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Logics for Data and Knowledge Representation

Logics for Data and Knowledge Representation. Semantic Matching. Outline. Introduction: Why matching? The matching problem Kinds of matching: Syntactic versus Semantic Steps in matching Kinds of schemas S-Match MinSMatch. Approaching the heterogeneity problem.

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Logics for Data and Knowledge Representation

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  1. Logics for Data and KnowledgeRepresentation Semantic Matching

  2. Outline • Introduction: • Why matching? • The matching problem • Kinds of matching: Syntactic versus Semantic • Steps in matching • Kinds of schemas • S-Match • MinSMatch

  3. Approaching the heterogeneity problem INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Knowledge can be represented using graphs. • These graphs can be very different: • Different structure (RDBs, OODB, XML, thesauri, formal ontologies) • Different conceptualizations: they reflect different visions of the world • They contain different terminology and polysemous terms • They have different degrees of specificity, scope and coverage • They can be expressed in different languages • Heterogeneity of these graphs demands the exposition of relations between them, such as semantically equivalent. 3

  4. The Matching Problem INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Matching Problem: given two finite graphs, finds all nodes in the two graphs that syntactically or semantically correspond to each other. • Given two graph-like structures (e.g., classifications, XML and database schemas, ontologies), a matching operator produces a mapping between the nodes of the graphs. • Solution: A possible solution consists in the conversion of the two graphs in input into lightweight ontologies and then matching them semantically. 4

  5. A Matching Problem ? ? ? INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH 5

  6. Kinds of Matching: Syntactic versus Semantic (I) INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Syntactic matching Matching of nodes as objects or strings (without meaning) “Virus” partially matches with “Computer virus” “Wives” is an exceptional form for “Wife” • Semantic matching Matching of nodes as concepts “Car” is equivalent to “Automobile” Car ≡ Automobile “Dog” is more specific than “Animal” Dog ⊑ Animal

  7. Kinds of Matching: Syntactic versus Semantic (II) INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Syntactic matching Relations are computed between labels at nodes. The similarity is given in a range [0,1]. Most of the tools developed so far are syntactic. • Semantic matching Relations are computed between concepts at nodes. They return a semantic similarity, namely R ∈ {,≡, ⊑, ⊒, ⊓}, sometimes with a confidence value in the range [0,1]. There are a few tools of this kind, but they are recently increasing. They return different kinds of semantic relations. 7

  8. Steps in matching INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • A matching problem can be decomposed in three steps: • Extract the graphs from the conceptual models under consideration; • Convert the graphs into formal ontologies • Match the formal ontologies • In the next slides we show some examples of step 1

  9. Relational DB Schemas INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Let us consider the following relational database (RDB) model, say “BANK”: • We can represent the RDB model “BANK” as a graph (i.e. a tree) with root “BANK”.

  10. Relational DB Schemas: Representation #1 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • The RDB model is first partitioned into relations, then attributes and data instances.

  11. Relational DB Schemas: Representation #2 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • The model is partitioned into relations, then into tuples, attributes and data instances.

  12. Relational DB Schemas: notes INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • 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) and then approximated into trees.

  13. OODB Schemas INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • 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:

  14. OODB Schemas: notes INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • OODB schemas capture more semantics than the RDBs. • 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.

  15. Semi-structured Data INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Neither RDBs nor OODBs capture all the features of semi-structuredorunstructured 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.

  16. XML Schemas INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • XML schemas can be represented as DAGs. • The graph from the RDB “BANK” could also be obtained from an XML schema.

  17. XML Schemas: notes INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • 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.

  18. Concept Hierarchies INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • A concept hierarchy is asemi-formal conceptualization of an application domain in terms of concepts and relationships.

