This paper studies the \emph{refuter} problems, a family of decision-tree $\mathrm{TFNP}$ problems capturing the metamathematical difficulty of proving proof complexity lower bounds. Suppose $\varphi$ is a hard tautology that does not admit any length-$s$ proof in some proof system $P$. In the corresponding refuter problem, we are given (query ... more >>>

We show that there is a language in $\mathrm{S}_2\mathrm{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds for the classes $\Sigma_2\mathrm{E}$, $(\Sigma_2\mathrm{E}\cap\Pi_2\mathrm{E})/_1$, and $\mathrm{ZPE}^{\mathrm{NP}}/_1$. Previously, only "half-exponential" circuit lower bounds for these ... more >>>

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time.

The *range avoidance problem*, denoted as $\mathcal{C}$-$\rm Avoid$, asks to find a non-output of a given $\mathcal{C}$-circuit $C:\{0,1\}^n\to\{0,1\}^\ell$ with stretch $\ell>n$. This problem has recently received much attention in complexity theory for its connections with circuit lower bounds and other explicit construction problems. Inspired by the Algorithmic Method for circuit ... more >>>

Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called *bounded relativization*. For a complexity class $C$, we say that a statement is *$C$-relativizing* if the statement holds relative ... more >>>

It is a long-standing open problem whether the Minimum Circuit Size Problem ($\mathrm{MCSP}$) and related meta-complexity problems are NP-complete. Even for the rare cases where the NP-hardness of meta-complexity problems are known, we only know very weak hardness of approximation.

In this work, we prove NP-hardness of approximating meta-complexity with ... more >>>

Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where:

* MCSP can be solved in deterministic polynomial time, but ... more >>>

We show that one-way functions exist if and only if there is some samplable distribution D such that it is hard to approximate the Kolmogorov complexity of a string sampled from D. Thus we characterize the existence of one-way functions by the average-case hardness of a natural \emph{uncomputable} problem on ... more >>>

A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity K^t is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures:

1. We show, perhaps surprisingly, that the ... more >>>

We prove that for all constants a, NQP = NTIME[n^{polylog(n)}] cannot be (1/2 + 2^{-log^a n})-approximated by 2^{log^a n}-size ACC^0 of THR circuits (ACC^0 circuits with a bottom layer of THR gates). Previously, it was even open whether E^NP can be (1/2+1/sqrt{n})-approximated by AC^0[2] circuits. As a straightforward application, ... more >>>