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# Logarithmic Space Verifiers on NP-complete

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 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)

How to cite:
Vega, F. Logarithmic Space Verifiers on NP-complete. *Preprints* **2019**, 2019080037 (doi: 10.20944/preprints201908.0037.v1).
Vega, F. Logarithmic Space Verifiers on NP-complete. Preprints 2019, 2019080037 (doi: 10.20944/preprints201908.0037.v1).

## Abstract

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.

## Supplementary and Associated Material

https://github.com/frankvegadelgado/VerifyReduction: GitHub repository

https://hal.archives-ouvertes.fr/hal-02199310: HAL Preprint

https://www.academia.edu/39973754/Logarithmic_Space_Verifiers_on_NP-complete: Academia Preprint

## Subject Areas

complexity classes; completeness; verifier; reduction; polynomial time; logar-ithmic space

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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