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Proving total correctness and generating preconditions for loop programs via symbolic-numeric computation methods

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Abstract

We present a symbolic-numeric hybrid method, based on sum-of-squares (SOS) relaxation and rational vector recovery, to compute inequality invariants and ranking functions for proving total correctness and generating preconditions for programs. The SOS relaxation method is used to compute approximate invariants and approximate ranking functions with floating point coefficients. Then Gauss-Newton refinement and rational vector recovery are applied to approximate polynomials to obtain candidate polynomials with rational coefficients, which exactly satisfy the conditions of invariants and ranking functions. In the end, several examples are given to show the effectiveness of our method.

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References

  1. McIver A K, Morgan C. Partial correctness for probabilistic demonic programs. Theoretical Computer Science, 2001, 266(1–2): 513–541

    Article  MATH  MathSciNet  Google Scholar 

  2. Kovacs L. Automated Invariant Generation by Algebraic Techniques for Imperative Program Verification in Theorema. PhD thesis, Johannes Kepler University Linz, Austria, 2007

    Google Scholar 

  3. Logozzo F. Automatic inference of class invariants. In: Proceedings of the 5th International Conference on Verification, Model Checking, and Abstract Interpretation. 2004, 2937: 211–222

    Article  Google Scholar 

  4. Rodríguez-Carbonell E, Kapur D. Automatic generation of polynomial loop invariants for imperative programs. URL: www.cs.unm.edu/moore/tr/03-10/invpaper3.pdf, 2003

    Google Scholar 

  5. Chen Y, Xia B, Yang L, Zhan N. Generating polynomial invariants with DISCOVERER and QEPCAD. Formal Methods and Hybrid Real-Time Systems, 2007, 67–82

    Chapter  Google Scholar 

  6. Kapur D. Automatically generating loop invariants using quantifier elimination. In: Proceedings of the 10th International Conference on Applications of Computer Algebra. 2006

    Google Scholar 

  7. Rodríguez-Carbonell E, Kapur D. Generating all polynomial invariants in simple loops. Journal of Symbolic Computation, 2007, 42(4): 443–476

    Article  MATH  MathSciNet  Google Scholar 

  8. Rodríguez-Carbonell E, Kapur D. Program verification using automatic generation of invariants. In: Proceedings of the 1st International Conference on Theoretical Aspects of Computing. 2004, 325–340

    Google Scholar 

  9. Sankaranarayanan S, Sipma H B, Manna Z. Non-linear loop invariant generation using Gröbner bases. ACM SIGPLAN Notices, 2004, 39(1): 318–329

    Article  Google Scholar 

  10. Colón M, Sipma H. Synthesis of linear ranking functions. Tools and Algorithms for the Construction and Analysis of Systems, 2001, 67–81

    Chapter  Google Scholar 

  11. Bradley A, Manna Z, Sipma H. Termination analysis of integer linear loops. CONCUR 2005-Concurrency Theory, 2005, 488–502

    Chapter  Google Scholar 

  12. Cook B, Gulwani S, Lev-Ami T, Rybalchenko A, Sagiv M. Proving conditional termination. In: Proceedings of the 20th International Conference on Computer Aided Verification. 2008, 328–340

    Chapter  Google Scholar 

  13. Podelski A, Rybalchenko A. A complete method for the synthesis of linear ranking functions. In: Proceedings of the 5th International Conference on Verification, Model Checking, and Abstract Interpretation. 2004, 2937: 465–486

    MathSciNet  Google Scholar 

  14. Cousot P. Proving program invariance and termination by parametric abstraction, Lagrangian relaxation and semidefinite programming. In: Proceedings of the 6th International Conference on Verification, Model Checking, and Abstract Interpretation. 2005, 3385: 1–24

    Article  MathSciNet  Google Scholar 

  15. Chen Y, Xia B, Yang L, Zhan N, Zhou C. Discovering non-linear ranking functions by solving semi-algebraic systems. In: Proceedings of the 4th International Conference on Theoretical Aspects of Computing. 2007, 34–49

    Google Scholar 

  16. Yang L, Zhou C, Zhan N, Xia B. Recent advances in program verification through computer algebra. Frontiers of Computer Science in China, 2010, 4(1): 1–16

    Article  Google Scholar 

  17. Dijkstra E W. A Discipline of Programming, Volume 1. New Jersey: Englewood Cliffs, 1976

    Google Scholar 

  18. Gulwani S, Srivastava S, Venkatesan R. Program analysis as constraint solving. ACM SIGPLAN Notices, 2008, 43(6): 281–292

    Article  Google Scholar 

  19. Leino K R M. Efficient weakest preconditions. Information Processing Letters, 2005, 93(6): 281–288

    Article  MATH  MathSciNet  Google Scholar 

  20. Barnett M, Leino K R M. Weakest-precondition of unstructured programs. In: Proceedings of the 6th ACM SIGPLAN-SIGSOFT Workshop on Pregram Analysis for Software Tools and Engineering. 2005, 82–87

    Chapter  Google Scholar 

  21. Hoare C A R. An axiomatic basis for computer programming. Communications of the ACM, 1969, 12(10): 576–580

    Article  MATH  Google Scholar 

  22. Bagnara R, Rodríguez-Carbonell E, Zaffanella E. Generation of basic semi-algebraic invariants using convex polyhedra. In: Proceedings of the 12th International Conference on Static Analysis. 2005, 19–34

    Google Scholar 

  23. Sankaranarayanan S, Sipma H B, Manna Z. Constraint-based linearrelations analysis. In: Proceedings of the 11th International Symposium on Static Analysis. 2004, 53–68

