1 Introduction

Assembly Theory (AT) is a hypothesis that has recently garnered significant attention. Although its hyperbolic claims have not been taken seriously, the scientific media has amplified the many misleading arguments of their authors. Responding to how open-ended forms can emerge from matter without a blueprint, AT purports to explain and quantify the presence of biosignatures, selection, and evolution, and has recently even suggested to be able to explain time, matter, the expansion of the universe and even cosmic inflation [1]. The assembly index (or MA, a variation applied to quantify the assembly index on molecules) is the proposed complexity measure used to allegedly distinguish abiotic from biotic matter, the central claim of AT as advanced in [2, 3]. According to the assembly index, objects with a high assembly index “are very unlikely to form abiotically”. This has been contested in [4], whose results “demonstrate that abiotic chemical processes have the potential to form crystal structures of great complexity”, exceeding the assembly index threshold proposed by AT’s authors. The existence of such abiotic objects would render AT’s methods prone to false positives, corroborating the predictions in [5]. The assembly index (or MA) was shown to perform equally well, or worse in some cases, relative to popular compression algorithms (including those of a statistical nature) [5], some of which have been applied before in [6–8].

In order to investigate the fundamental limitations and theoretical underpinnings of such a complexity measure, and to explain the above results, this article shows that the non-linear (or “tree”-like) structure of the minimum rooted assembly (sub)spaces (from which assembly indexes are calculated) is one of the ways to define a (compression) scheme that encodes the assembling process itself back into a linear sequence of codewords (or phrases). We demonstrate that the assembling process that results in the construction of an object according to AT is a compression scheme that belongs to the LZ family of compression methods, and therefore cannot perform better than Shannon Entropy in stochastic scenarios. All the detailed formalism and proofs are provided in the supporting information of the present article (S1 Appendix).

The LZ family includes many distinct compression methods, such as LZ77, LZ78, LZW, LZSS, and LZMA [9]. In the same manner as exemplified by the illustrative individual cases in later work in [10], one can trivially prove, for example, that the parsing according to the LZ78 differs from the parsing according to the LZ77, or that the one according to the LZMA differs from LZW, and so on. In accordance with the results earlier demonstrated in the present article, the examples presented in [10] further confirm that different parsings give rise to variations of compression schemes in the LZ family. The pointers in traditional LZ compression schemes, such as LZ77, LZ78 or LZW, refer to encoded tuples that have occurred before as recorded by the associated dictionary. Instead, for minimum rooted assembly subspaces, the pointers refer to other tuples that might not have occurred before in a particular assembling path, but are part of the minimum rooted assembly subspace, that is, part of a “tree”-like structure that is encoded in the dictionary, which is only built and queried during the encoding/decoding process.

The compressibility achieved by the LZ scheme that the assembly index calculation method is equivalent to depends on the length of the shortest assembling paths and the “simplicity” of the minimal assembly spaces. This notion of simplicity encompasses both how the assembly space (AS) structure differs from a single-thread (or linear) space and how many more distinct assembling paths can lead to the same object. Such an LZ scheme reduces the complexity (which the assembly index aims to quantify) of the assembling process of an object to the compressibility and computational resource efficiency of a compression method.

The generative/assembling process in an assembly space (AS) is strictly equivalent to a Context-Free Grammar (CFG), while the assembly index is equivalent to the size of such a compressing grammar. This implies that the assembly index can only be an approximation to the number of factors (or phrases) [11] in LZ schemes (e.g., LZW), which are statistical compression methods whose compression rates converge to that of Entropy. Thus, the assembly index calculation method is also a CFG-based compression method whose statistical compression rate is bounded by LZ schemes, such as LZ78 or LZW, and therefore by Entropy.

The theory and methods of AT describing a compression algorithm are, therefore, subsumed into Algorithmic Information Theory (AIT), which in turn has been applied in the same areas that AT has covered, from organic versus non-organic compound classification to bio- and technosignature detection and selection and evolution [6, 7, 12, 13]. This makes the assembly index and assembly number proposed by AT effectively approximations to algorithmic (Solomonoff-Kolmogorov-Chaitin) complexity [14–17].

