On the Decidability of Termination for Polynomial Loops

We consider the termination problem for triangular weakly non-linear loops (twn-loops) over some ring $\mathcal{S}$ like $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$. Essentially, the guard of such a loop is an arbitrary Boolean formula over (possibly non-linear) polynomial inequations, and the body is a single assignment $(x_1, \ldots, x_d) \longleftarrow (c_1 \cdot x_1 + p_1, \ldots, c_d \cdot x_d + p_d)$ where each $x_i$ is a variable, $c_i \in \mathcal{S}$, and each $p_i$ is a (possibly non-linear) polynomial over $\mathcal{S}$ and the variables $x_{i+1},\ldots,x_{d}$. We present a reduction from the question of termination to the existential fragment of the first-order theory of $\mathcal{S}$ and $\mathbb{R}$. For loops over $\mathbb{R}$, our reduction entails decidability of termination. For loops over $\mathbb{Z}$ and $\mathbb{Q}$, it proves semi-decidability of non-termination. Furthermore, we present a transformation to convert certain non-twn-loops into twn-form. Then the original loop terminates iff the transformed loop terminates over a specific subset of $\mathbb{R}$, which can also be checked via our reduction. This transformation also allows us to prove tight complexity bounds for the termination problem for two important classes of loops which can always be transformed into twn-loops.