What is complexity?

Complectere is constructed from the Latin root plectere which means “braid, link” to which the prefix com- is preceded, with which the word acquires the sense of the duality of two opposite elements that are intertwined intimately, but without canceling its duality. Thus, the term complexity that comes from Complectere, refers to what is intertwined by the effect of opposing and complementary forces: randomness and self-organization.

In general, these two forces act through very simple components that interact with each other and with their surroundings, often not linearly, and without central control. These forces give rise to a set of properties that we have identified common to most of what we call complex systems: the existence of heterogeneity, hierarchical order, openness, adaptation, memory, emergency and anticipation.

The consequences of this is that complex systems generate new information not contained in the original description of its parts, but that it arises from interactions. In addition, it is common for these interactions or the properties that emerge from them to change at an observable rate. This implies that in general, complex systems are unpredictable and uncontrollable.

But we have advanced too much towards this that would be our proposal of definition of complexity.

One of the most impressive features of complex systems is that they are very simple, in the sense that their components are very simple. Take for example what is known as the game of life.

The game of the life of the British mathematician John Horton Conway, was first presented in the October 1970 issue of Scientific American, in the column of mathematical games by Martin Gardner. From a theoretical point of view, it is interesting because it is equivalent to a universal Turing machine, that is, everything that can be computed (algorithmically) can be computed in the game of life.

You may have heard of Truing for the film that takes part of his life as the main plot, Enigma (2001). The film’s title was taken from the encryption machine used by the Nazis during World War II and was finally deciphered by the first computer built by Turing, Ultra (reminds them of Ultron?).

In the image a public domain photograph taken from Enigma’s Wikipedia.

Of course, this simplified version is very unfair with respect to the heroic work of the Poles before the appearance of turing who not only intercepted a non-military Enigma machine before the war but also from their study, Marian Rejewski, Jerzy Rozycki and Henryk Zygalski managed to break their code from a formal mathematical perspective. However, as the character Fat Tony would say in the monumental work on the role of randomness in life and decisions, the Uncertain Taleb, and that it is largely the source of inspiration for many parts of this work. “In theory there is no difference between theory and practice, in practice if there is one”

And it is that one thing is to solve a formal mathematical problem and another very different is to be able to perform the calculations to apply it in the real world. It was there that the genius of Turing and his collaborators (sorry for being systematically unfair) took a turn to the Second World War to effectively decode war messages encrypted with Enigma using his Ultra (mechanical) computer.

In honor of Turing, a device that manipulates symbols on a strip of tape according to a rule table is called “Turing machine”. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm. In the same way a Turing machine is defined as a device that is capable of simulating any other Turing machine. So when we say that the game of life is a universal turing machine, we are not saying anything.

An American-made version of the Bombe, a machine developed in Britain for decrypting messages sent by German Enigma cipher machines during World War II.

National Museum of the US Air Force (070918-F-1234S-006)

The amazing thing is not that it is a universal Turing machine, but the simplicity of the game of life.

The game of life is a zero player game, which means that its evolution is determined by the initial state and does not need any subsequent data entry. The game takes place on a virtual board designed as a rectangular flat mesh in the style of a chess board in which each square is called a “cell” and has a set of variables or properties defined in it. The status of all cells is taken into account to calculate the status of the cells the next turn, in a scheme of first neighbors (the eight cells adjacent to any one in particular). All cells are updated simultaneously at each turn, following these rules:

(1) a dead cell enters a live state, with exactly 3 live neighboring cells (that is, the next turn will be alive); (2)

a living cell with 2 or 3 living neighboring cells is still alive, otherwise it dies (due to “loneliness” or “overpopulation”).

The interested reader could explore this implementation of the game of life online: https://pmav.eu/stuff/javascript-game-of-life-v3.1.1/ or if you even want to learn to program or modify it, you can visit NETLOGO implementation that can be explored either online or by downloading it: https://www.netlogoweb.org/launch#https://www.netlogoweb.org/assets/modelslib/Sample%20Models/Computer%20Science/Cellular% 20Automata / Life.nlogo

In the image above taken from wikipedia with CC license, with these very simple rules we can observe how space-time patterns arise (or emerge) like these patterns that travel from the upper central part diagonally downwards to the left. These patterns known as sliders cannot be predicted based only on the initial state of the system.

Another great example of this is surfactants, for example sperm-shaped molecules that can have hydrophilic heads and hydrophobic tails such as those shown in the image on below taken from wikipedia with a CC license.

One can perfectly know all the physicochemistry of a surfactant molecule and we could not predict the emergence of spatial structures such as micelles (these spheres of surfactant molecules) once a certain critical concentration is exceeded from which these molecules are self-organize (obviously without central control).

A great Complexity Literacy, in which i had a little participation, created by the international community of Complexity scientist may be found inhere: https://complexityexplained.github.io/