An Algebra of Lightweight Ontologies: Implementation and Applications

Transcription

1 An Algebra of Lightweight Ontologies: Implementation and Applications Marco A. Casanova Departamento de Informática, PUC-Rio NII Shonan Meeting: Implicit and explicit semantics integration in proof based developments of discrete systems Shonan Village Center (SVC), Japan November 22-25, 2016

2 Topics Introduction A Formal Framework Operations over Ontologies Implementation of the Operations Applications Conclusions 2

3 Introduction Question How to design an export ontology to publish data on the Web so that an application can understand the data? 3

4 Introduction Question rephrased How to design an export ontology (and an application ontology) so that matching the application ontology with the export ontology becomes trivial? 4

5 Introduction Suggested Answer An export ontology (and an application ontology) should be a combination of fragments of one or more well-known domain ontologies so that matching the application ontology with the export ontology becomes trivial van Valckenborch, Lucas. The Tower of Babel Oil on panel cm Galerie de Jonckheere, Paris, France Standards for everything Domain ontologies Object Ids 5

6 Introduction New Question!! What is the meaning of a combination of fragments of one or more domain ontologies? van Valckenborch I, Marten. The Building of the Tower of Babel (more paintings of the Tower of Babel at 6

7 Introduction Others Questions How to compare two ontologies? How to compare two versions of the same ontology? How to design a mediated ontology? 7

8 Topics Introduction A Formal Framework Operations over Ontologies Implementation of the Operations Applications Conclusions 8

9 A Formal Framework Lightweight Constraints E F 9

10 A Formal Framework Constraint Graphs A Ç B = Æ iff A Í B iff B Í A 10

11 A Formal Framework A Decision Procedure The IMPLIES procedure A sound and complete procedure to test logical implication for lightweight constraints 11

12 A Formal Framework A Decision Procedure A Ç B = Æ iff A Í B iff B Í A 12

13 Constraint graphs inspired on a 2-SAT solver A Formal Framework A Decision Procedure Completeness proof Constructs a Herbrand model Uses constants to represent classes Uses function symbols to represent number restrictions Depends heavily on the fact that the left-hand side of a lightweight inclusion is a positive expression 13

14 Topics Introduction A Formal Framework Operations over Ontologies Implementation of the Operations Applications Conclusions 14

15 Operations over Ontologies Operations Create new ontologies, including their constraints, out of other ontologies Treat an ontology O=(V,S) as a theory, i.e., a set of constraints t[s] Constraints Constraints Operation Constraints 15

16 Operations over Ontologies Useful operations: Projection The projection of O 1 = (V 1,S 1 ) over W, denoted p[w](o 1 ), returns the ontology O P = (V P,S P ), where V P = W and S P is the set of constraints int[s 1 ] that use only classes and properties in W Deprecation The deprecation of Y from O 1 = (V 1,S 1 ), denoted d[y](o 1 ), returns the ontology O D = (V D,S D ), where V D = V 1 and S D = S 1 - Y 16

17 Operations over Ontologies Union The union of O 1 = (V 1,S 1 ) and O 2 = (V 2,S 2 ), denoted O 1 È O 2, returns the ontology O U = (V U,S U ), where V U = V 1 È V 2 and S U = S 1 È S 2 Intersection The intersection of O 1 = (V 1,S 1 ) and O 2 = (V 2,S 2 ), denoted O 1 Ç O 2, returns the ontology O N = (V N,S N ), where V N = V 1 Ç V 2 and S N = t[s 1 ] Ç t[s 2 ] Difference The difference of O 1 = (V 1,S 1 ) and O 2 = (V 2,S 2 ), denoted O 1 - O 2, returns the ontology O F = (V F,S F ), where V F = V 1 and S F = t[s 1 ] - t[s 2 ] 17

18 Operations over Lightweight Ontologies Computing the operations Union (must check if the new set of constraints implies e ^, for some e) Projection, Intersection implemented as variants of IMPLIES use the transitive closure of the constraint graphs Difference (To be further investigated) 18

19 Operations over Lightweight Ontologies Projection Input: an ontology O 1 = (V 1,S 1 ) and a vocabulary W Í V 1 Output: an ontology O P = (W,S P ), where S P is a set of constraints tautologically equivalent to the set of constraints in t[s 1 ] that use only symbols in W 1. Generate G(S 1 ), the constraint graph of S Compute the transitive closure G*(S 1 ) of G(S 1 ). 3. Mark all nodes of G*(S 1 ) that are labeled with expressions that use only symbols in W. 4. Generate a set of constraints S P that correspond to: a) Arcs of G*(S 1 ) connecting marked nodes; and b) Expressions (in W) that label the same marked node. 19