Algorithmic complexity has been shown to be profoundly connected to causality [18–21], and not only in strings. It has also been applied to images [13], networks [22–25], vectors, matrices, and tensors in multiple dimensions [26–28]; and to chemical structures [6]. In contrast to the statistical compression schemes on which AT is based, and whose assembly index method is a particular case of a compression scheme, exploring other resource-bounded computable approximations to algorithmic complexity [20, 29] that consider aspects other than traditional statistical patterns such as identical repetitions, may have more discriminatory power. This is because they may tell apart cases with a high assembly index and an expectedly low copy number (or frequency of occurrence in the ensemble) from those with low LZ compressibility and high Entropy, seeming to indicate that an object is more causally disconnected, independent, or statistically random, while actually, it is strongly causally dependent due to non-trivial rewriting rules not captured by statistical means [5, 30]. This is because AT is to statistical correlation, as measures of algorithmic complexity are to causation. Rarely the two are the same—correlation is not causation—and when they are, they are fully captured by other traditional statistical measures such as Shannon Entropy without the introduction of a methodological different framework that adds no more to the correlation problem beyond Shannon Entropy.

As proposed in [3], the assembly number is intended to measure the amount of selection and evolution necessary to produce the ensemble of (assembled) objects. That is, the assembly number aims to quantify the presence of constraints or biases in the underlying generative processes (e.g., those parts of the environment in which the objects were assembled) of the ensemble, processes that set the conditions for the appearance of the assembled objects. A higher assembly number—not to be conflated with the assembly index—would mean that more “selective forces” were in play as, e.g., environmental constraints or biases, in order to allow or generate a higher concentration of high-assembly-index elements. Otherwise, these high-assembly-index objects would not occur as often in the ensemble.

Consonant with the constraints and biases that the assembly number aims to quantify, though in fact, as we demonstrate, it constitutes a compression method subsumed into AIT, ensembles with higher assembly numbers are more compressible and would therefore diverge more markedly from those ensembles that are outcomes of (or constituted by) perfectly random processes. In turn, a more compressible ensemble implies that more constraints or biases played a role in generating more high-complexity objects more frequently than would have been the case in an environment with fewer constraints and biases (i.e., a more random or incompressible environment), thus increasing the frequency of occurrence of less compressible objects in this environment. Should one assume that the assembly number is indeed capable of measuring this feature or characteristic of the ensemble as a whole from the distribution of the assembled objects, then a higher assembly number would mean that more constraints or biases played a role in increasing the frequency of occurrence of high-assembly-index objects.

Conversely, under the same assumption, the presence of more biotic processes in an ensemble would imply a higher assembly number, which in turn imply that the ensemble is more compressible. This occurs, for example, in scenarios where there is a stronger presence of top-down (or downward) causation [21] behind the possibilities or paths that lead to the construction of the objects, while a less compressible ensemble would indicate a weaker presence (or absence) of top-down causation (see also Section 3).

We therefore conclude that AT cannot offer a different or better explanation of selection, evolution, or top-down causation than the connections already established [7, 18, 21], consistent with our previous position that a single, intrinsic scalar is unlikely to classify life or quantify selection in evolutionary processes independently of the environment and the perturbations it imposes on the objects (whether these are biotic or abiotic). Characterising the complexity of the phase transitions in complex systems has long been investigated within the scope of emergent and self-organising properties that either derive from or downwardly affect interactive relationships between local (micro-level or individual) systems [21, 31–33]. In order to understand the interactions between the system’s parts and the environment’s subsystems, it is necessary to devise sound theories that explain crucial aspects of complex systems in general, such as swarm intelligence [34], gene regulatory networks [35], self-organisation [31, 32, 36], and two-way causality effects in feedback loops between the system and the environment [21, 37, 38].

An intrinsic complexity measure, such as the one put forward by AT, would also face many challenges to explain phase transitions near the criticality, where systems are on the verge of collapsing ordered patterns into disorganised parts or stochastic randomness [36, 39]. This is because the investigation of collective dynamics that accounts for other extrinsinc factors, e.g. those brought about by causal effects between multiple subsystems [20], is a challenge to such intrinsic complexity measures. These would miss the quantification of phenomena resulting from network dynamics that are crucial to understand (abiotic or biotic) complex systems [6, 22, 40, 41], whose emergent increase in the collective/global computation capabilities has been shown to be fundamentally connected to network topological properties [21, 42, 43].

As shown in the following Sections 2.1, 2.2, 2.3, 2.4, and 2.5, and demonstrated in S1 Appendix, the claim advanced by its authors that AT unifies life and biology with physics [3, 44] relies on a circular argument and on the use of a popular compression algorithm, with no empirical or logical support to establish deeper connections to selection and evolution than those already known (such as high modularity and self-assembly), already made (in connection to complexity) [7, 21, 22], or previously investigated using information- and graph-theoretic approaches to chemical and molecular complexity [6, 45–47].

