From Gödel incompleteness to the consistency of circuit lower bounds

2026-04-28

AI Summary

This study investigates the consistency of the bounded arithmetic theory $S^1_2$ with the circuit lower bound statement $\mathrm{EXP} \not\subseteq \mathrm{P/poly}$. By introducing a logical analysis method based on Takeuti-style consistency statements, it pioneers the application of Gödelian incompleteness ideas to the consistency of circuit lower bounds, thereby establishing a deep connection between separations of formal theories and the provability of complexity lower bounds. The main contributions include proving the consistency of $S^1_2$ with $\mathrm{EXP} \not\subseteq \mathrm{P/poly}$, extending this result to other complexity classes such as $\mathrm{PSPACE}$, and presenting an amplification theorem that demonstrates the heightened difficulty of proving “almost-everywhere” circuit lower bounds. These findings illuminate both the expressive power and inherent limitations of weak arithmetic theories in capturing computational lower bounds.

Abstract

We prove that the bounded arithmetic theory $S^1_2$ is consistent with EXP $\not\subseteq$ P/poly. More generally, we show that certain separations of $V^1_2$ from a theory $T$ imply the consistency of $T$ with EXP $\not\subseteq$ P/poly. For $T=S^1_2$, Takeuti (1988) established such a separation using a variant of Gödel's consistency statement. Analogous results hold for PSPACE $\not\subseteq$ P/poly but the required separations of theories are yet unknown. Finally, we give magnification results for the hardness of proving almost-everywhere versions of these lower bounds.

Problem

• bounded arithmetic
• circuit lower bounds
• consistency
• complexity classes
• Gödel incompleteness

Innovation

• bounded arithmetic
• circuit lower bounds
• consistency
• EXP vs P/poly
• proof complexity