  19. S-Match: the matching problem INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Semantic Matching Given two graphs G and H, for any node niG and mj H, find the strongest semantic relation R holding between them • The strength of the relations is given by a partial order, that is: disjointness (), equivalence (≡), more/less specific (⊑,⊒), overlap (⊓). • A mapping elementis a 4-tuple <IDij, ni, mj , R>, where: • IDij is a unique identifier of the given mapping element; • ni is the i-th node of the first graph; • mjis the j-th node of the second graph; • R specifies a semantic relation between the concepts at the given nodes • A mapping is a set of mapping elements 19

  20. ⊑ Europe Images 1 1 ≡ Europe Pictures Wine and Cheese 2 2 3 ⊒ 5 4 4 3 Austria Italy Austria Italy Example INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH < ID22, 2, 2, ≡ > < ID21, 2, 1, ⊑ > < ID24, 2, 4, ⊒ > A mapping A mapping element 20

  21. S-Match: the algorithm INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH Four Macro Steps: Given two labeled trees T1 and T2, do: • For all labels in T1 and T2 compute concepts at labels • For all nodes in T1 and T2 compute concepts at nodes • For all pairs of labels in T1 and T2 compute relations between concepts at labels • For all pairs of nodes in T1 and T2 compute relations between concepts at nodes • Steps 1 and 2 constitute the preprocessing phase (off-line), and are executed once and each time after the schema/ontology is changed (graphs are converted into lightweight ontologies) • Steps 3 and 4 constitute the matching phase (on-line), and are executed every time the two schemas/ontologies are to be matched (the two lightweight ontologies are matched) 21

  22. Step 1: compute concepts at labels INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Natural language expressions are translated into a formal language • Concepts are computed by disambiguating among all possible senses of words in a label and their interrelations Computation: • Tokenization. Labels are parsed into tokens. • Lemmatization. Tokens are morphologically analyzed in order to find their possible basic forms. • Building atomic concepts. An oracle (WordNet) is used to extract senses of lemmatized tokens. • Building complex concepts. Prepositions, conjunctions, etc. are translated into logical connectives and used to build complex concepts out of the atomic concepts. 22

  23. Step 1: compute concepts at labels INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Tokenization. “Images and text” <Images, and, text>; • Lemmatization. “Images”  Image; • Building atomic concepts. “Image” has 8 senses in WordNet, 7 as a noun and 1 as a verb. Some might be filtered out analyzing the context. The rest are kept: “Image”  Image#1 ⊔ Image#3 ⊔ Image#7; • Building complex concepts  (Image#1 ⊔ Image#3 ⊔ Image#7) ⊔ text#2 23

  24. Step 2: compute concepts at nodes INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Concepts at labels are extended by taking into account the context of the node, i.e. the position of the node in the tree Computation: • The concept at a node for some node n is computed as the conjunction of the concepts at labels along the path from the root till the node n itself 24

  25. Step 2: compute concepts at nodes Europe 1 Pictures Wine and Cheese 2 3 5 4 Austria Italy INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH C4= CEurope⊓CPictures ⊓CItaly 25

  26. Step 3: compute relations between concepts at labels INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Exploit a priori knowledge, e.g., lexical, domain knowledge with the help of element level semantic matchers Computation: • Concepts at labels of each pair of nodes from the two trees are compared to determine semantic relations between them (without caring about their context) • The set of semantic relations computed with this step constitute the set of axioms that will be used to compute the mapping, i.e. our TBox! 26

  27. Step 3: compute relations between concepts at labels T1 T2 Europe Images 1 1 Wine and Cheese Pictures Europe 2 2 3 T1 T2 CEurope CPictures CItaly CAustria Italy CWine CCheese 4 5 Austria Austria Italy 3 4 CImages ≡ CEurope ≡ CAustria  ≡ CItaly ≡  INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH 27

  28. Step 4: compute relations between concepts at nodes INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • The matching problem is reduced to a set of node matching problems • Each node matching problem is reduced to a validity problem • TBox T: the relations between concepts at labels (from step 3) • Given the pair of nodes (n, m), deciding if a given semantic relation holds between them corresponds to verifying that a certain relation holds between corresponding concepts at node: • Subsumption: T ⊨ Cn⊑ Cm or T ⊨ Cn⊒ Cm • Equivalence: T ⊨ Cn⊑ Cm and T ⊨ Cn⊒ Cm • Disjointness: T ⊨ Cn ⊓ Cm⊑ • This is done by eliminating the TBox T and verifying that the negation of each formula is unsatisfiable. • The strongest relation holding between the two nodes is returned

