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Efficient Methods for Solving Finite

Satisfiability Problems in UML Class Diagrams

- Mira Balaban and Azzam Maraee

Motivation

Whats wrong with this ontology ?

Enzyme ? Protein ?

Protein ? Chemical

? Chemical ? Reaction ?

Reaction lt Enzyme

Enzyme lt Enzyme

Consistency Checking of Semantic Web

OntologiesKenneth Baclawski, Mieczyslaw M.

Kokar, Richard Waldinger, Paul A. Kogut

Agenda

- Background
- Reasoning Problems
- Previous works
- Research Contribution

- Conclusions and Future Work

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

Class Diagram Cardinality Constraint and Class

Hierarchy

1..

1

- Class Advisor and University
- Association Employment
- Cardinality Constraint
- Class Hierarchy

Legal Instance

Class Diagram Generalization Set Concept

1..

1

overlapping,incomplete

disjoint, complete

disjoint, incomplete

overlapping, complete

Agenda

- Background
- Reasoning Problems
- Previous works
- Research Contribution
- Finite Satisfiability Method
- Class Hierarchy and Generalization Set

Constraints

- Conclusions and Future Work

Reasoning Problems (1) Inconsistency

- Emptiness

disjoint

Reasoning Problems (2) Infinity Problem (1)

- Consider the pervious example

Infinity Problem (2)

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.

Reasoning Needs (2)

- Current case tools do not support reasoning

tasks. - Need for powerful CASE tools
- with reasoning capabilities.

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.

Consistent

The whole class diagram is consistent

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.

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

Agenda

- Background
- Reasoning Problems
- Previous works
- Research Contribution
- Finite Satisfiability Method.
- Class Hierarchy and Generalization Set

Constraints

- Conclusions and Future Work

Testing Finite Satisfiability

- Lenzerini and Nobili (92)
- ER diagrams without class hierarchy.
- Calvanese and Lenzerini (94)
- Extend with class hierarchy.

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.

rmin1a , rmax1a , rmin2b , rmax2b, a, b,

rgt0

Example

1

depend

Reaction

2

Example

1

depend

Reaction

2

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

Calvanese and Lenzerini (94) Extension with

class hierarchies (2)

Advisor,Master_Student

Advisor,PhD_Student

Advisor,Master_Student, PhD

Advisor

Calvanese and Lenzerini (94) Extension with

class hierarchies (3)

Advisor, Master, Ph.D

Advisor, PhD

Advisor

Advisor, Master

Advisor,Master , Ph.D

Advisor Master

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.

Hardness of Finite Satisfiability

- Berardi et al (2005) showed
- Lutz et al (2005) showed

?

Agenda

- Background
- Reasoning Problems
- Previous works
- Research Contribution
- Finite Satisfiability Method

- Conclusions and Future Work

Splitting the Problem

- Presents of Constraints
- Without
- With
- Hierarchy Structure
- Tree Structure.
- Acyclic Structure.
- Graph Structure.

Splitting the Problem Hierarchy Structure

- Tree Structure.

Splitting the Problem Hierarchy Structure

- Tree Structure.
- Acyclic Structure.

Splitting the Problem Hierarchy Structure

- Tree Structure
- Acyclic Structure
- Graph Structure.

Splitting the Problem

Hierarchy Structure

GS Constraint

Tree Acyclic Graph

Without With

Splitting the Problem

Hierarchy Structure

GS Constraint

Tree Acyclic Graph

Without With

Limited

Finite Satisfiabilty over Unconstrained Tree

Hierarchy

- Input
- Tree class hierarchy.
- Unconstrained generalization sets.
- Binary associations.
- Method
- A reduction to the algorithm of Lenzerini Nobili

Finite Satisfiabilty over Unconstrained Tree

Hierarchy

A class diagram that includes binary

associations and unconstrained tree hierarchy

Create the Lenzerini Nobili inequalities

system and Solve.

Class Diagram without hierarchy constraint

The Inequalities Systems

Finite Satisfiabilty over Unconstrained Tree

Hierarchy

dm

d2d

m2d

Claims

Finite Satisfiabilty over Unconstrained Tree

Hierarchy

- 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).

Proof Correctness Claim

Finite Satisfiabilty over Unconstrained Tree

Hierarchy

CD finitely satisfiable

CD finitely satisfiable

CD

CD

A class diagram that includes binary

associations and unconstrained tree hierarchy

Class Diagram without hierarchy constraint

Create the Lenzerini Nobili inequalities

system and Solve.

It is sufficient to show a reduction of the

finite satisfiability problem for CD to finite

satisfiability problem for CD without

generalization sets.

Reduction Proof

Finite Satisfiabilty over Unconstrained Tree

Hierarchy

CD

CD

Lenzerini Nobili inequalities

CD finitely satisfiable

CD finitely satisfiable

I

I

Extensions of Finite Satisfiabilty Method

- Applies properly also to the rest of the

unconstrained structured - Acyclic Structure.
- Graph Structure.

Splitting the Problem

Hierarchy Structure

GS Constraint

Tree Acyclic Graph

Without With

Limited

Finite Satisfiabilty Method Constrained Tree

Hierarchy

ma

Constrained Tree Hierarchy

disjoint, complete

advisor

Empty

Finite Satisfiabilty Method Constrained Tree

Hierarchy

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

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

Class Diagram without hierarchy constraint

OCL constraint Const

Single Constraints

Finite Satisfiabilty Method Constrained Tree

Hierarchy

- 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

Correctness Claim (Intuition)

CD finitely satisfiable

CD finitely satisfiable

A class diagram that includes binary associations

and constrained tree structure

Class Diagram without hierarchy constraint.

Expanded inequality system

OCL constraint Const

I

I

Correctness Claim (Intuition)

CD finitely satisfiable

CD finitely satisfiable

A class diagram that includes binary associations

and constrained tree structure

Class Diagram without hierarchy constraint.

Expanded inequality system

OCL constraint Const

I

Solution

Finite Satisfiabilty Method Constrained Tree

Hierarchy

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

Class Diagram without hierarchy constraint

OCL constraint Const

No Exclusive Inequalities

1) Create the Lenzerini and Nobili

inequalities system. 2) Expand the

inequalities and Solve.

Class Diagram without hierarchy constraint.

A class diagram that includes binary associations

and constrained tree structure

OCL constraint Const

The Inequalities System

Finite Satisfiabilty MethodStructure Extension

Hierarchy Structure

GS Constraint

Tree Acyclic Graph

Without With

Limited

?

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

Exploring the Limits

- Infinity

The Inequalities System

There is a solution

C21

C11

C12

C22

B

GS4

E

C21

C12

B

(No Transcript)

Implementation

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.

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,

Thank You

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