    Google Scholar 

  24. Colón M, Sankaranarayanan S, Sipma H. Linear invariant generation using non-linear constraint solving. In: Proceedings of the 15th International Conference on Computer Aided Verification. 2003, 420–432

    Chapter  Google Scholar 

  25. Tiwari A, Rueß H, Saïdi H, Shankar N. A technique for invariant generation. Tools and Algorithms for the Construction and Analysis of Systems, 2001, 113–127

    Chapter  Google Scholar 

  26. Xia B, Yang L, Zhan N. Program verification by reduction to semialgebraic systems solving. Communications in Computer and Information Science, 2008, 17: 277–291

    Article  Google Scholar 

  27. Xia B, Yang L, Zhan N, Zhang Z. Symbolic decision procedure for termination of linear programs. Formal Aspects of Computing, 2011, 23(2): 171–190

    Article  MATH  MathSciNet  Google Scholar 

  28. Yang L, Zhan N, Xia B, Zhou C. Program verification by using DISCOVERER. Lecture Notes in Computer Science, 2005, 4171: 528–538

    Article  Google Scholar 

  29. Parrilo P. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, 2000

    Google Scholar 

  30. Prajna S, Papachristodoulou A, Seiler P, Parrilo P A. SOSTOOLS: Sum of squares optimization toolbox for MATLAB, 2002. Available at http://www.cds.caltech.edu/sostools

    Google Scholar 

  31. Löfberg J. YALMIP: A toolbox for modeling and optimization in matlab. In: Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design. 2004, 284–289

    Google Scholar 

  32. Sturm J F. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 1999, 11/12: 625–653

    Article  MathSciNet  Google Scholar 

  33. Wu M, Yang Z. Generating invariants of hybrid systems via sums-of-squares of polynomials with rational coefficients. In: Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation. 2011, 104–111

    Chapter  Google Scholar 

  34. Bochnak J, Coste M, Roy M. Real Algebraic Geometry, Volume 36. Springer Verlag, 1998

    Google Scholar 

  35. Kaltofen E, Li B, Yang Z, Zhi L. Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars. In: Proceedings of the 21st International Symposium on Symbolic Algebraic Computation. 2008, 155–163

    Google Scholar 

  36. Kaltofen E, Li B, Yang Z, Zhi L. Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients. Journal of Symbolic Computation, 2012, 47(1): 1–15

    Article  MATH  MathSciNet  Google Scholar 

  37. Lagarias J C. The computational complexity of simultaneous diophantine approximation problems. SIAM Journal on Computing, 1985, 14(1): 196–209

    Article  MATH  MathSciNet  Google Scholar 

  38. Xia B. DISCOVERER: A tool for solving semi-algebraic systems. ACM Commun. Compute. Algebra, 2007, 41(3): 102–103

    Google Scholar 

  39. Mohab S E D. Raglib (real algebraic library maple package). Available at http://www-calfor.lip6.fr/?safey/RAGLib, 2003

    Google Scholar 

  40. Petter M. Berechnung von polynomiellen invarianten. Master’s thesis, Fakultät für Informatik, Technische Universität München, 2004

    Google Scholar 

  41. Dai L, Xia B, Zhan N. Generating non-linear interpolants by semidefinite programming. Lecture Notes in Computer Science, 2013, 8044: 364–380

    Article  Google Scholar 

  42. Dai L, Gan T, Wang B Y, Xia B, Zhan N, Zhao H. Non-linear interpolant generation and its applications to program verification. http://cav2013.forsyte.at/files/naijun-zhan.pdf, 2013

    Google Scholar 

  43. Shen L, Wu M, Yang Z, Zeng Z. Generating exact nonlinear ranking functions by symbolic-numeric hybrid method. Journal of Systems Science and Complexity, 2013, 26(2): 291–301

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Wang Lin, Min Wu & Zhengfeng Yang

  2. College of Mathematics and Information Science, Wenzhou University, Zhejiang, 325035, China

    Wang Lin

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Zhenbing Zeng

Authors
  1. Wang Lin
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  2. Min Wu
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  3. Zhengfeng Yang
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  4. Zhenbing Zeng
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Corresponding author

Correspondence to Min Wu.

Additional information

Wang Lin received the PhD in technology of computer application of East China Normal University, China. His research interests are program verification, analysis and verification of hybrid systems, and symbolic-numeric computation.

Min WU received her PhDs in mathematics from Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China and Université de Nice-Sophia Antipolis, France in 2005. She is currently an associate professor at Software Engineering Institute of East China Normal University, China. Her research interests are in the area of symbolic computation, trustworthy computing and their applications.

Zhengfeng Yang received the PhD degree from Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 2006. He is currently an associate professor at Software Engineering Institute of East China Normal University, China. His research interests include symbolic computation, symbolic-numeric computation, and program verification.

Zhenbing Zeng received the BS from Northwestern China Normal University, China in 1984, the MS from Chengdu Institute ofMathematical Sciences of Chengdu Branch of the Chinese Academy of Sciences in 1987, and the PhD from Bielefeld University, Germany in 1993. He is currently a professor of mathematics and computer science at East China Normal University, China. His research interest includes mathematics mechanization, symbolic computation, and artificial intelligence.

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Lin, W., Wu, M., Yang, Z. et al. Proving total correctness and generating preconditions for loop programs via symbolic-numeric computation methods. Front. Comput. Sci. 8, 192–202 (2014). https://doi.org/10.1007/s11704-014-3150-6

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  • DOI: https://doi.org/10.1007/s11704-014-3150-6

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