Assembly theory (AT) quantifies selection using the assembly equation and identifies complex objects that occur in abundance based on two measurements, assembly index and copy number, where the assembly index is the minimum number of joining operations necessary to construct an object from basic parts, and the copy number is how many instances of the given object(s) are observed. Together these define a quantity, called Assembly, which captures the amount of causation required to produce objects in abundance in an observed sample. This contrasts with the random generation of objects. Herein we describe how AT's focus on selection as the mechanism for generating complexity offers a distinct approach, and answers different questions, than computational complexity theory with its focus on minimum descriptions via compressibility. To explore formal differences between the two approaches, we show several simple and explicit mathematical examples demonstrating that the assembly index, itself only one piece of the theoretical framework of AT, is formally not equivalent to other commonly used complexity measures from computer science and information theory including Shannon entropy, Huffman encoding, and Lempel-Ziv-Welch compression. We also include proofs that assembly index is not in the same computational complexity class as these compression algorithms and discuss fundamental differences in the ontological basis of AT, and assembly index as a physical observable, which distinguish it from theoretical approaches to formalizing life that are unmoored from measurement.

翻译：组装理论通过组装方程量化选择过程，并基于组装指数与拷贝数这两个测量指标识别大量存在的复杂对象。其中，组装指数指从基础部件构建对象所需的最小连接操作数，拷贝数则指观测到给定对象的实例数量。二者共同定义了称为“组装量”的度量，该度量表征了在观测样本中大量产生对象所需的因果作用量。这与对象的随机生成形成对比。本文阐述组装理论以选择作为复杂性生成机制的研究取向，相较于计算复杂度理论侧重于通过可压缩性实现最小描述的研究范式，提供了独特的研究路径并回应了不同的问题。为探究两种方法的形式差异，我们通过若干简洁的数学实例证明：作为组装理论框架组成部分的组装指数，在形式上并不等同于计算机科学与信息论中常用的复杂度度量（包括香农熵、霍夫曼编码及Lempel-Ziv-Welch压缩算法）。我们进一步给出形式证明，表明组装指数与上述压缩算法不属于同一计算复杂度类别，并讨论组装理论本体论基础的根本差异——组装指数作为物理可观测量的特性，使其区别于那些脱离实际测量的生命形式化理论进路。