2007)

XML Schemas: NOTEs Often XML schemas represent hierarchical data

models. In this case the only relationships between the elements

are {is-a}.

Attributes in XML are used to represent extra information about data. There are no strict rules telling us when data should be represented as elements, or as attributes.

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Concept Hierarchies Def. A concept hierarchy is a semi-formal

conceptualization of an application domain in terms of concepts and relationships.

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(Giunchiglia & Shvaiko, 2007)

Concept Hierarchies: NOTEs Examples are classification hierarchies, e.g.,

and directories (catalogs). Classification hierarchies / Web directories are

sometimes referred to as lightweight ontologies (Uschold & Gruninger, 2004). However: They are not ontologies, as they lack of a formal

semantics (semi-formal vs formal.) They don’t formalize class instances.

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Semantic Matching Introduction

Matching Problems Graphs Matching

Syntactic vs. Semantic Semantic Relations

Semantic Matching via SAT in ClassL

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How to deal with these heterogeneity?

Matching: given two graph-like structures (e.g., concept hierarchies or ontologies), produce a mapping between the nodes of the graphs that semantically correspond to each other

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Syntactic vs. Semantic

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Matching

Semantic Matching

Syntactic Matching

Relations are computed between labels at nodes

R = {x[0,1]}

Relations are computed between concepts at nodes

R = { =, ⊑, ⊒, , ⊓}

Note: all previous systems are syntactic…

Note: needed for proper labeling of context mappings

Semantic Matching Mapping element is a 4-tuple < IDij, n1i, n2j, R >, where

IDij is a unique identifier of the given mapping element;

n1i is the i-th node of the first graph;

n2j is the j-th node of the second graph;

R specifies a semantic relation between the concepts at the given nodes

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Semantic Matching: Given two graphs G1 and G2, for any node n1i G1, find the strongest semantic relation R’ holding with node n2j G2

Computed R’s, listed in the decreasing binding strength order: equivalence { = }; more general/specific {⊒,⊑}; mismatch { }; overlapping {⊓}.

< ID21, 2, 1, ⊑ >

Example

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< ID22, 2, 2, = >

=

?

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< ID22, 2, 2, = >

< ID24, 2, 4, ⊒ >

Step 4

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Images

Europe

ItalyAustria

2

3 4

1

Italy

Europe

Wine and Cheese

Austria

Pictures

1

2 3

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S-Match AlgorithmFour Macro StepsGiven two labeled trees T1 and T2, do:1. For all labels in T1 and T2 compute concepts at labels 2. For all nodes in T1 and T2 compute concepts at nodes3. For all pairs of labels in T1 and T2 compute relations between

concepts at labels4. For all pairs of nodes in T1 and T2 compute relations between

concepts at nodes

Steps 1 and 2 constitute the preprocessing phase, and are executed once and each time after the schema/ontology is changed (OFF- LINE part)

Steps 3 and 4 constitute the matching phase, and are executed every time the two schemas/ontologies are to be matched (ON - LINE part)

Step 1: compute concepts at labels The idea:

Translate natural language expressions into internal formal language Compute concepts based on possible senses of words in a label and their

interrelations Preprocessing:

Tokenization. Labels (according to punctuation, spaces, etc.) are parsed into tokens. E.g., Wine and Cheese <Wine, and, Cheese>;

Lemmatization. Tokens are morphologically analyzed in order to find all their possible basic forms. E.g., Images Image;

Building atomic concepts. An oracle (WordNet) is used to extract senses of lemmatized tokens. E.g., Image has 8 senses, 7 as a noun and 1 as a verb;

Building complex concepts. Prepositions, conjunctions, etc. are translated into logical connectives and used to build complex conceptsout of the atomic concepts

E.g., CWine and Cheese = <Wine, U(WNWine)> ⊔ <Cheese, U(WNCheese)>,

where ⋃ is a union of the senses that WordNet attaches to lemmatized tokens

Step 2: compute concepts at nodes The idea: extend concepts at labels by capturing the knowledge

residing in a structure of a graph in order to define a context in which the given concept at a label occurs

Computation (basic case): Concept at a node for some node n is computed as an intersection of concepts at labels located above the given node, including the node itself

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Wine and Cheese

Italy

Europe

Austria

Pictures

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2 3

4 5

C4 = Ceurope ⊓CPictures⊓CItaly

Step 3: compute relations between concepts at labels

The idea: Exploit a priori knowledge, e.g., lexical, domain knowledge with the help of element level semantic matchers

Results of step 3:

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Italy

Europe

Wine and

Cheese

Austria

Pictures

1

2 3

4 5

Europe

Italy Austria

2

3 4

1

Images

T1 T2

T2

=CItaly

=CAustria

=CEurope

=CImages

CAustriaCItalyCPicturesCEuropeT1

CWine CCheese

Step 4? Wait! What we have done in Step 3?