  29. T1 T2 C21 C22 C23 C24 C25 C11 C12 = C13  C14 =  Step 4: compute relations between concepts at nodes INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • Suppose we want to check if C12≡C22 (C1Images  C2Pictures)  (C1Europe  C2Europe)  (C12  C22 ) Goal Context (the relevant part of the TBox)

  30. T1 T1 T1 T1 T2 T2 T2 T2 Europe Europe Europe Europe Images Images Images Images 1 1 1 1 1 1 1 1 Wine and Cheese Wine and Cheese Wine and Cheese Wine and Cheese Pictures Pictures Pictures Pictures Europe Europe Europe Europe 2 2 2 2 2 2 2 2 3 3 3 3 Italy Italy Italy Italy 4 4 4 4 5 5 5 5 Austria Austria Austria Austria Austria Austria Austria Austria Italy Italy Italy Italy 3 3 3 3 4 4 4 4 The final result INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH ≡ 

  31. Problems with S-Match INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH JOURNALS journals#1 ⊑ PROGRAMMING AND DEVELOPMENT programming#2 ⊔ development#1 A D ⊒ ⊑ ⊑ ⊑ DEVELOPMENT AND PROGRAMMING LANGUAGES (development#1 ⊔ programming#2) ⊓ languages#3 ⊓ journals#1 LANGUAGES languages#3 ⊓ (programming#2 ⊔ development#1) ⊑ B E ⊒ ⊑ ⊑ ⊑ JAVA (development#1 ⊔ programming#2) ⊓ languages#3 ⊓ journals#1 ⊓ Java#3 JAVA Java#3⊓ languages#3 ⊓ (programming#2 ⊔ development#1) C F ⊑ ≡ G MAGAZINES Magazines#1 ⊓ Java#3⊓ languages#3 ⊓ (programming#2 ⊔ development#1) • It is slow. Can we make the algorithm faster? • Too many relations between nodes are returned by S-Match. • Are there some relations which are “more important” than others? 31

  32. (1) (3) (4) (2) A A A D A D D D ≡ ≡ ⊑ ⊒ ⊥ ≡ B E B E B E B E ⊒ ⊑ ⊥ ⊥ ⊥ F C F C C F C F Redundancy patterns INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH • There are some axioms (the dashed arrows) which can be derived from others (the solid ones) • Note that (4) is a combination of (1) and (2) • Note that (1) and (2) are still true in case of equivalence 32

  33. A redundant mapping element is a mapping element which can be derived from the others using the patterns. A redundant mapping is a set containing redundant mapping elements The minimal mapping is the biggest subset of mapping elements among those without redundant elements The minimal mapping always exists and it is unique Advantages in visualization, validation and maintenance The mapping of maximum size is the set containing the maximum number of mapping elements it can be obtained from the propagation of the elements in the minimal set using the intuitions encoded in the patterns. Minimal and redundant mappings INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH 33

  34. The minimal mapping INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH JOURNALS journals#1 ⊑ PROGRAMMING AND DEVELOPMENT programming#2 ⊔ development#1 A D ⊒ ⊑ ⊑ ⊑ DEVELOPMENT AND PROGRAMMING LANGUAGES (development#1 ⊔ programming#2) ⊓ languages#3 ⊓ journals#1 LANGUAGES languages#3 ⊓ (programming#2 ⊔ development#1) ⊑ B E ⊒ ⊑ ⊑ ⊑ JAVA (development#1 ⊔ programming#2) ⊓ languages#3 ⊓ journals#1 ⊓ Java#3 JAVA Java#3⊓ languages#3 ⊓ (programming#2 ⊔ development#1) C F ⊑ ≡ G MAGAZINES Magazines#1 ⊓ Java#3⊓ languages#3 ⊓ (programming#2 ⊔ development#1) 34

  35. Computing the minimal mapping M: function TreeMatch(tree T1, tree T2) { TreeDisjoint(root(T1),root(T2)); direction := true; TreeSubsumedBy(root(T1),root(T2)); direction := false; TreeSubsumedBy(root(T2),root(T1)); TreeEquiv(); }; Computing the set of maximum size: function Propagate(M) MinSMatch: the algorithm INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH (3) (1) (2) (4), from (1) and (2) 35

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