- P = NP means that for every problem that has an efficiently verifiable solution, one can find that solution efficiently as well.
- Conversely, P ≠ NP means that even though a given problem has an efficiently verifiable solution, one cannot find that solution efficiently.
Using this Problem Definition, let us consider how P = NP is physically possible:
How would a perfectly rational entity (e.g. a powerful super-computer); fully endowed with and/or given free access to all known information to mankind as of present day approach any given problem?
Please note that all known information as stated above also includes the solution to all solved problems as of present day which implies 2 things:
i) That the solution to each of those problems can be verified efficiently.
ii) Those solutions therefore now become templates/guidelines/foundations for other similar problems which when solved (assuming that none of them break any known laws of the known universe) can also be verified efficiently.
Step 1: Is the solution to this problem (i.e. the desired state) physically possible (i.e. does the desired state violate any of the known laws of the known universe such as the law of conservation of mass & energy and so on)?
Step 2: If this perfectly rational entity determines that no known laws of the known universe are broken, it then visualizes that desired state (e.g. as a diagram).
Step 3: Having visualized the desired state, it then visualizes 1 step less than that desired state and so on until it visualizes the initial problem presented to it. In doing so, the perfectly rational entity checks to ensure that every step it visualizes does not itself violate any known laws of the known universe. It is important to note here that none of the steps need to be evaluated for economic feasibility because that is only an artificial / a man-made limitation.
Having thus shown that a perfectly rational entity (e.g. a powerful super-computer); fully endowed with and/or given free access to all known information to mankind as of present day can arrive step-wise (albeit backwards) at a solution to a problem provided no known laws of the known universe are broken; if the economics of each such step can be supported; then the solution can be developed.
It is then a relatively routine matter to independently verify that solution.