2 Results

2.4 Lack of control experiments and absence of supporting empirical data beyond current domain knowledge

A critical examination of the AT methodology reveals significant shortcomings. Firstly, proponents of AT failed to conduct basic control experiments, a foundational aspect of introducing a new scientific metric. Benchmarking against established indices, particularly in coding and compression algorithms, is crucial to validating any new metric in the domain. Previous work on AT has never included meaningful experimental comparisons of the assembly index with other existing measures on false grounds that their measure is completely different [10, 54] (see Figs 1 and 2). Yet, we have shown that other algorithms, such as RLE, Huffman coding (the first dictionary-based universal compression algorithm), and other compression algorithms based on dictionary-based methods and Shannon Entropy produced equivalent or superior results compared to the results published by the authors of AT [5].

AT also introduces a cutoff value in its index, purported to offer a unique perspective on molecular complexity, distinct from those based on algorithmic complexity. This value indicates when an object is more likely to be organic, alive, or a product of a living system. However, various indices and compression algorithms tested have yielded equivalent cutoff values (see [5]). In fact, such an Assembly Index threshold replicates previous results on molecular separation that employ algorithmic complexity. Thus, this overlap in results challenges the notion that AT provides a unique tool for distinguishing between organic and non-organic matter. For instance, similar cutoff values have been derived in other studies, such as those mentioned in [6], effectively separating organic from non-organic molecular compounds. While the authors of AT ignored decades of research in chemical complexity based on graph theory and the principles of Entropy [45–47, 61], the depth of the purported experimental validation of AT has been notably limited, especially when contrasted with more comprehensive studies. For instance, a more exhaustive and systematic approach, with proper control experiments involving over 15,000 chemical compounds [6] and employing algorithms from the LZ family (or Entropy) and others of greater sophistication (BDM), demonstrated the ability to separate organic from non-organic molecular compounds.

Turing’s motivation was to explore what could be automated in the causal mechanisation of operations in Fig 1B, for example, stepwise by hand using paper and pencil. The shortest among all the computer programs of this type is an upper bound approximation to K. In other words, K cannot be longer than the length of this diagram. The concept of pathway complexity and the algorithm for copy number instantiated by the assembly index formally represent a restricted version of a Turing machine or finite automaton. As also discussed in Section 2.2, it is a mistake to think of a Turing machine as a physical object or an object with particular properties or features (such as a head or a tape). All the algorithms in assembly theory are Turing machines.

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Fig 1.

A: The authors of AT have suggested that algorithmic complexity (K) would be proven to be contained in AT [55]. This Venn diagram shows how AT is connected to and subsumed within algorithmic complexity and within the group of statistical compression as proven in this paper (see S1 Appendix) by a simple template argument representing the very restricted type of complexity that is able to capture. B: Causal transition graph of a Turing machine with number 3019 (in Wolfram’s enumeration scheme [36]) with an empty initial condition found by using a computable method (e.g. CTM [62]) to explain how the block-patterned string 111000111000 was assembled step-by-step based on the principles of algorithmic complexity describing the state, memory, and output of the process as a fully causal mechanistic explanation. A Turing machine is simply a procedural algorithm and any algorithm can be represented by a Turing machine. By definition, this is a mechanistic process as originally intended by Alan Turing himself, and as physical as anything else (first computers were human), not an ‘abstract’ or ‘unrealisable’ process as the authors of AT have suggested [54] misunderstanding a basic concept.

https://doi.org/10.1371/journal.pcsy.0000014.g001

2.5 A circular argument cannot unify physics and biology

Central to Assembly Theory (AT) and the public claims made through the authors’ university press releases [44] is the connection that they make to selection and evolution, maintaining that AT unifies physics and biology and explains and quantifies selection and evolution, both biotic and abiotic [3, 10]. To demonstrate this connection, assuming selectivity in the combination of linear strings (or, equivalently, objects) (P), the authors compare two schemes for combining linear strings Q, one random versus a non-random selection. This experiment yields observations of differences between the two (R), and the authors conclude that selectivity (S) exists in molecular assembly, and therefore that AT can explain it.

Observing that a random selection differs from a non-random selection of strings, the authors use this as evidence for “the presence of selectivity in the combination process between the polymers existing in the assembly pool”.

