Abstract
Termination analysis of loop programs is very important in many applications, especially in those of safety critical software. In this paper, the termination of programs with polynomial guards and linear assignments is simplified to decide solvability of semi-algebraic systems(SAS). If the number of functions are finite or the functions are integer periodic, then the termination of programs is decidable. The discussion is based on simplifying the linear loops by its Jordan form. And then the process to find the nonterminating points for general polynomial guards is proposed. For avoiding floating point computations in the process, a symbolic algorithm is given to compute the Jordan form of a matrix.
This research was supported in part by NSFC(No. 90718041).
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References
Floyd, R.W.: Assigning meanings to programs. In: Schwartz, J.T. (ed.) Mathematical Aspects of Computer Science. Proceedings of Symposia in Applied Mathematics, vol. 19, pp. 19–32. American Mathematical Socity (1967)
Hoare, C.A.R.: An axiomatic basis for computer programming. Communications of ACM 12(10), 576–580 (1969)
Turing, A.: On computable numbers, with an application to the entscheidungsproblem. London Mathematical Socity 42(2), 230–265 (1936)
Colón, M., Sipma, H.: Synthesis of linear ranking functions. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 67–81. Springer, Heidelberg (2001)
Colón, M., Sipma, H.: Practical methods for proving program termination. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 442–454. Springer, Heidelberg (2002)
Podelski, A., Rybalchenko, A.: A complete method for the synthesis of linear ranking functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 239–251. Springer, Heidelberg (2004)
Bradley, A.R., Manna, Z., Sipma, H.B.: Termination analysis of integer linear loops. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 488–502. Springer, Heidelberg (2005)
Cousot, P.: Proving program invariance and termination by parametric abstraction, lagrangian relaxation and semidefinite programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 1–24. Springer, Heidelberg (2005)
Chen, Y., Xia, B., Yang, L., Zhan, N., Zhou, C.: Discovering non-linear ranking functions by solving semi-algebraic systems. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) ICTAC 2007. LNCS, vol. 4711, pp. 34–49. Springer, Heidelberg (2007)
Bradley, A.R., Manna, Z., Sipma, H.B.: Termination of polynomial programs. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 113–129. Springer, Heidelberg (2005)
Babic, D., Hu, A.J., Rakamaric, Z., Cook, B.: Proving termination by divergence. In: SEFM 2007: Software Engineering and Formal Methods, London, England, UK, pp. 93–102. IEEE, Los Alamitos (2007)
Leue, S., Wei, W.: A region graph based approach to termination proofs. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 318–333. Springer, Heidelberg (2006)
Wu, B., Bi, Z.: Termination of nested loop. In: ISCSCT 2008: International Symposium on Computer Science and Computational Technology, Shanghai, China, vol. 2, pp. 536–539. IEEE, Los Alamitos (2008)
Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS 2004: Logic in Computer Science, Turku, Finland, pp. 32–41. IEEE, Los Alamitos (2004)
Tiwari, A.: Termination of linear programs. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 70–82. Springer, Heidelberg (2004)
Braverman, M.: Termination of integer linear programs. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 372–385. Springer, Heidelberg (2006)
Bi, Z., Shan, M., Wu, B.: Non-termination analysis of linear loop programs with conditionals. In: ASEA 2008: Advanced Software Engineering and Its Applications, Hainan Island, China, pp. 159–164. IEEE, Los Alamitos (2008)
Smart, D.R.: Fixed Piont Theorems. Cambridge University Press, Cambridge (1980)
Hoffman, K., Kunze, R.: Linear algebra, 2nd edn. Prentice-Hall, New Jersey (1971)
Xia, B., Zhang, Z.: Termination of linear programs with nonlinear constraints (2009), http://arxiv.org/abs/0904.3588v1
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Wu, B., Shen, L., Bi, Z., Zeng, Z. (2010). Termination of Loop Programs with Polynomial Guards. In: Taniar, D., Gervasi, O., Murgante, B., Pardede, E., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2010. ICCSA 2010. Lecture Notes in Computer Science, vol 6019. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12189-0_42
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