Assembly Theory is an approximation to algorithmic complexity based on LZ compression that does not explain selection or evolution

We formally prove the equivalence between Assembly Theory (AT) and Shannon Entropy via a method based upon the principles of statistical compression that belongs to the LZ family of popular compression algorithms. Such popular lossless compression algorithms behind file formats such as ZIP and PNG have been shown to empirically reproduce the results that AT considers its cornerstone. The same results have also been reported before AT in successful application of other complexity measures in the areas covered by AT such as separating organic from non-organic molecules and in the context of the study of selection and evolution. We demonstrate that the assembly index is equivalent to the size of a minimal context-free grammar. The statistical compressibility of such a method is bounded by Shannon Entropy and other equivalent traditional LZ compression schemes, such as LZ77 and LZW. We also demonstrate that AT, and the algorithms supporting its pathway complexity, assembly index, and assembly number, define compression schemes and methods that are subsumed into algorithmic information theory. We conclude that the assembly index and the assembly number do not lead to an explanation or quantification of biases in generative (physical or biological) processes, including those brought about by (abiotic or biotic) selection and evolution, that could not have been arrived at using Shannon Entropy, or that have not been already reported before using classical information theory or algorithmic complexity.Author summary: Assembly Theory (AT) has recently been proposed in order to investigate the distinction between abiotic from biotic matter, while explaining and quantifying the presence of biosignatures, selection, and evolution. We previously have shown that AT cannot rule out false positives and that it has equal or worse performance in comparison to popular compression algorithms at counting exact copies in data without evidence that their compression mechanics are favoured over others. This article investigates these limitations and the many challenges of the theoretical foundations of AT. We demonstrate that AT’s complexity measures (both for individual assembled objects and ensembles of objects) are subsumed into algorithmic information theory. The calculated assembly index for an object in AT is equivalent to the size of a compressing context-free grammar, and its calculation method is an LZ compression scheme that cannot perform better than Shannon Entropy in stochastic scenarios and cannot deal with non-stochastic (generative/causal) ones. Although AT may contribute with a graph-like pedagogical approach to LZ compression in application to molecular complexity, this article disproves hyperbolic claims raised by the authors of AT that introduce AT as a novel method, fundamentally different from other complexity indexes, or as a breakthrough. Instead, the principles behind AT are known elementary principles of complexity rehashed but introduced high logical inconsistency. AT lacks empirical evidence that it is different from or outperforms other complexity indexes in connection to selection, evolution or any of the applications in which the authors of AT have promoted it as capable of explaining physical and biological phenomena.