# Functional Hypothesis of Complexity Classes

## Keywords:

computational complexity, algorithms, equality## Abstract

This work describes the hypothesis of the relation between the classes of complexity: for this purpose we define the functions over algorithms or state machines for which the equality holds true and, thus, the decision can be made towards polynomial reduction of the computational complexity of algorithms. The specific class of impractical or exponential measures of complexity against the polynomial ones is also discussed – for this case we divide these classes according to the discrete numbers which are known to the present time. We also present the approximate algorithm for the classical NP-complete problem like Traveling Salesman using the memory construction. The question of P and NP equality is important in decision-making algorithms which commonly decide inequality of these classes – we define the memory factor which is exponential and space consumption is non-deterministic. The memory consumption problem within the memorization principle or dynamic programming can be of varying nature giving us the decision to build the approximation methods like it’s shown on the example of Traveling Salesman problem. We also give the notion of the past work in theory of complexity which, in our opinion, is of the same consideration in most cases when the functional part is omitted or even isn’t taken into account. The model theorem with its proof of the equality of classes over congruent function is also given in the end of this article.

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*ADVANCED TECHNOLOGIES AND COMPUTER SCIENCE*, (3), 4–9. Retrieved from https://atcs.iict.kz/index.php/atcs/article/view/61

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