Yet, assuming selectivity in the combination of strings and proceeding to say that a selection algorithm is expected to differ from a random one makes for a circular argument. Furthermore, the conclusion (S) reaffirms the initial assumption (P), resulting in a circular argument. In a formal propositional chain, it can be represented as P ⇒ Q ⇒ R ⇒ S ⇒ P, where the conclusion merely restates the initial assumption, lacking any validation or verification, empirical or logical, beyond a self-evident tautology.

Whether this sequential reasoning is relevant to how molecules are actually assembled is also unclear, given the overwhelming evidence that objects are not constructed sequentially and that object complexity in living systems is clearly not driven by identical copies only [63]. See also Section 3.

Nevertheless, the question of random versus non-random selection and evolution in the context of approximations to algorithmic complexity, including copy counting, was experimentally tested, empirically supported, and reported before in [7] following proper basic principles such as a literature search and review, control experiments (comparison to other measures), and validation against existing knowledge in genetics and cell biology.

Such circular arguments, lacking a foundation in chemistry, biology, or empirical evidence, lead the authors to propose the concept of “assembly time”. The suggestion is a time-scale separation between molecule production (assembly time) and discovery time, aiming to unify physics and biology. Time, however, has always been fundamental in evolutionary theory, and claims about an object’s historical contingencies are the foundations of evolutionary theory. Thus, AT revisits existing complexity science concepts described in the timeline provided in Fig 2 but without acknowledgment or attribution.

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Fig 2.

A timeline of results in complexity science relevant to the claims and results of AT, which renames several concepts, e.g. dictionary trees as ‘assembly (sub)spaces’; relies heavily on algorithmic probability in its reduction of combinatorial space arguments, without attribution; and, as demonstrated, the assembly index is an LZ compression scheme (proofs provided in the S1 Appendix).

https://doi.org/10.1371/journal.pcsy.0000014.g002

3 Discussion

The most popular examples used by the authors of Assembly Theory (AT) in their papers as illustrations of the way their algorithms work, such as ABRACADABRA and BANANA, are traditionally used to teach LZ77, LZ78, or LZW compression in Computer Science courses at university level. Dictionary trees have been used for pedagogic purposes in computer science for decades. In addition, beyond a notation ‘c(x)’ for the formalisation of the assembly indexes (and assembly spaces) [54] that resembles that of the number of phrases (or factors) in the parsings of LZ78 in the proof presented in [60], they also share underlying key ideas in the relationship between these parsings and the probability of the phrases into which the objects are decomposed. This is because one of the central motivations for the assembly index as a complexity measure is that the probability of assembling paths should decrease as the respective assembly index (or, in the case of molecules, MA) of the object increases [2], unless there are environmental constraints or an extrinsic agent responsible for increasing the frequency of occurrence of high-assembly-index objects. Similarly, this appears as the key idea in the proof that LZ78 achieves optimal compression [60] for stationary and ergodic stochastic processes: sequences with a larger number of distinct phrases to which the pointers necessarily recur less often would have lower probability, and therefore they are less compressible; while sequences with fewer distinct phrases corresponds to a higher probability, and therefore they can be compressed more efficiently.

At first glance, the above aspects suggest some similarity between AT and compression algorithms, particularly those of a statistical nature based on recursion to previous states, copy counting, and the re-usage of patterns and repetitions. Indeed, such an interdisciplinary approach between physics, chemistry, and computer science is very beneficial and fruitful for science in general, specially in the field of complex systems science. However, in contrast to AT’s authors claims, we have shown in the present article that the calculation of the assembly index—by finding a minimal assembly (sub)space (AS) rooted in the basis objects—in fact is a compression scheme belonging to the LZ family of compression algorithms, rather than merely being similar. The present study also theoretically establishes and extends the limitations previously investigated in [5].

We have formally proved that the minimum rooted AS from which the assembly index is calculated is a compression scheme (namely, ‘LZAS’) that belongs to the LZ family of compression algorithms.

This result also mathematically proves how the compression rates achieved by the assembly index calculation method depend on shortest assembly paths, assembly indexes, and assembly spaces. Additionally, it demonstrates that the assembly index calculation method is a dictionary-based compression method whose encoded dictionary enables one to generate/assemble the objects, for example via a context-free grammar (CFG). In fact, we also prove that the assembly index itself is equivalent to the size of a compressing CFG. Diving into the details of AT’s algorithms and methods, we have revealed that they are equivalent to LZ encoding, and therefore to (Shannon) Entropy-based methods. All of the results in this article are fully detailed in the S1 Appendix.

