For a complexity class $C$ and language $L$, a constructive separation of $L
\notin C$ gives an efficient algorithm (also called a refuter) to find
counterexamples (bad inputs) for every $C$-algorithm attempting to decide $L$.
We study the questions: Which lower bounds can be made constructive? What are
the consequences of constructive separations? We build a case that
"constructiveness" serves as a dividing line between many weak lower bounds we
know how to prove, and strong lower bounds against $P$, $ZPP$, and $BPP$. Put
another way, constructiveness is the opposite of a complexity barrier: it is a
property we want lower bounds to have. Our results fall into three broad
categories.
1. Our first set of results shows that, for many well-known lower bounds
against streaming algorithms, one-tape Turing machines, and query complexity,
as well as lower bounds for the Minimum Circuit Size Problem, making these
lower bounds constructive would imply breakthrough separations ranging from
$EXP \neq BPP$ to even $P \neq NP$.
2. Our second set of results shows that for most major open problems in lower
bounds against $P$, $ZPP$, and $BPP$, including $P \neq NP$, $P \neq PSPACE$,
$P \neq PP$, $ZPP \neq EXP$, and $BPP \neq NEXP$, any proof of the separation
would further imply a constructive separation. Our results generalize earlier
results for $P \neq NP$ [Gutfreund, Shaltiel, and Ta-Shma, CCC 2005] and $BPP
\neq NEXP$ [Dolev, Fandina and Gutfreund, CIAC 2013].
3. Our third set of results shows that certain complexity separations cannot
be made constructive. We observe that for all super-polynomially growing
functions $t$, there are no constructive separations for detecting high
$t$-time Kolmogorov complexity (a task which is known to be not in $P$) from
any complexity class, unconditionally.