Assembly theory and its relationship with computational complexity

Assembly theory (AT) quantifies selection using the assembly equation, identifying complex objects through the assembly index, the minimal steps required to build an object from basic parts, and copy number, the observed instances of the object. These measure a quantity called Assembly, capturing causation necessary to produce abundant objects, distinguishing selection-driven complexity from random generation. Unlike computational complexity theory, which often emphasizes minimal description length via compressibility, AT explicitly focuses on the causation captured by selection as the mechanism behind complexity. We illustrate formal distinctions through mathematical examples demonstrating that the assembly index is fundamentally distinct from complexity metrics like Shannon entropy, Huffman encoding, and Lempel-Ziv-Welch compression. We provide proofs showing that the assembly index belongs to a different computational complexity class compared to these measures and compression algorithms. Additionally, we highlight AT's unique ontological grounding as a physically measurable framework, setting it apart from abstract theoretical approaches to formalizing life that lack empirical measurement foundations.