Our results prove that an object with a low assembly index has high LZ compressibility (i.e., it is more compressible according to the LZ scheme), and therefore would necessarily display low Entropy when generated by i.i.d. stochastic processes (or low Entropy rate in ergodic stationary processes in general). In the opposite direction, an object with a high assembly index will have low LZ compressibility, and therefore high Entropy. The notion of how complex an assembling process needs to be in order to construct an object is equivalent to how much less compressible (by the LZ scheme to which the assembly index calculation method is equivalent) and more computationally demanding (according to this encoding-decoding LZ scheme) this process is. This interdependence between compressibility and computational efficiency characterises the notion of complexity that the assembly index intends to measure, but that in fact pertains, as we have demonstrated, to an LZ compression scheme.

Thus, the methods based on AT, Shannon Entropy, and LZ compression algorithms are indistinguishable from each other with regard to the quantification of complexity of the assembling process, and the claim that these other approaches are incapable of dealing with AT’s type of data (capturing ‘structure’ or anything that could supposedly not be characterised by Shannon Entropy or LZW) is inaccurate.

Kempes et al. [10] argue in later work that the assembly index is distinct from LZW and Huffman encoding by presenting counterexamples in which their parsings differ. These examples highlight parsing differences already subsumed into the general-case proofs in the S1 Appendix. The LZ family tree of compression schemes includes many distinct schemes, including LZ77, LZ78, LZW, LZSS, and LZMA [9], and as demonstrated in the present article it also includes the assembly index calculation method (to which we refer as ‘LZAS’). These LZ schemes may differ not only in the parsing (or decomposition into factors or phrases) of the object, but also in performance. All of these compression methods are defined by schemes that resort to a (static or dynamic) dictionary containing tokens, indexes, pointers, or basic symbols from which the decoder or decompressor can retrieve the original raw data from the received encoded/compressed form according to the respective LZ scheme [9, 60]. Thus, raising an argument such as the one that the assembly index calculation method is not LZ compression because its parsings differ from those of LZW is as pertinent as arguing that LZW is not LZ compression because its parsings differ from those of LZ77.

Computing the exact value of the assembly index (i.e., the size of the minimum rooted assembly space) is known to be intractable in the general case so that one needs to employ approximation methods, such as the split-branch version [54, 56], to speed up the process by constraining the search over the possible assembling spaces or paths. Because the assembly index calculation method is an LZ compression scheme, our results also demonstrate that such an intractability not only arises in potential applications of AT to coding and compression, but also that it is the very intractability which in fact is intrinsic to AT, occurring whenever one tries to calculate the assembly index for an assembly space—a process for which AT had to employ approximation techniques, such as the split-branch assembly space.

Different computationally efficient implementations of these approximation methods will produce varying results that eventually diverge (i.e., they are suboptimal with respect to what AT formalised and proposed as the ideally true value) from the optimal/ideal assembly indexes in the general case (which remains computationally intractable). As we have demonstrated, each of these more efficient approximation methods defines a variation of the (computationally inefficient) LZAS scheme, variation which is in turn equivalent to a more tractable compression scheme in the LZ family. Thus, the more computationally efficient an empirical implementation of the assembly index approximation method is, the more efficient the LZ compression scheme that such a method defines, and hence is equivalent to.

Since previous work in [5] (and also further investigated in [53]) has shown that the assembly index method performs equally well, or worse in some cases, an extensive evaluation of the performance of their algorithms in contrast to other established methods in the literature is necessary in future research, should the proponents of the assembly theory aim to argue that their specific implementations outperform established, computationally efficient compression algorithms like LZW for their specific purposes.

We have also demonstrated that AT is subsumed into the theory and methods of Algorithmic Information Theory (AIT) as illustrated in Fig 1A and mathematically proven in the S1 Appendix. In addition, both theoretical concepts and empirical results in [2, 3, 54, 64] have been reported previously in earlier work in relation to chemical processes in [6], and to biology in [6, 7, 12, 13] but with comparison to other measures, including methods that go beyond traditional statistical compression and are connected to the concept of causal discovery [20, 22].

In cases which AT displays discriminatory power [2, 64], it is because of its connection to Shannon Entropy and statistical compression in general. Our results first demonstrate that in the case of pure stochastic processes, the assembly number (not to be conflated with the assembly index) is either a suboptimal or an optimal compression method with respect to what the noiseless source coding theorem establishes, therefore adding no advantage in comparison to (Shannon) entropy-based methods. Secondly, in the general case—including the pure stochastic one but also when the degree of stochasticity or determinism of the generative processes of the ensembles is unknown—, they demonstrate that the assembly number is a compression method, and thus an approximation to algorithmic complexity (or, equivalently, to algorithmic probability), but a suboptimal measure with respect to the more general methods from algorithmic information theory.

