Efficient Craig Interpolation for Linear Diophantine DisEquations and Linear Modular Equations 1st edtion by Himanshu Jain, Edmund Clarke, Orna Grumberg ISBN 3540705437 9783540705437

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Product details:

ISBN 10:       3540705437
ISBN 13: 978-3540705437
Author:        Himanshu Jain, Edmund Clarke, Orna Grumberg 

The use of Craig interpolants has enabled the development of powerful hardware and software model checking techniques. Efficient algorithms are known for computing interpolants in rational and real linear arithmetic. We focus on subsets of integer linear arithmetic. Our main results are polynomial time algorithms for obtaining interpolants for conjunctions of linear diophantine equations, linear modular equations (linear congruences), and linear diophantine disequations. We show the utility of the proposed interpolation algorithms for discovering modular/divisibility predicates in a counterexample guided abstraction refinement (CEGAR) framework. This has enabled verification of simple programs that cannot be checked using existing CEGAR based model checkers.

Efficient Craig Interpolation for Linear Diophantine (Dis)Equations and Linear Modular Equations 1st Table of contents:

Chapter 1: Introduction
1.1 Overview of Craig Interpolation
1.2 Motivation for Applying Craig Interpolation to Linear Diophantine Equations
1.3 Challenges in Linear Modular Equations
1.4 Objective of the Paper
1.5 Structure of the Paper

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Chapter 2: Background and Preliminaries
2.1 Craig Interpolation: Definition and Properties
2.2 Linear Diophantine Equations
2.3 Linear Modular Equations
2.4 Previous Work on Craig Interpolation in Theoretical Computer Science
2.5 Connection Between Craig Interpolation and Automated Theorem Proving

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Chapter 3: Linear Diophantine (Dis)Equations
3.1 Definition of Linear Diophantine Equations
3.2 Existence of Solutions for Linear Diophantine Equations
3.3 Methods for Solving Linear Diophantine Equations
3.4 Applications of Linear Diophantine (Dis)Equations
3.5 Challenges in Applying Interpolation to Linear Diophantine Equations

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Chapter 4: Linear Modular Equations
4.1 Introduction to Modular Arithmetic and Modular Equations
4.2 Solving Linear Modular Equations
4.3 Properties and Challenges in Linear Modular Equations
4.4 Modular Interpolation and Its Relevance
4.5 Applications of Linear Modular Equations in Cryptography and Computing

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Chapter 5: Craig Interpolation for Linear Diophantine (Dis)Equations
5.1 Definition and Relevance of Craig Interpolation for Linear Diophantine Equations
5.2 Formalizing the Interpolation Procedure
5.3 Algorithm for Efficient Craig Interpolation in Linear Diophantine Equations
5.4 Complexity Analysis of the Interpolation Process
5.5 Applications of Interpolated Linear Diophantine Equations

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Chapter 6: Craig Interpolation for Linear Modular Equations
6.1 Adapting Craig Interpolation to Modular Equations
6.2 Interpolation in the Context of Modular Arithmetic
6.3 Efficient Algorithms for Interpolation in Linear Modular Equations
6.4 Performance and Complexity Considerations
6.5 Applications of Interpolation in Modular Equations

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Chapter 7: Algorithmic Approaches for Efficient Interpolation
7.1 Overview of Computational Techniques for Interpolation
7.2 Efficient Algorithms for Handling Linear Diophantine and Modular Equations
7.3 Optimizing Interpolation Algorithms
7.4 Comparison with Traditional Methods
7.5 Practical Considerations in Algorithm Implementation

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Chapter 8: Applications of Craig Interpolation
8.1 Applications in Automated Theorem Proving
8.2 Applications in Cryptography and Security
8.3 Applications in Algebraic Systems and Computation
8.4 Applications in Verification and Model Checking
8.5 Case Study: Application in Integer Programming and Optimization

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Chapter 9: Challenges and Open Problems
9.1 Limitations in Current Craig Interpolation Methods
9.2 Challenges in Extending Craig Interpolation to More Complex Equations
9.3 Open Problems in Algorithmic Efficiency and Scalability
9.4 Research Directions in Craig Interpolation
9.5 Integration with Other Computational Techniques

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Chapter 10: Conclusion
10.1 Summary of Key Contributions
10.2 Impact of Efficient Craig Interpolation on the Study of Diophantine and Modular Equations
10.3 Final Thoughts on Future Developments in Interpolation Techniques
10.4 Concluding Remarks on Practical Applications and Theoretical Insights

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