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Abstract

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.