An algebraic telic problem is a decision problem in $\textsf{NP}_\mathbb{R}$ formalizing finite-time reachability questions for one-dimensional dynamical systems. We prove that the existence of "natural" mapping reductions between algebraic telic problems coming from distinct dynamical systems implies the two dynamical systems exhibit similar behavior (in a precise sense). As a consequence, we obtain explicit barriers for algorithms solving algebraic telic problems coming from complex dynamical systems, such as those with positive topological entropy. For example, some telic problems cannot be decided by uniform arithmetic circuit families with only $+$ and $\times$ gates.

翻译：代数目的问题是一类决策问题，属于 $\textsf{NP}_\mathbb{R}$ 类，它形式化地描述了一维动力系统有限时间可达性问题。我们证明，对于来自不同动力系统的代数目的问题，若存在“自然”的映射归约，则意味着这两个动力系统表现出相似的行为（在精确的意义上）。由此，我们为求解来自复杂动力系统（例如具有正拓扑熵的系统）的代数目的问题的算法，获得了明确的障碍。例如，某些目的问题无法仅使用 $+$ 和 $\times$ 门的统一算术电路族来判定。