P versus NP

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 NL-reduced 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.

The content was simplified. In this way, the abstract and content of the paper were changed and improved.

This paper is a breakthrough in computer science and mathematics. Indeed, this is a solution for the well-known 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 NP-complete 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.

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 one-way. 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.

We forgot to verify in the Algorithm 1 whether K < m or not. We added this new verification in this version.

We create a new version with new content and abstract, but the conclusion is the same result.

We update the abstract and content of the paper, but the result is still the same.

We update the abstract, keywords and content of the paper, but the result is still the same.

We update content and abstract.

We improve the definitions and theory with this version.

I improved the last theorem with the concluded result.

We improve the paper: abstract, keywords and content were changed.

Minor details: we changed abstract and content.

We remove a citation from the abstract and add the Conclusion section.

Minor changes: abstract and content were changed.

Included the Acknowledgments section.

Improved theorem 12.