scholarly journals P vs NP: P is Equal to NP: Desired Proof

2021 ◽  
pp. 1-11
Author(s):  
Zulfia A. Chotchaeva
Keyword(s):  
Artificial Intelligence ◽  
Computer Science ◽  
Structure Prediction ◽  
Complexity Classes ◽  
Open Problems ◽  
Exponential Time ◽  
Running Time ◽  
Algorithmic Problems ◽  
Np Problems ◽  
Np Problem

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.

scholarly journals Logarithmic Space Verifiers on NP-complete

2019 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Complexity Class ◽  
Complexity Classes ◽  
Input Tape ◽  
Open Problems ◽  
Logarithmic Space ◽  
Np Completeness ◽  
Np Complete ◽  
Definition Of ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have f ailed. NP is the complexity class of languages defined b y p olynomial t ime v erifiers M su ch th at wh en th e in put is an el ement of the language with its certificate, then M outputs a string which belongs to a single language in P. Another major complexity classes are L and NL. The certificate-based definition of NL is based on logarithmic space Turing machine with an additional special read-once input tape: This is called a logarithmic space verifier. NL is the complexity class of languages defined by logarithmic space verifiers M s uch t hat when t he i nput i s a n e lement o f t he l anguage with i ts c ertificate, th en M outputs 1. To attack the P versus NP problem, the NP-completeness is a useful concept. We demonstrate there is an NP-complete language defined by a logarithmic space verifier M such that when the input is an element of the language with its certificate, then M outputs a s tring which belongs to a single language in L. In this way, we obtain if L is not equal to NL, then P = NP. In addition, we show that L is not equal to NL. Hence, we prove the complexity class P is equal to NP.

scholarly journals P versus NP

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
United States ◽  
National Security ◽  
Computer Science ◽  
The United States ◽  
Fundamental Question ◽  
Complexity Classes ◽  
National Security Agency ◽  
Open Problems ◽  
Precise Statement ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity classes are L and NL. Whether L = NL is another fundamental question that it is as important as it is unresolved. We demonstrate that every problem in NP could be NL-reduced to another problem in L. In this way, we prove that every problem in NP is in NL with L Oracle. Moreover, we show the complexity class NP is equal to NL, since it is well-known that the logarithmic space oracle hierarchy collapses into NL.

scholarly journals P versus NP

2019 ◽  
Author(s):  
Frank Vega
Keyword(s):  
United States ◽  
National Security ◽  
Computer Science ◽  
The United States ◽  
Fundamental Question ◽  
Complexity Classes ◽  
National Security Agency ◽  
Open Problems ◽  
Precise Statement ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity classes are L and NL. Whether L = NL is another fundamental question that it is as important as it is unresolved. We demonstrate that every problem in NP could be NL-reduced to another problem in L.

scholarly journals Sparse complete sets for coNP: Solution of the P versus NP problem

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Fundamental Question ◽  
Complexity Class ◽  
Open Problems ◽  
Precise Statement ◽  
Complete Sets ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP.

scholarly journals P versus NP

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complexity Class ◽  
Decision Problems ◽  
Open Problems ◽  
Precise Statement ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? The precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is P-Sel. P-Sel is the class of decision problems for which there is a polynomial time algorithm (called a selector) with the following property: Whenever it’s given two instances, a “yes” and a “no” instance, the algorithm can always decide which is the “yes” instance. It is known that if NP is contained in P-Sel, then P = NP. We claim a possible selector for 3SAT and thus, P = NP.

scholarly journals Sparse complete sets for coNP: Solution of the P versus NP problem

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Fundamental Question ◽  
Complexity Class ◽  
Open Problems ◽  
Precise Statement ◽  
Complete Sets ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP.

scholarly journals Sparse complete sets for coNP: Solution of the P versus NP problem

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Fundamental Question ◽  
Complexity Class ◽  
Open Problems ◽  
Precise Statement ◽  
Complete Sets ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP.

scholarly journals Sparse complete sets for coNP: Solution of the P versus NP problem

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Fundamental Question ◽  
Complexity Class ◽  
Open Problems ◽  
Precise Statement ◽  
Complete Sets ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is coNP. Whether NP = coNP is another fundamental question that it is as important as it is unresolved. In 1979, Fortune showed that if any sparse language is coNP-complete, then P = NP. We prove there is a possible sparse language in coNP-complete. In this way, we demonstrate the complexity class P is equal to NP.

scholarly journals A Scalable Scheme for Counting Linear Extensions

2018 ◽  
Author(s):  
Topi Talvitie ◽  
Kustaa Kangas ◽  
Teppo Niinimäki ◽  
Mikko Koivisto
Keyword(s):  
Artificial Intelligence ◽  
Partial Order ◽  
Polynomial Time ◽  
State Of The Art ◽  
Hard Problem ◽  
Approximate Counting ◽  
Linear Extensions ◽  
Exponential Time ◽  
Running Time ◽  
Cnf Formulas

Counting the linear extensions of a given partial order not only has several applications in artificial intelligence but also represents a hard problem that challenges modern paradigms for approximate counting. Recently, Talvitie et al. (AAAI 2018) showed that an exponential time scheme beats the fastest known polynomial time schemes in practice, even if allowing hours of running time. Here, we present a novel scheme, relaxation Tootsie Pop, which in our experiments exhibits polynomial characteristics and significantly outperforms previous schemes. We also instantiate state-of-the-art model counters for CNF formulas; two natural encodings yield schemes that, however, are inferior to the more specialized schemes.

Bounds on the partition dimension of convex polytopes

2020 ◽  
Vol 23 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Faisal Nadeem ◽  
Mohammad Azeem
Keyword(s):  
Combinatorial Optimization ◽  
Computer Science ◽  
Facility Location ◽  
Robot Navigation ◽  
Convex Polytopes ◽  
Pharmaceutical Chemistry ◽  
Resolving Set ◽  
Minimum Number ◽  
Partition Dimension ◽  
Np Problem

Aims and Objective: The idea of partition and resolving sets plays an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Method: In a graph to obtain the exact location of a required vertex which is unique from all the vertices, several vertices are selected this is called resolving set and its generalization is called resolving partition, where selected vertices are in the form of subsets. Minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension a graph is nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that, due to the complexity of computing the exact partition dimension we provides the bounds and show that all the graphs discussed in results have partition dimension either less or equals to 4, but it cannot been be greater than 4.

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