Constructive Separations and Their Consequences

TL;DR

The paper investigates whether standard complexity lower bounds can be made constructive by outputs (refuters) that efficiently produce counterexamples to any given algorithm within a model ${\cal M}$. It shows that making many known lower bounds constructive would imply major breakthroughs (e.g., $EXP\neq BPP$, $P\neq NP$), while also proving conditions under which constructive separations automatically follow for broad classes of separations. Conversely, it identifies limits, proving that some tasks (e.g., high-time Kolmogorov complexity) resist constructive separations, and under plausible hypotheses there exist NP languages with no constructive separations. Overall, the work reveals a divide between “weak” lower bounds that would become powerful if constructive and strong open separations that inherently demand new techniques, while connecting to hardness magnification and explicit obstructions as guiding principles. These results broaden the roadmap for proving stronger lower bounds by emphasizing explicit, constructive counterexample-generation as a central objective.

Abstract

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.

Theorems & Definitions (43)

• definition 1.1: Refuters and constructive separation
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• proposition 1.7
• Theorem 1.8
• definition 2.1: $\mathsf{MCSP}\xspace$