The computational power of a compute model determines the class of problems it can solve. Automata theory allows describing the computational power of abstract machines (automata) and the problems they can solve. At the top of the Chomsky hierarchy of formal languages and grammars are Turing machines, the most powerful automata, which resemble the concept on which most modern computers are built. Here, we investigate the computational power of particle methods, a well-established class of algorithms with applications in scientific computing and computer simulation. Although particle methods can be interpreted as automata based on their formal definition, their computational power has so far not been studied. We address this by analyzing Turing completeness of particle methods. In particular, we prove two sets of restrictions under which a particle method is still Turing complete, and we show when it loses Turing completeness. This contributes to understanding the theoretical foundations of particle methods and provides insight into the powerfulness of computer simulations.

翻译：计算模型的计算能力决定了它能解决的问题类别。自动机理论允许描述抽象机器（自动机）的计算能力及其所能解决的问题。在形式语言与文法的乔姆斯基层级顶端是图灵机，这是最强大的自动机，它类似于现代计算机所基于的概念。本文中，我们研究了粒子方法的计算能力——粒子方法是一类成熟的算法，广泛应用于科学计算和计算机模拟。尽管粒子方法基于其形式化定义可被视为自动机，但其计算能力至今尚未被研究。我们通过分析粒子方法的图灵完备性来解决这一问题。具体而言，我们证明了两组限制条件，在这些条件下粒子方法仍保持图灵完备性，并展示了何时它失去图灵完备性。这有助于理解粒子方法的理论基础，并为计算机模拟的强大能力提供洞见。