Proving that there are problems in PNP that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in MAEXP. Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that NP is easy infinitely often? Motivated by this and related
questions 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≥1 it is
consistent with theory T that computational class C⊆i.o.SIZE(nk), where (T,C) is one of
the pairs: T=T21 and C=PNP, T=S21 and C=NP, T=PV and
C=P. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory PV already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with PV