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*Correspondence:

This article was submitted to the journal Frontiers in Neurorobotics.

Edited by: Pierre-Yves Oudeyer, Institut national de recherche en informatique et en automatique, France

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Brains learn much better than computers, this has been discussed in a number of reviews on artificial intelligence, pattern recognitions, and neural networks (Perlovsky,

The reason for combinatorial complexity can be explained as follows: consider first, an example of a simple problem requiring no combinatorial complexity for learning: recognition of a single isolated object, which always appears exactly the same. Learning consists in storing in memory the object's image. Recognition consists in matching the stored image to a newly presented image: match or no match. The complexity of this algorithm approximately equals the number of pixels in an image. But in a real situation the object is not always exactly same; the algorithm has to account for variations in viewing angles, distance, color, etc. In addition, other objects are present with their variabilities. Combinations of various objects with their variabilities lead to combinatorial complexity. Combinations of all pixels in the field of view should be considered. A human eye senses ~10,000 pixels 10 times a second. Today, sensors measure millions of pixels each second (or more). The number of combinations of these pixels is “practically infinite”; combinations of 100 pixels (a relatively simple problem) are 100^{100}; this number is close to all of the interactions of all elementary particles in the entire life of the Universe.

Before considering how brains perceive objects, let us consider a parallel to the above complexity problem: the Gödel theory (Gödel,

In developing his theory Gödel demonstrated that all logical statements are equivalent in some way to all sequences of zeros and ones. It was essential for Gödel to consider infinite sequences of zeros and ones. The entire collection of such sequences is infinite and contains all infinite combinations of zeros and ones. The number of such combinations is a ^{N}^{100}, the “practically infinite” number discussed above. In both cases of finite and infinite sequences the number of combinations turns out to be significantly larger than the original sequence length. If Gödel's arguments are applied to any finite system, such as a computer, or a brain, and only finite combinations are considered, Gödel's proof of the existence of unprovable statements would not stand. A different difficulty would be faced, the practically infinite number of possible statements. No system, the mind or a computer would ever be able to count these statements; no algorithm would be able to execute so many operations. Similar to the Gödelian case, the number of combinations is “much larger” than the initial complexity. The combinatorial complexity of the logical algorithms considered previously is related to Gödel's argument when applied to a finite system.

As discussed above, machine learning and mathematical models of the mind face algorithmic difficulties related to combinatorial complexity. These difficulties are related to the use of logic in algorithms similar to the existence of unprovable statements in the Gödel theory. We face the possibility that the combinatorial complexity encountered since the 1950s is of similar fundamental origin as the Gödel theory, the fundamental limitation of logic.

Nevertheless, the mind works, visual systems perceive objects. To understand mathematically how this is possible, mathematicians have to consider the consequences of the Gödel theory and the Lucas–Penrose argument in full honesty. The mind is not a logical system. To make machines capable of learning and to model mathematically the learning abilities of the mind, new types of algorithms are needed that avoid combinatorial complexity. Several mathematical approaches have been proposed to overcome the limitations of logic; however, learning algorithms still have to use logical statements as a part of learning (Perlovsky,

Dynamic logic algorithms model uncertainty by using similarity functions among representations of concepts and incoming data. Often, these similarity functions are modeled functionally similar to probability densities. The dynamic logic idea “from vague-to-crisp” is implemented by initiating probability density functions with large variances. In the iterative dynamic logic processes variances might be reduced to small values, resulting in logic-like very narrow pdfs.

Dynamic logic algorithms have overcome the limitations of logic, have solved previously unsolvable problems, and have not only reached, but exceeded the performance of the human mind (Perlovsky,

Brain imaging experiments demonstrating the vague-to-crisp perception in neural mechanisms (Bar et al.,

A popular machine learning approach is statistical learning theory (SLT, Vapnik,

An exciting parallel with dynamic logic is explored in Wrede et al. (

Certain principles of Gestalt psychology are confirmed in contemporary neuroscience. They are mathematically modeled by dynamic logic. For example, top-down and bottom-up signal interaction is reminiscent of a Gestalt idea (that objects in their entirety are perceived before their parts). Gestalt goals to maintain stable percepts in a noisy world are modeled via models-representations. However, these ideas are not specific to dynamic logic. Dynamic logic has modeled them mathematically, overcoming the problem of complexity.

Dynamic logic is computable. Operations used by computers implementing dynamic logic algorithms are logical. But these logical operations are at a different level than human thinking. Compare the text of this article as stored in your computer and the related computer operations to the human understanding of this article. The computer's operations are logical, but on a different level from your “logical” understanding of this article. A computer does not understand the meaning of this article the way a human reader does. The reader's logical understanding is on top of 99% of the brain's operations that are not “logical” at this level. Our logical understanding is an end state of many illogical and unconscious dynamic logic processes.

The mind's “first principles” do not include logic. Nature uses different “first principles” at its different levels of organization. Thermodynamics is not based on Newton's laws, and this was a subject of special fascination to Einstein, who emphasized that thermodynamics is a physical science with its own first principles defined at an intermediate level of organization (Einstein,

I am thankful for the discussions with my colleagues Moshe Bar and Angelo Cangelosi.