Abstract
This paper presents a hybrid symbolic-numeric algorithm to compute ranking functions for establishing the termination of loop programs with polynomial guards and polynomial assignments. The authors first transform the problem into a parameterized polynomial optimization problem, and obtain a numerical ranking function using polynomial sum-of-squares relaxation via semidefinite programming (SDP). A rational vector recovery algorithm is deployed to recover a rational polynomial from the numerical ranking function, and some symbolic computation techniques are used to certify that this polynomial is an exact ranking function of the loop programs. At last, the authors demonstrate on some polynomial loop programs from the literature that our algorithm successfully yields nonlinear ranking functions with rational coefficients.
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This research was supported in part by the National Natural Science Foundation of China under Grant Nos.10901055, 61021004, 91118007, by NKBRPC 2011CB302802, 2011CB706901, and by the Fundamental Research Funds for the Central Universities under Grant No. 78210043.
This paper was recommended for publication by Editor DAI Yuhong.
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Shen, L., Wu, M., Yang, Z. et al. Generating exact nonlinear ranking functions by symbolic-numeric hybrid method. J Syst Sci Complex 26, 291–301 (2013). https://doi.org/10.1007/s11424-013-1004-1
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DOI: https://doi.org/10.1007/s11424-013-1004-1