AT may pedagogically contribute introducing a graph-like representational approach to approximating LZ compression in the specific context of molecular complexity potentially making it more accessible to a broader and less technically-minded community. However, as proven here, it is fundamentally not different either methodologically or fundamentally from Shannon Entropy and lossless compression and is inaccurate and a bad practice to ignore, omit or imply the absence of previous work (e.g. [6–8, 57]). We believe that it is wrong to belittle and devalue the theories and ideas (Shannon Entropy, compression, and algorithmic complexity) which AT’s indexes and measures are based on, and make disproportionate public claims related to the contributions of AT (with the media and the authors themselves going so far as to call it a theory that ‘unifies biology and physics’ and a ‘theory of everything’ and so on, e.g. [1]).

Both the assembly number and the assembly index are currently defined in such a way that prevents AT to quantify, even in principle, the downward effects of environmental influence and constraints on the assembling process of the objects, thus preventing it to quantify the presence of top-down causation. For example, these effects have been shown before to have a fundamental relationship with the notion of complexity [21, 32, 65, 66]. This is because the assembly index (not to be confused with the assembly number) of an object in its current mathematical formulation is an intrinsic measure not sensible to increases or decreases in the “complexity” of the ensemble as a whole (or, as AT purports, in the skewness toward high-“complexity” objects in the ensemble). The assembly number in its current state can only quantify the global variations of “complexity” resulting from individual local variations of the objects’ “complexities” (i.e., as proposed by AT, their assembly indexes), or variations in the distribution of these individual values—therefore, in principle, only bottom-up causation. However, it cannot quantify the other way around, that is, quantify local variations of “complexity” resulting from global variations of “complexity”.

In order to take into account top-down causation, the assembly index of an object itself would need to also depend on the multi-agent interaction dynamics (or organisation) of extrinsic factors, e.g. environmental catalytic conditions that may play a role in adding or reinforcing biases toward the formation of higher-assembly-index objects through less likely assembly pathways. Otherwise, the assembly index may classify a low-complexity molecule as being constructed by a more complex extrinsic agent, which is in fact of a much simpler nature (e.g. a naturally occurring phenomenon or the interaction of many abiotic subsystems in the environment).

That is, in case sufficiently complex environmental catalytic conditions play the role of this extrinsic factor (which increases the bias toward the construction of a more complex molecule), such a level of complexity would be completely missed by the capabilities of a simplistic measure such as the assembly index, thereby rendering it prone to false positives [4, 5].

This kind of influence (whether top-down or bottom-up) in the interplay between global and local scales is e.g. encompassed by complexity measures based on algorithmic probability (or, equivalently, algorithmic complexity) [18, 21, 22, 67]: when one approximates the algorithmic probability of either an individual or a collective (e.g., an object or a set), a greater quantity of possible underlying generative processes of different nature (whether intrinsic or extrinsic) are taken into account with the purpose of estimating the most probable candidate generative model.

Thus, a pervasive comparison in future research between assembly number and algorithmic-probability-based measures for group-controlled ensembles sampled from distinct environments may help quantify the presence of AT’s false positives. As the assembly number (or, as we have demonstrated in this article, the compressibility) of an abiotic ensemble sufficiently increases for certain environmental conditions, this relative increment of global “complexity” may be sufficient to in turn facilitate the formation of abiotic molecules with relatively higher assembly index. By empirically measuring the presence of such an effect on the assemblage of individual objects, future research is necessary to quantify the presence of top-down causation—a problem also raised in [68]—, and therefore it will help corroborate or falsify the AT’s formalism and assumption of a bottom-up-only intrinsic complexity measure being able to distinguish the transition from non-life to life.

We agree with AT’s authors on the importance of understanding the origin and mechanisms of the emergence of an open-ended generation of novelty in complex systems [21, 69]. However, these efforts by the authors are undermined by hyperbolic claims, fallacious arguments, and lack of attribution and comparison with previous results in the literature.

We argue that the results and claims made by AT, in fact, highlight a working hypothesis that information and computation underpin the concepts necessary to explain the processes and building blocks of physical and living systems, concepts which the methods and frameworks of AT have been heavily inspired by [70, 71].

Supporting information