Beyond Natural Proofs: Hardness Magnification and Locality

Abstract

Hardness magnification reduces major complexity separations (such as \(\mathsf {\mathsf {EXP}} \nsubseteq \mathsf {NC}^1 \) ) to proving lower bounds for some natural problem Q against weak circuit models. Several recent works [11, 13, 14, 40, 42, 43, 46] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than Q , while Q itself is susceptible to lower bounds but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon, and investigate the prospects of proving new lower bounds using this approach. In particular, we consider the following essential questions associated with the hardness magnification program: – Does hardness magnification avoid the natural proofs barrier of Razborov and Rudich [51] ? – Can we adapt known lower bound techniques to establish the desired lower bound for Q ? We establish that some instantiations of hardness magnification overcome the natural proofs barrier in the following sense: slightly superlinear-size circuit lower bounds for certain versions of the minimum circuit size problem \({\sf MCSP} \) imply the non-existence of natural proofs. As the non-existence of natural proofs implies the non-existence of efficient learning algorithms, we show that certain magnification theorems not only imply strong worst-case circuit lower bounds but also rule out the existence of efficient learning algorithms. Hardness magnification might sidestep natural proofs, but we identify a source of difficulty when trying to adapt existing lower bound techniques to prove strong lower bounds via magnification. This is captured by a locality barrier : existing magnification theorems unconditionally show that the problems Q considered above admit highly efficient circuits extended with small fan-in oracle gates, while lower bound techniques against weak circuit models quite often easily extend to circuits containing such oracles. This explains why direct adaptations of certain lower bounds are unlikely to yield strong complexity separations via hardness magnification.

A Finer Analysis of Multi-Structural Games and Beyond

“…Therefore, it may be worth investigating the existence of methodological barriers to using combinatorial games for this purpose. In complexity theory, formal barriers (such as relativization [20], natural proofs [21], arithmetization [22], and locality [23]) identify a precise feature F of certain arguments, such that no proof with F can obtain complexity separations. Some progress towards barriers for the combinatorial games approach was made in [19] by showing (roughly) that no proof of P = NP via EF games can involve efficiently-constructible structures.…”

On the Consistency of Circuit Lower Bounds for Non-deterministic Time

“…The term magnification has been coined in [27] in the context of circuit lower bounds where such results are currently intensively investigated (cf. [9]). In proof complexity such results are rare so far.…”

Complexity Lower Bounds from Algorithm Design

“…The area of computational complexity lower bounds is chock full of bad news. Not only is "bad news" the goal of complexity lower bounds (we want to prove that interesting tasks cannot be solved efficiently) but the desired bad news has its own bad news: there are substantial collections of barrier results (such as relativization [1], natural proofs [2], algebrization [3], and locality [4]) demonstrating broadly how various methods in complexity theory are not simultaneously subtle enough and powerful enough to prove theorems along the lines of P = NP (and significantly weaker results). In short, we cannot prove lower bounds, and we can prove that we can't prove lower bounds without significantly new ideas.…”