Dense Complete Set For NP

Frank Vega

doi:10.20944/preprints201908.0037.v8

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preprints.org > computer science and mathematics > computer science > doi: 10.20944/preprints201908.0037.v8

Preprint Article Version 8 Preserved in Portico This version is not peer-reviewed

Dense Complete Set For NP

Frank Vega

Version 1 : Received: 1 August 2019 / Approved: 5 August 2019 / Online: 5 August 2019 (03:31:23 CEST)
Version 2 : Received: 26 November 2019 / Approved: 29 November 2019 / Online: 29 November 2019 (07:28:14 CET)
Version 3 : Received: 4 April 2020 / Approved: 6 April 2020 / Online: 6 April 2020 (12:57:55 CEST)
Version 4 : Received: 15 April 2020 / Approved: 16 April 2020 / Online: 16 April 2020 (10:17:03 CEST)
Version 5 : Received: 18 September 2020 / Approved: 19 September 2020 / Online: 19 September 2020 (09:51:02 CEST)
Version 6 : Received: 10 February 2021 / Approved: 11 February 2021 / Online: 11 February 2021 (11:56:37 CET)
Version 7 : Received: 19 August 2021 / Approved: 27 August 2021 / Online: 27 August 2021 (14:09:47 CEST)
Version 8 : Received: 20 October 2021 / Approved: 26 October 2021 / Online: 26 October 2021 (11:07:59 CEST)
Version 9 : Received: 7 March 2024 / Approved: 8 March 2024 / Online: 8 March 2024 (11:06:11 CET)
Version 10 : Received: 14 March 2024 / Approved: 14 March 2024 / Online: 14 March 2024 (10:11:13 CET)
Version 11 : Received: 14 March 2024 / Approved: 15 March 2024 / Online: 18 March 2024 (08:29:48 CET)
Version 12 : Received: 29 March 2024 / Approved: 1 April 2024 / Online: 1 April 2024 (17:14:22 CEST)
Version 13 : Received: 6 April 2024 / Approved: 6 April 2024 / Online: 8 April 2024 (06:04:04 CEST)
Version 14 : Received: 11 April 2024 / Approved: 11 April 2024 / Online: 12 April 2024 (04:51:11 CEST)

How to cite: Vega, F. Dense Complete Set For NP. Preprints 2019, 2019080037. https://doi.org/10.20944/preprints201908.0037.v8 Vega, F. Dense Complete Set For NP. Preprints 2019, 2019080037. https://doi.org/10.20944/preprints201908.0037.v8

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MDPI and ACS Style

Vega, F. Dense Complete Set For NP. Preprints 2019, 2019080037. https://doi.org/10.20944/preprints201908.0037.v8

APA Style

Vega, F. (2021). Dense Complete Set For NP. Preprints. https://doi.org/10.20944/preprints201908.0037.v8

Chicago/Turabian Style

Vega, F. 2021 "Dense Complete Set For NP" Preprints. https://doi.org/10.20944/preprints201908.0037.v8

Abstract

A sparse language is a formal language such that the number of strings of length $n$ is bounded by a polynomial function of $n$. We create a class with the opposite definition, that is a class of languages that are dense instead of sparse. We define a dense language on $m$ as a formal language (a set of binary strings) where there exists a positive integer $n_{0}$ such that the counting of the number of strings of length $n \geq n_{0}$ in the language is greater than or equal to $2^{n - m}$ where $m$ is a real number and $0 < m \leq 1$. We call the complexity class of all dense languages on $m$ as $DENSE(m)$. We prove that there exists an $\textit{NP--complete}$ problem that belongs to $DENSE(m)$ for every possible value of $0 < m \leq 1$.

Keywords

Complexity Classes; Complement Language; Sparse; Completeness; Polynomial Time

Subject

Computer Science and Mathematics, Computer Science

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Comments (1)

Comment 1

Received: 26 October 2021
Commenter: Frank Vega
Commenter's Conflict of Interests: Author

Comment: We remove the claim of P=NP and as a consequence, we remove some theorems. We also changed the title and abstract.

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