1. Efficient Methods for Solving Finite Satisfiability Problems in UML Class Diagrams Mira Balaban and Azzam Maraee

2. 2 Motivation [Enzyme]  [Protein]  [Protein]  [Chemical]  [ Chemical]  [Reaction]  [Reaction] < [Enzyme] [Enzyme] < [Enzyme] What’s wrong with this ontology * ? * Consistency Checking of Semantic Web Ontologies Kenneth Baclawski, Mieczyslaw M. Kokar, Richard Waldinger, Paul A. Kogut

3. 3 Agenda Background Reasoning Problems Previous works Research Contribution Conclusions and Future Work

4. 4 UML UML is now widely accepted as the standard modeling language for software construction. UML 2.0 updates came from 54 companies. Class diagrams are widely used software and language. specification, database. ontology engineering

5. 5 Class Diagram: Cardinality Constraint and Class Hierarchy Class: Advisor and University Association: Employment –Cardinality Constraint Class Hierarchy 1..*1 Legal Instance:

6. 6 Class Diagram: Generalization Set Concept 1..* 1 {disjoint, incomplete} {disjoint, complete} {overlapping,incomplete} {overlapping, complete}

7. 7 Agenda Background Reasoning Problems Previous works Research Contribution –Finite Satisfiability Method: Class Hierarchy and Generalization Set Constraints Conclusions and Future Work

8. 8 Reasoning Problems (1): Inconsistency disjoint Emptiness

9. 9 Reasoning Problems (2): Infinity Problem (1) Consider the pervious example:

10. 10 Infinity Problem (2)

11. 11 Relevance of Reasoning It is important to guarantee that models provide a reliable support for the designed systems. –Implementability : A class diagram is implemented into a running system. –Project cost: Early detection of problems. –Used with MDA: Precise and consistent enough to be used within MDA.

12. 12 Reasoning Needs (2) Current case tools do not support reasoning tasks. Need for powerful CASE tools – with reasoning capabilities.

13. 13 Consistency notions (1) Berardi et al (2005), distinguish two cases: –Consistency of class diagram – has an instantiation with at-least one non-empty class extension. –Class consistency – there is an instantiation in which the class extension is non-empty. rest of the diagram Consistent The whole class diagram is consistent

14. 14 Consistency notions (2) All class consistency of a class diagram – every class is consistent. Full consistency of a class diagram – has an instance in which all class extensions are non-empty.

15. 15 Finite satisfiability notions All class finite satisfiability of a class diagram – every class there is a finite instance in which the class extension is non-empty (strong satisfiability, Lenzerini and Nobili, 1990). Finite satisfiability of a class diagram – it has a finite instance in which all class extensions are non-empty. Lenzerini & Nobili 92 Research Contribution UML Class Diagram

16. 16 Agenda Background Reasoning Problems Previous works Research Contribution –Finite Satisfiability Method. Class Hierarchy and Generalization Set Constraints Conclusions and Future Work

17. 17 Testing Finite Satisfiability Lenzerini and Nobili (92): –ER diagrams without class hierarchy. Calvanese and Lenzerini (94): –Extend with class hierarchy.

18. 18 Lenzerini and Nobily (92). Method: Transform cardinality constraints into an inequalities system. Solve the system. Result: The diagram is finitely satisfiable iff the inequalities system has a solution. min 1, max 1 min 2, max 2 r≥min 1 ∙a, r≤max 1 ∙a, r≥min 2 ∙b, r≤max 2 ∙b, a, b, r>0

19. Example 19 Reaction 1 2 depend

20. Example 20 Reaction 1 2 depend

21. 21 Calvanese and Lenzerini (94) – Extension with class hierarchies (1) Calvanese and Lenzerini extend the method of Lenzerini and Nobily (92) to apply to schemata with class hierarchy. The expansion is based on the assumption that class extensions may overlap. –Compound Class. –Compound relationship:

22. 22 Advisor,Master_Student Advisor,PhD_Student Advisor Advisor,Master_Student, PhD Calvanese and Lenzerini (94) – Extension with class hierarchies (2)

23. 23 Calvanese and Lenzerini (94) – Extension with class hierarchies (3) Advisor Advisor, Master, Ph.D Advisor, PhD Advisor, Master Advisor,Master, Ph.D Advisor: Master

24. 24 Calvanese and Lenzerini (94) – Extension with class hierarchies (2) For a class diagram with three classes and one association: 12 variables and 26 inequalities An exponential number of variables.

25. 25 Hardness of Finite Satisfiability Berardi et al (2005) showed : Lutz et al (2005) showed:

26. 26 Agenda Background Reasoning Problems Previous works Research Contribution –Finite Satisfiability Method Conclusions and Future Work

27. 27 Splitting the Problem Presents of Constraints: –Without –With Hierarchy Structure: Tree Structure. Acyclic Structure. Graph Structure.

28. 28 Splitting the Problem : Hierarchy Structure Tree Structure.

29. 29 Splitting the Problem : Hierarchy Structure Tree Structure. Acyclic Structure.

30. 30 Splitting the Problem : Hierarchy Structure Tree Structure Acyclic Structure Graph Structure.

