Consistency of circuit lower bounds with bounded theories

Abstract

Proving that there are problems in P NP 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(n k ), where (T, C) is one of the pairs: