Searching Large Spaces: Displacement and the No Free Lunch Regress

Searching for small targets in large spaces is a common problem in the sciences. Because blind search is inadequate for such searches, it needs to be supplemented with additional information, thereby transforming a blind search into an assisted search. This additional information can be quantified and indicates that assisted searches themselves result from searching higher-level search spaces–by conducting, as it were, a search for a search. Thus, the original search gets displaced to a higher-level search. The key result in this paper is a displacement theorem, which shows that successfully resolving such a higher-level search is exponentially more difficult than successfully resolving the original search. Leading up to this result, a measure-theoretic version of the No Free Lunch theorems is formulated and proven. The paper shows that stochastic mechanisms, though able to explain the success of assisted searches in locating targets, cannot, in turn, explain the source of assisted searches.