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.

翻译：我们考虑的是一些环上的 $\ mathbb+$, $\ mathbb+$, $\ mathb+$, 或$\ mathb{R} 美元等三角非线性环形( twn- loops) 的终止问题。 基本上, 这种环形的守护是一个任意的布利恩公式( 可能非线性) 多式对齐, 身体是一个单项任务 $ (x_ 1, \ ldot, x_ d) 的终止问题 。 圆形( c_ 1\ cdx_ 1 + p_ 1,\ ldots, c_ ddbb+ p_ d) 美元 的终止问题 。 每一项美元是( 可能非线性) 超线性 多元性 { {S& mathcal_ cal_ cal_ fal_ 问题 。 我们能证明 美元 =xxxxxxxxxxxxxxxxxxlal_ 变换 。