Find semantic relations. What for?

To build the semantic relation bases for further matching.

What is a ‘relation base’ in the logic sense?

TBox!

We are building the TBox!

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Step 4: compute relations between concepts at nodes

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The idea: Reduce the matching problem to a validity problem

Context: the relations between concepts at labels (from step 3)

Wffrel (C1i, C2j): relation to be proved (with: C1i in Tree1 and C2j in Tree2)

Translate into propositional logic: C1i = C2j is translated into C1i C2j C1i subsumes C2j is translated into C1i C2j C1i C2j is translated into ¬ (C1i C2j)

Prove Context Wffrel (C1i, C2j) is valid with Rel={=,⊑,⊒,⊥,⊓}

A propositional formula is valid iff its negation is unsatisfiable (SAT deciders are sound and complete…)

Pseudo code of Step4 CHANGE LINE15 1. i, j, N1, N2: int; 2. context, goal: wff; 3. n1, n2: node; 4. T1, T2: tree of (node); 5. relation = {=, ⊑ , ⊒ , }; 6. ClabMatrix(N1, N2), CnodMatrix(N1, N2), relation: relation

7. function mkCnodMatrix(T1, T2, ClabMatrix) { 8. for (i = 0; i < N1; i++) do 9. for (j = 0; j < N2; j++) do 10. CnodMatrix(i, j):=NodeMatch(T1(i),T2(j), ClabMatrix)}

11. function NodeMatch(n1, n2, ClabMatrix) { 12. context:=mkcontext(n1, n2, ClabMatrix, context); 13. foreach (relation in < =, ⊑ , ⊒ , >) do { 14. goal:= w2r(mkwff(relation, GetCnod(n1), GetCnod(n2)); 15. if VALID(mkwff(, context, goal)) 16. return relation;} 17. return IDK;}

Validity Reasoning

Contex|=goal?

Example Example. Suppose we want to check if C12 = C22

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T2

=C14

C13

=C12

C11

C25C24C23C22C21T1

(C1Images C2Pictures) (C1Europe C2Europe) (C12 C22 )

Context Goal

Examples

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Italy

Europe

Wine and

Cheese

Austria

Pictures

1

2 3

4 5

Europe

Italy Austria

2

3 4

1Images

T1 T2=

Italy

Europe

Wine and

Cheese

Austria

Pictures

1

2 3

4 5

Europe

Italy Austria

2

3 4

1Images

T1 T2

Italy

Europe

Wine and

Cheese

Austria

Pictures

1

2 3

4 5

Europe

Italy Austria

2

3 4

1Images

T1 T2

Italy

Europe

Wine and

Cheese

Austria

Pictures

1

2 3

4 5

Europe

Italy Austria

2

3 4

1Images

T1 T2

Another Example: Concept Hierarchies

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(Serafini et. al., 2003)

Example (cont’) Suppose we want to discover the relation R between

Chat and Forum in the Google directory (left) and Chat and Forum in the Yahoo directory (right):

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Example (cont’) Step 1 : transformation of nodes

n1 = Chat and Forum and n2 = Chat and Forum to propositions, P and Q; selection of the portion T of knowledge (background theory) relevant to the application of a SAT solver. WordNet is used at this step to build T.

Step 2 : select the relation R between n1 and n2 among the semantic relations of interest.

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Example (cont’) Step 3 : The question “Is Chat and Forum less

general than Chat and Forum?” becomes the SAT problem “Is T |= P ⊑ Q?” where:

P = (art#1 ⊓ literature#2 ⊓(chat#1 ⊔ forum#1)),Q = (art#1⊔humanities#1)⊓humanities#1⊓(chat#1⊔ forum#1),

T = {art#1⊑ humanities#1, humanities#1⊒ literature#2}

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References( Preprints available at http://dit.unitn.it/~ldkr/#Biblio )

F. Giunchiglia, P. Shvaiko, “Semantic matching.” Knowledge Engineering Review, 18(3):265-280, 2003.

L. Serafini, P. Bouquet, B. Magnini, S. Zanobini, “An Algorithm for Matching Contextualized Schemas via SAT.” IRST Technical Report 0301-06, ITC, January 2003.

F. Giunchiglia, M. Marchese, I. Zaihrayeu. “Encoding Classifications into Lightweight Ontologies.” J. of Data Semantics VIII, Springer-Verlag LNCS 4380, pp 57-81, 2007.

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