Termination Analysis of Polynomial Programs with Equality Conditions

In this paper, we investigate the termination problem of a family of polynomial programs, in which all assignments to program variables are polynomials, and test conditions of loops and conditional statements are polynomial equations. Our main result is that the non-terminating inputs of such a polynomial program is algorithmically computable according to a strictly descending chain of algebraic sets, which implies that the termination problem of these programs is decidable. The complexity of the algorithm follows immediately from the length of the chain, which can be computed by Hilbert’s function and Macaulay’s theorem. To the best of our knowledge, the considered family of polynomial programs should be the largest one with a decidable termination problem so far. The experimental results indicate the efficiency of our approach.