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Using the effective complexity measure, proposed by M. Gell-Mann and S. Lloyd, we give a quantitative definition of an emergent property. We use several previous results and properties of this particular information measure closely related to the random features of the entity and its regularities.

The Latin word

It is worth of mentioning that there have been very recent efforts to quantify emergence [

Also relatively recently, see [

physical systems that goes from the transparency of the water (or other liquid), phase transitions, and the so-called self-organized criticality state in granular systems, on one side, to the emergence of space-time at the other end of the physical scope;

biological systems, like the multicellular construct in a given organism, ending ultimately in organs, and the morphogenesis phenomena;

social organization observed in insects, mammals, and in general in every biological system consisting of agents (notice how the combination and interaction of all these subsystems also establish a higher level of emergent phenomena, as one can see, for example, in the biosphere).

In

To have the character of an emergent phenomenon, as we have mentioned, this new feature must not be reducible to the basis; it should be unexpected [

Before the twentieth-century, it was a consensus that large degrees of freedom were a necessary condition of unpredictable behavior. Probably the most beautiful (and maybe unfortunate) text written on these lines was by Laplace in his remark: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” Notice here the explicitly strong deterministic view of all natural phenomena, where no room for surprise or novel properties are possible since the power of calculation of the Laplace Demon envisions all future outcomes. Nowadays, we are certain that deterministic low-dimensional systems can exhibit very complicated behavior, and moreover, they have imprinted the very idea of unpredictability (if lack of infinite power computation) when strong chaos is present.

“Prediction is difficult, especially the future” disputably attributed to Niels Bohr, and “The next great era of awakening of human intellect may well produce a method of understanding the qualitative content of equations. Today we cannot. Today we cannot see that the water flow equations contain such things as the barber pole structure of turbulence that one sees between rotating cylinders. Today we cannot see whether Schroedinger’s equation contains frogs, musical composers, or morality—or whether it does not” [

Niels Bohr, most probably, was inspired by quantum physics, concerning (putting it very simply) the uncertainty principle. Under this circumstance there is a range of novelties (or different outcomes) that a physical system can have, and the capacity for the observer to know the future is reduced to a given set of probabilities even using the

Regarding these comments, it is important to address the question of how well the theory describes the phenomenon, or in other words, the coarse-graining at which the prediction’s theory works. We should be satisfied in saying that the theory works for a given phenomenon if described at the level in which the regularities that one wants to study are better explained. An example of this can be seen in classical mechanics at the level of describing the elliptical orbits, but we cannot give a complete answer to more complicated and detailed patterns like the anomalous precession of Mercury’s perihelion using this theory. Of course, this is not the only aspect considering a physical theory. For a given phenomenon the explanation should be simple,

There is somehow a strong connection, at least in a huge part of the community discussing emergent phenomena and emergent properties, between complex systems and emergence [

Much literature has been produced trying to define the concept (or from now on the measure) of complexity of a given system [

Some of these measures have been proposed to study different characteristics of a given system [

A perceptive reader should be noticing one very important aspect of the theory developed by Gell-Mann and Lloyd, that the effective complexity of an entity is dependent on the context and the subjectivity. We will give a

In order to introduce the effective complexity of an entity [

The explanation should be

The explanation should

The total information, ∑, will be the sum of the ensemble’s entropy

Now, a good theory requires that the previously defined total information should be as small as possible. This means (like Lemma 3 in [

Clearly, there should be a requirement to choose the best theory from the ones satisfying the last inequality. Following M. Gell-Mann and S. Lloyd, we will say that the best theory is the simplest theory, which in terms of this discussion means the ensemble

We will introduce a control parameter

The effective complexity of a string

where

A property,
_{c}_{c}

with

In this work, we have introduced a novel concept related to the very idea of an emergent property, and how it can be quantitatively described. We paid some attention to motivate the discussion on the important aspects of the role of a given theory to explain phenomena (any discussion related to emergence should address this issue). Another point that arises in our presentation, which is also of particular importance to understand how the topic depends on subjectivity and context, is how to distinguish regular features of the entity (

Under this framework, the concept

The epistemological point of view of the definition presented here is clear. Its consequences and possible future direction of research will be communicated elsewhere.

We would like to conclude by quoting a philosophical remark, very much within the lines of our results, by Carl G. Hempel and Paul Oppenheim [

The author deeply thanks Tanya Elliott for her constructive insight and criticisms, CONICTY Project: Anillo en Complejidad Social SOC-1101, FONDECYT 1140278, and Juniper Lovato for her review of a previous draft of this work.

The authors declare no conflict of interest.

Sketch of an emergent property. At level I interactions occur. When describing the entity at level II, a new property appears. This property must have an associated set of data