1. Logics for Data and Knowledge Representation 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 2

3. Approaching the heterogeneity problem  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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

4. The Matching Problem  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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

6. Kinds of Matching: Syntactic versus Semantic (I)  Syntactic matching Matching of nodes as objects or strings (without meaning) 6 “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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

7. Kinds of Matching: Syntactic versus Semantic (II)  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. 7  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. INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

8. Steps in matching  A matching problem can be decomposed in three steps: 1. Extract the graphs from the conceptual models under consideration; 2. Convert the graphs into formal ontologies 3. Match the formal ontologies  In the next slides we show some examples of step 1 8 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

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

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

12. 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) and then approximated into trees. 12 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

14. OODB Schemas: notes  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. 14 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

15. 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. 15 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

17. 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. 17 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

19. S-Match: the matching problem  Semantic Matching Given two graphs G and H, for any node n i  G and m j  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 element is a 4-tuple, where:  ID ij is a unique identifier of the given mapping element;  n i is the i-th node of the first graph;  m j is 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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

21. S-Match: the algorithm Four Macro Steps: Given 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 nodes 3. For all pairs of labels in T1 and T2 compute relations between concepts at labels 4. 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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

22. Step 1: compute concepts at labels  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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

23. Step 1: compute concepts at labels  Tokenization. “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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

24. Step 2: compute concepts at nodes  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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

26. Step 3: compute relations between concepts at labels  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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

27. Step 3: compute relations between concepts at labels 27 Italy Europe Wine and Cheese Austria Pictures 1 23 45 Europe Italy Austria 2 34 1 Images T1T2  ≡ C Italy ≡  C Austria ≡ C Europe ≡ C Images C Austria C Italy C Pictures C Europe T1 C Wine C Cheese INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

28. Step 4: compute relations between concepts at nodes 28  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 ⊨ C n ⊑ C m or T ⊨ C n ⊒ C m  Equivalence: T ⊨ C n ⊑ C m and T ⊨ C n ⊒ C m  Disjointness: T ⊨ C n ⊓ C m ⊑   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 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

29. Step 4: compute relations between concepts at nodes  Suppose we want to check if C1 2 ≡ C2 2 29 T2  = C1 4  C1 3 = C1 2 C1 1 C2 5 C2 4 C2 3 C2 2 C2 1 T1 (C1 Images  C2 Pictures )  (C1 Europe  C2 Europe )  (C1 2  C2 2 ) Context (the relevant part of the TBox) Goal INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

30. The final result 30 Italy Europe Wine and Cheese Austria Pictures 1 23 45 Europe Italy Austria 2 34 1 Images T1T2 ≡ Italy Europe Wine and Cheese Austria Pictures 1 23 45 Europe Italy Austria 2 34 1 Images T1T2 Italy Europe Wine and Cheese Austria Pictures 1 23 45 Europe Italy Austria 2 34 1 Images T1T2  Italy Europe Wine and Cheese Austria Pictures 1 23 45 Europe Italy Austria 2 34 1 Images T1T2 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

31. A BE D JOURNALS journals#1 C DEVELOPMENT AND PROGRAMMING LANGUAGES (development#1 ⊔ programming#2) ⊓ languages#3 ⊓ journals#1 JAVA (development#1 ⊔ programming#2) ⊓ languages#3 ⊓ journals#1 ⊓ Java#3 PROGRAMMING AND DEVELOPMENT programming#2 ⊔ development#1 F G LANGUAGES languages#3 ⊓ (programming#2 ⊔ development#1) JAVA Java#3 ⊓ languages#3 ⊓ (programming#2 ⊔ development#1) 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? Problems with S-Match 31 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

32. A BE D C F ⊑ ⊑ A BE D CF ⊒ ⊒ A BE D CF ⊥ ⊥ ⊥ A BE D CF ≡≡ ≡ (1) (2) (3)(4) ⊥  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 Redundancy patterns 32 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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 33 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

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

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) (3) (1) (2) (4), from (1) and (2) MinSMatch: the algorithm 35 INTRODUCTION :: KINDS OF MATCHING :: STEPS :: KINDS OF SCHEMAS :: S-MATCH :: MINSMATCH

36. S-Match Lab 36 S-MATCH LAB Java: JRE or JDK (preferred) Java IDE: Eclipse, IDEA, NetBeans, … Text Editor