20 Operations over Lightweight Ontologies Optimization Problem: The transitive closure G*(S 1 ) contains redundancies! Solution: The problem is equivalent to finding the minimum equivalent graph (MEG) of a graph G, defined as the graph G with the minimum set of edges such that the transitive closure of G and G are equal Finding the minimum equivalent graph has a polynomial solution for acyclic graphs and is NP-hard for strongly connected graphs 20

21 Topics Introduction A Formal Framework Operations over Ontologies Implementation of the Operations Applications Conclusions 21

22 Implementation of the Operations OntologyManagerTab Implemented as a Protégé Plugin Works with lightweight ontologies Offers a friendly user interface Examples: A projection of the FOAF Ontology The intersection of FOAF and the Music Ontology 22

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25 Topics Introduction A Formal Framework Operations over Ontologies Implementation of the Operations Applications Conclusions 25

26 Applications Question How to design an export ontology? How to compare two ontologies? How to design a mediated ontology? Operations Projection, Union, Deprecation Intersection Difference Intersection 26

27 Applications Design of an Export Ontology Example: Goal: Create a new ontology about music artists, solo artists, music groups and (record) labels Strategy: Design the new ontology as a projection of the Music Ontology 27

28 Applications Design of an Export Ontology 28

29 Applications Design of an Export Ontology 29

30 Applications Design of an Export Ontology 30

31 Applications Design of an Export Ontology owl:disjointwith 31

32 Applications Design of an Export Ontology 32

33 Applications Design of an Export Ontology A Ç B = Æ iff A Í B iff B Í A 33

34 Applications Design of an Export Ontology 34

35 Applications Example: Reconstruction of a fragment of the Music Ontology 35

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37 O 1 = p[w](foaf) where W = { foaf:person, foaf:agent, foaf:organization } 37

38 O 2 = (V 2,S 2 ), where V 2 = { mo:group, mo:musicartist, mo:corporatebody, mo:solomusicartist, mo:musicgroup, mo:label } S 2 = {mo:solomusicartist mo:musicartist, mo:musicgroup mo:musicartist, } O 1 = p[w](foaf) where W = { foaf:person, foaf:agent, foaf:organization } 38

39 O 3 = s[f](p[w](foaf) È O 2 ) W = { foaf:person, foaf:agent, foaf:organization } O 2 = (V 2,S 2 ), where V 2 = { mo:group, mo:musicartist, mo:corporatebody, mo:solomusicartist, mo:musicgroup, mo:label } S 2 = {mo:solomusicartist mo:musicartist, mo:musicgroup mo:musicartist, } F = {mo:solomusicartist foaf:person, mo:musicartist foaf:agent, mo:group foaf:agent, mo:corporatebody foaf:organization, $mo:member_of foaf:person, $ mo:member_of - mo:group } 39

40 Topics Introduction A Formal Framework Operations over Ontologies Implementation of the Operations Applications Conclusions 40

41 Conclusions Ontology Design Design of an export ontology a combination of fragments of one or more domain ontologies (So that matching the application ontology with the export ontology becomes trivial) Operations over ontologies create new ontologies, including their constraints, out of other ontologies Implementation based on a structural proof procedure that works for lightweight constraints 41

42 Conclusions Further Work Data Mashups as Default Theories Maximally consistent data mashup extension of the Default Theory Computation of maximally consistent data mashups based on the structural proof procedure, extended to assertions 42

43 Example Set S of constraints: s 1 : ( 1 p ) D s 4 : A C s 2 : C ( 1 p) s 5 : B C s 3 : C ( 2 p) s 6 : A B Set D of simple defaults: d 1. : B(a) / B(a) d 2. : p(a,b) / p(a,b) d 3. : p(a,c) / p(a,c) Set A of assertions obtained by firing all defaults: 1. B(a) (obtained by firing default d 1 ) 2. p(a,b) (obtained by firing default d 2 ) 3. p(a,c) (obtained by firing default d 3 ) Set S of the maximal subsets of the consequents of the defaults in D: S = { {B(a), p(a,b)}, {B(a), p(a,c)}, {p(a,b), p(a,c)} } 43

44 Thank You! For this presentation and associated references, search for marco antonio casanova puc-rio 44