计算能力可通过为计算装置赋予代数结构来度量。本文将康威生命游戏（Conway's Game of Life）的一个小区域转化为一个变换半群（transformation semigroup）。该转化不仅捕获时间演化，还涵盖交互操作，从而使该细胞自动机可直接编程。完成此度量后，我们对该所得代数对象应用分层分解，以增进对其理解。这些分解基于受统计力学启发的宏观/微观态划分。然而，细胞自动机具有大量全局状态，因此我们聚焦于状态空间的划分，并构建可作为宏观层次描述的同态像（morphic images）近似。本文所发展的方法不仅适用于细胞自动机，亦可推广至更一般的离散动力系统。

Computational power can be measured by assigning an algebraic structure to a computational device. Here, we convert a small patch of Conway's Game of Life into a transformation semigroup. The conversion captures not only time evolution but also interactive operations. In this way, the cellular automaton becomes directly programmable. Once this measurement is made, we apply hierarchical decompositions to the resulting algebraic object as a way of understanding it. These decompositions are based on a macro/micro-state division inspired by statistical mechanics. However, cellular automata have a large number of global states. Therefore, we focus on partitioning the state space and creating morphic images approximations that can serve as macro-level descriptions. The methods developed here are not limited to cellular automata; they apply more generally to discrete dynamical systems.