PCSY-D-24-00068

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

PLOS Complex Systems

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1. Does this manuscript meet PLOS Complex Systems’s publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: No

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: No

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Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: No

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5. Review Comments to the Author

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Reviewer #1: This paper rigorously analyzes the Assembly Theory approach, demonstrating that its complexity measures can be reduced to well-established compression algorithms like LZ formulations. In disproving Assembly Theory's claims of uniqueness, the study highlights the fallacy of reductionist binary perspectives of life vs non-life when examining truly complex natural phenomena or systems. By advancing the theoretical understanding of life's complexity beyond simplistic frameworks, this research makes an important contribution to complex systems science. Therefore, I highly reccomend publication of this research, after a minor revision addressing the following comments:

Here are some suggestions for improving the paper from the perspective of a scientific peer reviewer:

1. The paper could acknowledged that distinguishing life from non-life exists on a spectrum rather than rigid binary categories. Accounting for critical systems (edge of chaos/phase-transitions) or ambiguous cases would strengthen the analysis. When criticizing Assembly Theory's assumptions about sequential assembly processes, the paper could note that some collective or emergent behaviors may exhibit features of both sequential and parallel dynamics. Recognizing these complex interaction effects would provide a more nuanced perspective. For instance, swarm intelligence or flocking behaviors, or collective dynamics of gene-regulatory networks exhibit emergent patterns that cannot be confined to assembly theory, but could benefit from LZ compression algorithms (and other algorithmic complexity measures). Addressing this power of LZ-like measures over assembly theory could benefit the paper's arguments.

2. Incorporating discussion of additional algorithms or models beyond compression approaches could further enrich the literature. A prospective studies and limitations section could benefit readership, in this regard. For example, as described above, comparing LZ-like methods and Assembly Theory to network-based models of molecular interactions (network science) could further enhance discussions. For instance, properties like small-worldness, clusters, hubs, and multi-nestedness/recursion, etc. have relevance to complex emergent dynamics. While not necessarily needed to make their core arguments, incorporating any of these complementary perspectives from complexity science could strengthen the interdisciplinary impact of the paper. Note: This point is not as compulsory as 1 and 3. But overall the work is clear, rigorous, and makes a strong scientific contribution as is.

3. The conclusions would benefit from some recommendations for how Assembly Theory could be refined or augmented, using the insights gained from LZ- rather than solely critiquing its limitations. Constructive suggestions aimed at advancing the scientific debate would make the feedback more positive and impactful.

Overall, this research performs a valuable service by rigorously analyzing claims made in the complex realm where life, physical processes, and computation intersect. While offering important criticisms, the paper also reinforces how reductionist binary approaches like Assembly Theory often fall short when examining truly complex, emergent phenomena. Advancing dialogue between different theories, through open exchange of viewpoints and models as this paper does, is crucial for making continued progress in science. Most importantly, it grounds Assembly Theory as a subset of the great pioneering works in computational and algorithmic complexity theory that it has failed to acknowledge. I would recommend publishing this work after considering the minor points given above.

Reviewer #2: This is a very well-written and timely manuscript. It applies the Occam razor in the sense that it shows that the newly branded Assembly Theory is nothing new, and it is absolutely equivalent to the well-known compression algorithms (in fact, it does not really matter, just any theoretically reasonable compression algorithm would do). Thus, the AT is nothing new and is subsumed by the classical Algorithmic Information Theory (first introduced by Solomonoff and Kolmogorov). The so-called assembly index is nothing new but a computable approximation to Kolmogorov complexity (that overshoots the latter by an arbitrary constant). Therefore, one has many better ways to estimate the Kolmogorov complexity of an object (be it a molecule, or a gene sequence). The body of the paper lays out strong and convincing arguments against the AT being any different from the AIT. There are may be advantages of using the AIT, but it is nothing comparable to a new "breakthrough" theory. In fact, it appears to me (personally) as overcomplicated while more shallow. The authors did a good job here.

The most important part is the mathematical addendum that contains all the proofs. I urge the authors to rewrite it a bit more carefully. For example, explicitly define K(x) and |x|, since all the necessary notation from the AT is carefully defined, it will not hurt repeating the classical notation used too (bear in mind that this paper should be read also by non-experts in the AIT who may be actually deceived by the AT).

In the proof of Theorem 7 I implore the authors to use \\itemize to describe the grammar's construction and separate the grammar rules into groups rather than give them as a contiguous chunk of text (that is dense to read). For example, the proof of Theorem 9, however technical, is written in a sparser manner, and is therefore more readable.

Other than these minor technicalities, I think that the paper should be accepted and published.

Reviewer #3: The authors claim to "prove" the equivalence between Assembly Theory (AT) and Shannon Entropy using methods/principles of compression/complexity/algorithmic information theory. While this thesis is plausible, interesting and relevant, I find the paper lacking massively in the following ways:

1. There is no rigorous, formal proof of any kind in the paper. Algorithmic Information Theory supplies enough formalism to clearly state and prove this equivalence.

2. What are the assumptions under which the equivalence is valid? Notions of stationarity and ergodicity are assumed to invoke Shannon–McMillan–Breiman theorem to define the existence of the notion of entropy limit for sources of information.

3. The authors should also clearly state in a rigorous and formal manner the AT assumptions, statements and claims and how they are trying to show that it is equivalent to Entropy. In what sense is it equal? For what kind of sources are they equal?

4. The paper contains merely (at best) philosophical arguments and is not backed by any rigorous, formal proofs of any kind.

5. Even logically, the authors do not show why AT is circular or fallacious. The treatment on AT is very scarce and difficult to follow. It is not clear what exactly they are saying regarding the drawbacks of AT.

6. Causality is also a very tricky concept with several operational definitions in engineering and sciences. It has a rich philosophical background as well. What concept of causality are the authors referring to? And what is the position of AT on this?

On the whole, I find the paper critically lacking in mathematical rigor need to justify the claim that "Assembly Theory (AT) and Shannon Entropy" are equivalent. A proof is not shown in this paper and hence I am inclined to reject the paper.

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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

PCSY-D-24-00068R1

Dear Dr. Abrahão,

We are pleased to inform you that your manuscript 'Assembly Theory is an approximation to algorithmic complexity based on LZ compression that does not explain selection or evolution' has been provisionally accepted for publication in PLOS Complex Systems.

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Hocine Cherifi

Editor-In-Chief

PLOS Complex Systems

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Reviewer Comments (if any, and for reference):