## Combinatorics and Graph Theory   ## P vs NP Problem Solutions Generalized

Authors: A. A. Frempong

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly and completely, the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.

Comments: 8 Pages. Copyright © by A. A. Frempong. Reference: P vs NP:Solutions of NP Problems,viXra:1408.0204 by A. A. Frempong

### Submission history

[v1] 2016-04-05 23:26:53
[v2] 2016-04-11 03:15:58
[v3] 2016-05-06 03:21:13
[v4] 2016-05-07 19:06:06