We consider the problem of proving termination for triangular weakly non-linear loops (twn-loops) over some ring $\mathcal{S}$ like $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$. The guard of such a loop is an arbitrary quantifier-free Boolean formula over (possibly non-linear) polynomial inequations, and the body is a single assignment of the form $(x_1, \ldots, x_d) \longleftarrow (c_1 \cdot x_1 + p_1, \ldots, c_d \cdot x_d + p_d)$ 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 show that the question of termination can be reduced to the existential fragment of the first-order theory of $\mathcal{S}$. For loops over $\mathbb{R}$, our reduction implies 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. Moreover, we formalize a technique to linearize (the updates of) twn-loops in our setting and analyze its complexity. Based on these results, we prove complexity bounds for the termination problem of twn-loops as well as tight bounds for two important classes of loops which can always be transformed into twn-loops. Finally, we show that there is an important class of linear loops where our decision procedure results in an efficient procedure for termination analysis, i.e., where the parameterized complexity of deciding termination is polynomial.
翻译:我们考虑在某个环上证明三角非线性环状(twn-loops) 的终止问题,比如 $\ mathbb+$, $\ mathbb+$, 或者 $\ mathbb{R} 美元。 此环的守护是一个任意的量化无布利安公式, 可能是非线性多式的多式方程式。 而每张美元是形式 $( x_ 1, rdotots, x_d) 的单一任务。 liftrowrows (c_ 1\ cdockot x_ + p_ 1, ldots, c_dd\ d_ p_d) 的终止问题。 在( $xxxxxxxxxxxxxxxxxxxxxxxx) 的解算法中, 每张平价的解算可以让我们的解析。