Abstract
Computations and computational complexity are fundamental for mathematics and all computer science, including web load time, cryptography (cryptocurrency mining), cybersecurity, artificial intelligence, game theory, multimedia processing, computational physics, biology (for instance, in protein structure prediction), chemistry, and the P vs. NP problem that has been singled out as one of the most challenging open problems in computer science and has great importance as this would essentially solve all the algorithmic problems that we have today if the problem is solved, but the existing complexity is deprecated and does not solve complex computations of tasks that appear in the new digital age as efficiently as it needs. Therefore, we need to realize a new complexity to solve these tasks more rapidly and easily. This paper presents proof of the equality of P and NP complexity classes when the NP problem is not harder to compute than to verify in polynomial time if we forget recursion that takes exponential running time and goes to regress only (every problem in NP can be solved in exponential time, and so it is recursive, this is a key concept that exists, but recursion does not solve the NP problems efficiently). The paper’s goal is to prove the existence of an algorithm solving the NP task in polynomial running time. We get the desired reduction of the exponential problem to the polynomial problem that takes O(log n) complexity.
Highlights
Another mention of the underlying problem occurred in a ... letter written by Kurt Gödel to John von Neumann
Methodology a) Definition of the Task Any NP class problem can be solved by exhaustive search of all instances, i.e., by brute force search that requires exponential execution time, this is unacceptable in practice, we need to solve the NP problems in polynomial time, and if one of these NP problems is solved in polynomial time, the others will be solved in polynomial time
Let us take, for example, a set with n elements, where we need to find all subsets of this set
Summary
Another mention of the underlying problem occurred in a ... letter written by Kurt Gödel to John von Neumann. The NP elements of an array are the complements of the n1 and tasks do not require making two recursive calls when n2 elements, the sbasis=10, suppose we need to find growth doubles with each addition to the input data set, the exponential values of these elements, when n1n= 35 we have a new path to solve this problem in polynomial and n2n= 75, the bases and the exponents of the nn running time using a sequence of matrix loops that uses 9 elements are taken arbitrarily, and the current sbasis is a sorted array and takes O(log n) complexity. Adelman was recently proved the stronger result that every set accepted in polynomial time by a randomized Turing machine has small circuits (Lipton & Karp, 1980).” ‘I see complexity as the intricate and exquisite interplay between computation (complexity classes) and applications (that is, problem) (Papadimitriou, 1994).” “We do not know of polynomialtime algorithms for these problems, and we cannot prove that polynomial-time algorithms exist...These are the NP-complete problems... (Kleinberg & Tardos, 2006).” “...there is a strictly ascending sequence with a minimal pair of upper bounds to the sequence...if P≠NP there are members of NP-P that are not polynomial complete (Ladner, 1975).” “...minimal propositional logic corresponds to dependent typed-calculus... (Sorensen & Urzyczyn, 1998).” ‘Practical problems requiring polynomial time are almost solvable in an amount of time that we can tolerate, while those that require exponential time generally cannot be solved except for small instances (Hopcroft, Motwani, & Ullman, 2001).” “Some success was had by causing the machine to systematically eliminate the redundancy; but the problem of total length increasing rapidly still remained when more complicated problems were attempted (Davis, Logemann, & Loveland, 1961).” “Gȍdel and others went on to show that various other mathematically interesting statements, besides the consistency statement, are undecidable by P, assuming it to be consistent... (Boolos, Burgess, & Jeffrey, 2007).” “There has been much work in getting the number of variables needed for an undecidability result to be small (Gasarch, 2021).”
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