# —if a solution is easy to check, is it easy to find?
> thoughts on P vs NP
—why might a solution be easier to check than to find?
For solvable problems, consider the idea that "the solution (to our problem) already exists, before we have found it"
I think it can be useful to think of an undiscovered solution as "existing already", within a special kind of "problem-relative abstract space" — just as physical-objects exist within a physical-place — and just as with physical-places, an "abstract problem-space" can also be explored to search-for and find whatever is contained within
- like physical-places, some abstract problem-spaces are small and uncluttered — which makes the task of finding whatever solution is contained within easier
- like physical-places, some abstract problem-spaces are large, and overflow with all manner of miscellaneous bric-a-brac and junk (and at times, might seem to be full of everything-other than the thing we want to find...) — which makes finding solutions harder
For some challenging problems, the thing we search for (our as-yet undiscovered solution) might be broken up into fragments — only found by a more extensive search throughout the entire problem-space:-
1. sometimes like a jigsaw puzzle, whereby each fragment is recognisable in its own right;
2. and on other occasions, sought-for fragments might be individually unrecognisable — until that-is some critical-mass, sufficient for recognisable form to be composed, is found.
On those occasions (having found sufficient fragments to compose the recognisable form, of our of now-discovered solution), the task of re-discovering the same solution within the same problem-space is made easier, because we now know what we are looking for, and we recognise it (our solution, and fragments-thereof) more easily.
In this way, we might notice that solutions to problems are often easier to "rediscover" than to "discover" — because, when we know more about "what-it-is-we-are-looking-for", (whether in whole or in part), we spend less time inspecting "all-that-we-aren't"
> intuitively then, we might say that "exploration costs less, when examination costs less"
—but is this all there is to P vs NP?
1/n
#pnp #pvsnp
