P versus NPEasyChair Preprint no. 3061, version historyVersion  Date  Pages  Version notes 

1  March 29, 2020  8  —  2  April 4, 2020  9  Maybe, it was not clear enough the outstanding result that implies the first version. The original submission claims that every problem in NP could be NLreduced to another problem in L. However, this might be not explicit at all. Indeed, that result implies that every problem in NP is in NL with L Oracle. Neil Immerman proved that NL = coNL. His result provides a direct proof and improvement of the main result in (Buss, Samuel R and Cook, Stephen A and Dymond, Patrick W and Hay, Louise with paper: The Log Space Oracle Hierarchy Collapses), by collapsing the log space oracle hierarchy into NL. In this way, we have that the complexity class NP is equal to NL. This new revision makes more evident this result.  3  April 15, 2020  8  The content was simplified. In this way, the abstract and content of the paper were changed and improved.  4  April 23, 2020  8  This paper is a breakthrough in computer science and mathematics. Indeed, this is a solution for the wellknown problem P vs NP. Moreover, using the knowledge introduced in this version with the another paper [1], then it is possible to find a practical and feasible algorithm for the NPcomplete problems. It has been discussed that a P = NP solution might be useful for the cure of cancer. For that reason, it is very important this version may be visible in a public site, since this result could be used for worldwide battle that we are engage for the cure of coronavirus. The abstract and content of the paper were changed and improved. [1] Samuel R. Buss, Stephen A. Cook, Patrick W. Dymond, and Louise Hay. The Log Space Oracle Hierarchy Collapses, July 1987. In Citeseer at http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=C115F3BE20DF7E237DF40CB849AEF37C?doi=10.1.1.5.760&rep=rep1&type=pdf.  5  April 25, 2020  8  Unfortunately, we introduced a flaw in the last version with the counting of the number of clauses from the Boolean formula. Certainly, if we initially count the number of clauses, then the logarithmic space verifier will not be in oneway. However, this is not a fatal error, since this could be fixed adding the number of clauses as a part of the input and checking at the end the number of clauses. This was what we changed in the content of this version.  6  April 26, 2020  8  We forgot to verify in the Algorithm 1 whether K < m or not. We added this new verification in this version.  7  June 24, 2020  14  We create a new version with new content and abstract, but the conclusion is the same result.  8  June 29, 2020  14  We update the abstract and content of the paper, but the result is still the same.  9  July 10, 2020  13  We update the abstract, keywords and content of the paper, but the result is still the same.  10  July 11, 2020  13  We update content and abstract.  11  July 18, 2020  13  We improve the definitions and theory with this version.  12  July 20, 2020  14  I improved the last theorem with the concluded result.  13  September 19, 2020  13  We improve the paper: abstract, keywords and content were changed.  14  October 3, 2020  13  Minor details: we changed abstract and content.  15  October 4, 2020  14  We remove a citation from the abstract and add the Conclusion section.  16  October 6, 2020  14  Minor changes: abstract and content were changed.  17  December 7, 2020  14  Included the Acknowledgments section.  18  December 28, 2020  14  Improved theorem 12. 
Keyphrases: completeness, complexity classes, logarithmic space, oneway, polynomial time, reduction 
