Let's say I reduce the problem $ A in L $ at $ B in K $ , with a function $ f: Sigma ^ {*} rightarrow Gamma ^ {*} $ such as $ w in L Leftrightarrow f (w) in K $ . I know if I want to solve $ A $, given a polynomial time algorithm for $ B $, I just have to transform $ A $ at $ B $ and solve $ B $. This can be thought of as:

The reduction must be carried out from an arbitrary instance of $ A $ to a legal authority $ B $

My question is, should I reduce to *arbitrary* example of $ B $ or *some* example of $ B $? That is to say. TQBNF reduction to generalized geography is performed in a valid graph instance, but there are many other valid generalized geography instances.