Consistency of circuit lower bounds with bounded theories

Abstract

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that requireboolean circuits of super-linear size is a major frontier in complexity theory.While such lower bounds are known for larger complexity classes, existingresults only show that the corresponding problems are hard on infinitely manyinput lengths. For instance, proving almost-everywhere circuit lower bounds isopen even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty ofproving lower bounds that hold for all large input lengths, we ask thefollowing question: Can we show that a large set of techniques cannot provethat $\mathsf{NP}$ is easy infinitely often? Motivated by this and relatedquestions about the interaction between mathematical proofs and computations,we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it isconsistent with theory $T$ that computational class ${\mathcal C} \not\subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one ofthe pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T= \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establishinfinitely often circuit upper bounds for the corresponding problems. This isof interest because the weaker theory $\mathsf{PV}$ already formalizessophisticated arguments, such as a proof of the PCP Theorem. These consistencystatements are unconditional and improve on earlier theorems of [KO17] and[BM18] on the consistency of lower bounds with $\mathsf{PV}$