31. 31 Splitting the Problem Tree Acyclic Graph Hierarchy StructureGS Constraint Without With

32. 32 Splitting the Problem Tree Acyclic Graph Hierarchy StructureGS Constraint Without With Limited

33. 33 Finite Satisfiabilty over Unconstrained Tree Hierarchy Input: –Tree class hierarchy. –Unconstrained generalization sets. –Binary associations. Method: –A reduction to the algorithm of Lenzerini Nobili:

34. 34 Finite Satisfiabilty over Unconstrained Tree Hierarchy sub 0..1 sub 0..1 super 11 ISA 1 ISA 2 Class Diagram without hierarchy constraint A class diagram that includes binary associations and unconstrained tree hierarchy Create the Lenzerini & Nobili inequalities system and Solve.

35. 35 The Inequalities Systems AdvisorMa_StudentPh.DStudentAdviceISA_1ISA_2 dmpadisa 1 isa 2 m=2d d≥m d≥2d Finite Satisfiabilty over Unconstrained Tree Hierarchy

36. 36 Claims Claim [correctness]: A class diagram with unconstrained tree hierarchy is finitely satisfiable iff there exists a solution to the inequalities system. Claim [Complexity]: Unconstrained Tree Hierarchy finite satisfiabilty method adds to the Lenzerini and Nobili method an O(n) time complexity, where n is the size of the class diagram (including associations, classes and class hierarchy constraints). Finite Satisfiabilty over Unconstrained Tree Hierarchy

37. 37 Proof [Correctness Claim] Class Diagram without hierarchy constraint Create the Lenzerini & Nobili inequalities system and Solve. A class diagram that includes binary associations and unconstrained tree hierarchy CDCD’ It is sufficient to show a reduction of the finite satisfiability problem for CD to finite satisfiability problem for CD’ without generalization sets. CD : finitely satisfiable CD’ : finitely satisfiable Finite Satisfiabilty over Unconstrained Tree Hierarchy

38. 38 Reduction Proof I I’ CD’ : finitely satisfiable CD : finitely satisfiable CDCD’ Lenzerini & Nobili inequalities Finite Satisfiabilty over Unconstrained Tree Hierarchy

39. 39 Extensions of Finite Satisfiabilty Method Applies properly also to the rest of the unconstrained structured: –Acyclic Structure. –Graph Structure.

40. 40 Splitting the Problem Tree Acyclic Graph Hierarchy StructureGS Constraint Without With Limited

41. 41 Finite Satisfiabilty Method : Constrained Tree Hierarchy Empty Constrained Tree Hierarchy advisor ma Master_Student { disjoint, complete }

42. 42 …………….. Const 0..1 1 1 Const …... Class Diagram without hierarchy constraint. 1) Create the Lenzerini and Nobili inequalities system. 2) Expand the inequalities and Solve. A class diagram that includes binary associations and constrained tree structure OCL constraint: Const Finite Satisfiabilty Method : Constrained Tree Hierarchy

43. 43 Class Diagram without hierarchy constraint 1) Create the Lenzerini and Nobili inequalities system. 2) Expand the inequalities and Solve. A class diagram that includes binary associations and constrained tree structure OCL constraint: Const Single Constraints Finite Satisfiabilty Method : Constrained Tree Hierarchy

44. 44 Claim [Correctness]: A class diagram with constrained tree structure is finitely satisfiable iff there exists a solution for the expanded inequalities system Claim [Complexity]: adds to the Lenzerini and Nobili method an O(n) time complexity, where n is the size of the class diagram (including associations, classes and class hierarchy constraints Finite Satisfiabilty Method : Constrained Tree Hierarchy

45. 45 Correctness Claim (Intuition) A class diagram that includes binary associations and constrained tree structure Class Diagram without hierarchy constraint. Expanded inequality system OCL constraint: Const I I’ CD : finitely satisfiable CD’ : finitely satisfiable

46. 46 Correctness Claim (Intuition) A class diagram that includes binary associations and constrained tree structure Class Diagram without hierarchy constraint. Expanded inequality system OCL constraint: Const I’ CD’ : finitely satisfiable Solution CD’: finitely satisfiable

47. 47 Class Diagram without hierarchy constraint 1) Create the Lenzerini and Nobili inequalities system. 2) Expand the inequalities and Solve. A class diagram that includes binary associations and [T-C]-GS OCL constraint: Const Finite Satisfiabilty Method: Constrained Tree Hierarchy

48. 48 No Exclusive Inequalities Class Diagram without hierarchy constraint. 1) Create the Lenzerini and Nobili inequalities system. 2) Expand the inequalities and Solve. A class diagram that includes binary associations and constrained tree structure OCL constraint: Const

49. 49 The Inequalities System AdvisorMa_StudentPh.DStudentAdviceISA_1ISA_2 admpdisa 1 isa 2

50. 50 Finite Satisfiabilty Method Structure Extension Tree Acyclic Graph Hierarchy StructureGS Constraint Without With Limited

51. 51 Finite Satisfiabilty Method Extension to Graph Structure. Succeeds in determining non-finite satisfiabilty. Succeeds in determining finite satisfiabilty: –overlapping and incomplete Fails in: – Complete, disjoint

52. 52 Exploring the Limits Infinity

53. 53 The Inequalities System There is a solution

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55. 55 E

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57. 57 Implementation

58. 58 Conclusion Introduced graph structure based methods. Have a complexity advantage over existing methods that require an exponential time in the worst case. Simplicity: Our methods are simple and sufficient for reducing the satisfiabilty problem.

59. 59 Future Work Extensions: –Graph structure with disjoint, complete. –N-ary. –Qualifier. –Association class. –Aggregation. –Identification and Fixing (Sven Hartman, 2001) –integrate the implementation as add-in in existing case tool,

60. 60