There are phenomena in nature that we can describe with such precision that we can predict their future behavior over very long periods of time. However, certain dynamics, such as that described by the Darwinian theory of evolution, are unpredictable.
If mathematical prediction becomes impossible, what can we say about a phenomenon that, by its very nature, cannot be predicted?
Fig.1 Illustration of an example devolution.
In this article, we'll be talking about "open evolution" (OE). This is the process of creating novelties typical of living systems, as described by the naturalist Charles Darwin in the second half of the 19th century.
Darwinian evolution is based on the principle of generating new living species. This is achieved through the iteration of the phenomena of variation, reproduction and selection.
Not only does it represent a cutting-edge field of research in theoretical biology and theoretical physics. It also has important implications for the study of complex systems. Last but not least, it influences a large number of technological applications.
Novelties and creativity
The dynamics majority of known natural phenomena is, by nature, predictable thanks to mathematical mathematical methods. These methods are not, however, sufficient to predict the behavior of certains living systems which are distinguished from inanimate systems by their strong dependence on theeur history ; the result of the choices made by the individual and his social group.
Variation as change of change
This historicity explains why it is impossible to predict changes using current mathematical tools. Indeed, the sequence of events makes the properties of two living systems, initially identical, singular.
For example, the events in the lives of twinned people will make them two fundamentally different individuals, even though there was no physical difference between them at birth. Darwin would call this adaptation.
In physics, we study objects called *generic* [1], because the laws governing them remain unchanged throughout their history. In biology, on the other hand, we speak of *specific* objects [1], because their determination depends specifically on their history.
We can redefine EO as a process of emergence of novelties playing an a priori unforeseeable role. These are functional innovations. These innovations irreversibly modify the system's functions and, consequently, its laws.
Fig.2 Graphical representation of the principle of up to third order order.
Lhe laws that described the system before the appearance of something new are no longer sufficient because the latter has undergone a functional irreversible functional change.
Small mathematical excursus
We now turn to a small example of the mathematical difficulties involved in describing these phenomena. Readers frightened by mathematics can save themselves the pain by skipping ahead to the next section.
The inaccessibility of the laws of evolution
Consider the evolution of a set of laws Ω that regulate the temporal dynamics of a certain system. The introduction of a novelty in the laws Ω implies an unpredictable functional variation 𝑓1 of Ω at some point in its history. We call this change 𝑓1(Ω). Taking our cue from Fig. (2), the new laws will themselves also be subject to another change 𝑓2. Iterating this reasoning, we obtain for the n-th variation an expression of the type :
Thus written, the evolution of the system could therefore be interpreted as the result of a hierarchy of functions 𝑓𝑖 defining the dynamics of the evolution of the laws.
However, it can be shown [1] that the only case in which EO is found is when, at any time, a new function 𝑓 can appear in any hierarchical order. Under these assumptions, variation is incalculable and no prediction is possible. In short, if a dynamic follows EO, then it is not describable by an enumerable set of equations.
The notion of creativity
But then, is it possible to predict the arrival of something new if it's unpredictable?
In other words, what's the minimum we can say about what we can't talk about? Is there a way of predicting at least the arrival of something new, without going into detail about it? Can we determine when a system is capable of open evolution, without necessarily specifying in which direction it will evolve?
To answer these questions, we'll be using the *Kolmogorov complexity*, or algorithmic complexity. This quantity, defined by the Russian mathematician Andrei Kolmogorov, corresponds to the size of the smallest possible algorithm capable of generating an object.
When studying a particular system, we can approximate this complexity by considering the size of the smallest known algorithm.
Evolving means boosting complexity
In the case of living systems, we can speak of the complexity of the description of specific objects, which, after the appearance of a functional novelty, must increase. However, a series of novelty appearances a a greater impact than a simple series of random events. It is on this assumption that recent theoretical developments [2] argue that OE is found in systems where the complexity K increases over time t more than a random chain. Since the complexity of a random chain increases linearly over time, we can say that the growth in complexity in the case of EO increases at least super-linearly. In mathematical terms [2]:
We will call 𝛾 creativity. A creative system is therefore a system capable of generating novelties. In other words, we can say that EO generates changes that follow a logic of historical dynamics as natural selection does, and are therefore more than just a random process.
Other ways of estimating the complexity of an object can be developed ad hoc for each system, as is the case, for example, with the Assembly theory [3,4] recently developed for chemical systems. This theory proposes to evaluate the complexity of a chemical molecule using the number of reactions required to synthesize it.
To open... you have to close!
On the other hand, open evolution cannot operate without certain conditions.
For example, an evolving species is also a species with an evolutionary history. This means that open evolution corresponds to the subsequent variation of a species whose system laws had already undergone other variations.
