One of the main problems in AI can be the lack of a mathematical definition of intelligence. The concept of intelligence is not related to optimization. Optimization is usually an automatic process based on mathematical algorithms. With the development of artificial intelligence we have found a link between them, however, this must be researched a little more.

The very first attempts to introduce intelligence in control systems were done through logic algorithms. Logic can be expressed in mathematical functions and then, logical reasoning can be considered a mathematical one.

It is true that AI algorithms can be used to solve some problems of optimization: however, they cannot provide an exact solution in mathematical terms for many problems.

As artificial intelligence has evolved from automation, we usually look at intelligence as a process of automation. But if we look in deeper, we can find that the process of intelligence usually tries to escape from automation algorithms. AI in control systems has been mostly used to provide supervisory control. I am going to put an example in order to illustrate this concept.

Imagine a mobile robot or a car driven by AI. The optimal route from a point A to a point B is a straight line. We do not need AI in order to make that the device reaches the point B. AI is required when we cannot define the optimal route. If there is an obstacle in route of the mobile robot, it must change the automatic calculation in order to describe a new path. The intelligence of the device lets to escape from the optimal path because it can incorporate new information from the environment.

Intelligence is not the capability to solve formal problems only but the capability to adapt itself to different scenarios too. In this point, we can link intelligence with complexity. A complicated system has a lot of interconnected parts and its state is deterministic. A complex system has a lot of interconnected parts and its state have a certain degree of uncertainty.

To consider that intelligence is logical reasoning is to focus it on the support of the analysis of complicated systems; however, complex systems require more things than logical reasoning.

These problems arose in the early days of artificial intelligence. The answer for the analysis of uncertainty was the definition of new logic, known as fuzzy logic, that lets to provide a single solution for a problem where state variables are under uncertainty. Tools like experts systems were adapted to use of fuzzy logic.

Complex systems are characterized because they can change sharply their behavior under unexpected events. Intelligence offers solutions to those changes trying to preserve the system under control. In the example of the mobile robot, AI would calculate a new path to the point B after the detection of an unexpected obstacle in the way. In this case, AI is not optimizing the path but abandoning it after the unexpected event. The system is, then, under control because it is following a new path although it cannot follow the previously defined one. On the other hand, a probabilistic system would have calculated the initial path considering the probability of finding an obstacle in the way but this probability is impossible of knowing in most situations.

This concept can be very important when we are using big data to determine the behavior of a system. We use big data in order to define probabilities that can be incorporated to the algorithms. We are assuming that we can avoid any unexpected event increasing the amount of information, what is very vain as I am going to explain now.

When we play the dice, we assume that the probability of getting a certain result is 1/6 because we assume that it was perfect and homogeneously constructed from a physical viewpoint. If we accept that it was not perfectly built, we can use the big data. Once it has been thrown out many times, we can get different probabilities from the historic data that would adjust to its defects. But it does not provide information about a new dice. We cannot define a probability for the new dice, only the possible results without a probability for any of them. The result must be seen as uncertainty instead of risk.

Intelligence is a way to cope with complex systems and environments. Great geniuses of science have been people that have been able look at certain problems in a way far from the mean. Intelligence never has been a matter of statistics calculation but a matter of changing models of analysis. It has been related to the enlargement of the system domain to provide more general solutions and to apply it to a particular one. It is a matter of better identification of problems instead of simple logical or probabilistic reasoning. Perhaps, complexity analysis can provide a good starting point to produce new AI algorithms from a mathematical viewpoint.