TITLE: Question about the P versus NP Problem
QUESTION [5 upvotes]: It seems to be an accepted belief based on decades of experience that naive algorithms are not 
adequate to solve NP-complete problems in a reasonable amount of time. Even those who believe P = NP seem to be looking for an algorithm with a very clever representation, so far without success.
Can the observations above be formalized? Let's use the full, unrestricted Clique problem as an example. Suppose it could be shown that no clever representation exists for the Clique problem. That is, suppose it could be proven for every algorithm that either:

the algorithm is correct and uses an internal representation such that no two distinct input cliques    lead to the same internal representation,
or 
the algorithm is incorrect. 

Would this be a worthwhile result? How important would it be? Or is it wrong?
Is there already a proof that CLIQUE or SAT, for example, cannot be solved in polynomial time by performing operations on cliques or Boolean expressions respectively?
References with your answer would be helpful. Thanks

REPLY [6 votes]: You ask: 

suppose it could be proven for every
  algorithm that either:

the algorithm is correct and uses an internal representation such
  that no two distinct input cliques
  lead to the same internal
  representation, or
the algorithm is incorrect.

Would this be a worthwhile result? How
  important would it be? Or is it wrong?

It's wrong. You can always take a correct algorithm for clique and add a preprocessing step that removes edges which are clearly not in any maximal clique. This will still be a correct algorithm for clique, and obviously does not satisfy either (1) or (2).