The appearance of a new variation is an emergent property. It's as if it represents a hidden creative "talent".
For emergence to occur, it is essential for the system's constituents to organize themselves. They must select the processes best suited to the specific context and history in which they are embedded. Darwin calls this selection.
The concept of a fence
In a multi-agent system, i.e. one composed of several individuals, open evolution is causally linked to the notion of *closure* [4,5]. This notion suggests interdependence and inter-activation of components over a certain time interval.
The existence of some components of the system becomes essential to the survival of others. In the long term, this creates a context in which specific objects are selected. These objects gradually become more and more complex.
An example of closure can be found in chemical reaction systems, the basis of all living systems and, more broadly, of life itself. Take a system of catalytic reactions, where the speed of reactions increases in the presence of a molecule called a catalyst. In this case, we can demonstrate the emergence of open evolution when reactions self-organize to form an autocatalytic group. This group is a set of catalytic reactions that produce their own catalysts [4].
In short, the more the components of the system are organized according to the principle of enclosure, the more selective forces intensify. In this context, species with better adaptive capacities will have a greater probability of generating novelties, and therefore of surviving.
Applications
Examples of EO in our daily lives include the evolution of languages, music and countless cultural phenomena. The study of methods for stimulating the creativity of a system is at the root of various types of fundamental research, because these kinds of phenomena are an integral part of our lives.
Ubiquitous applications
Artificial intelligence is one of the main fields of application, especially since the introduction of generative AI. In the field of AI, the idea is to push this technology towards the autonomous generation of novelties [6-7]. Indeed, controlling a system's creativity will be increasingly important in fields such as cryptography or defense, where predicting and possibly controlling creativity can compensate for the absence of specific information about future events. Even the videogame industry has recently begun to develop creative games capable not only of generating new side quests in total autonomy, but also of evolving the game world with its characters. This feature would push the game engine to generate other games that increase the complexity and longevity of the main experience [e.g. Minecraft, Elder Scrolls 6, GTA VI, etc.].
In basic research, scientists draw on EO theory to try and develop a formal mathematical framework capable of better describing life and its evolution. However, life and its evolution are still unexplainable phenomena. Even if great scientists have tried to make sense of them (cf. [8]), the laws of life still seem to elude our understanding.
Conclusions
Thanks to this article, we have discovered the notion of open evolution, a process of introducing functional novelties typical of living systems. These novelties are, by nature, unpredictable, which makes them very difficult to study. We have seen that it is possible to evaluate the creativity of a system, in order to develop methods that determine when a system is openly evolving. However, OE is not a phenomenon that emerges free of charge. It is for this reason that this type of evolution appears in systems that are highly dependent on their history, and which are better able to adapt to change. As those who manage to adapt to selective forces are also those who benefit from selection, this theoretical framework is well represented by the notion of closure.
Moral of the story
In the end, we can say that the key to open evolution lies in a compromise that seems as contradictory as it should be obvious to those who have followed me so far: if we want our system to evolve openly, it must first be closed!
If you'd like to find out more, or if you're interested in one of our experts as part of your team, please don't hesitate to contact us here. We'd be delighted to provide you with further details and discuss customized solutions tailored to your needs. Our specialists are at your disposal to explore collaboration opportunities together and guarantee excellent results.
Bibliography
[1] Maël Montevil, Matteo Mossio, Arnaud Pocheville, and Giuseppe Longo, "Theoretical principles for biology: Variation. Progress in Biophysics and Molecular Biology", 122(1):36-50, October 2016.
[2] Bernat Corominas-Murtra, Luıs F. Seoane, and Ricard Sole, "Zipf's Law, unbounded complexity and open-ended evolution", Journal of The Royal Society Interface, 15(149):20180395, December 2018.
[3] Abhishek Sharma, Daniel Czégel, Michael Lachmann, Christopher P. Kempes, Sara I. Walker, and Leroy Cronin, "Assembly theory explains and quantifies selection and evolution", Nature, 622(7982):321-328, October 2023.
[4] Marco Faggian, "Evidence of causality between Closure and Open-Ended Evolution in the Kauffman model", bioRxiv, November 2024
[5] Alvaro Moreno and Matteo Mossio. Biological autonomy: a philosophical and theoretical enquiry, volume 12. Springer, 2015.
[6] Tijn Van der Zant, Matthijs Kouw, Lambert Schomaker, "Generative Artificial Intelligence", Springer Nature
[7] "Open-Ended Intelligence", 9th International Conference, AGI 2016, New York, NY, USA, July 16-19, 2016, Proceedings
[8] Erwin Schrödinger, "What is life?"
Marco FAGGIAN
Data Scientist Consultant
