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.

翻译：对于复杂性类$C$与语言$L$，$L \notin C$的构造性分离给出了一个高效算法（称为反驳者），能够为每个试图判定$L$的$C$-算法找到反例（不良输入）。我们研究以下问题：哪些下界可以被构造化？构造性分离的后果是什么？我们论证"构造性"是许多已知可证明的弱下界与针对$P$、$ZPP$和$BPP$的强下界之间的分界线。换言之，构造性是复杂性障碍的对立面：它是我们希望下界具备的性质。我们的结果分为三大类：1. 第一组结果表明，对于流算法、单带图灵机和查询复杂性的许多著名下界，以及最小电路规模问题的下界而言，将这些下界构造化将意味着从$EXP \neq BPP$到$P \neq NP$的突破性分离。2. 第二组结果表明，对于针对$P$、$ZPP$和$BPP$的大多数主要开放下界问题（包括$P \neq NP$、$P \neq PSPACE$、$P \neq PP$、$ZPP \neq EXP$和$BPP \neq NEXP$），任何分离的证明将进一步蕴含构造性分离。该结果推广了之前关于$P \neq NP$[Gutfreund, Shaltiel, and Ta-Shma, CCC 2005]和$BPP \neq NEXP$[Dolev, Fandina and Gutfreund, CIAC 2013]的结论。3. 第三组结果表明某些复杂性分离无法被构造化。我们观察到，对于所有超多项式增长函数$t$，不存在从任何复杂性类中检测高$t$-时间柯尔莫哥洛夫复杂度（已知不在$P$中的任务）的构造性分离，这一结论无条件成立。