Bounded-Depth Frege Lower Bounds for Weaker Pigeonhole Principles

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle $PHP^{m}_n$ where $m= (1+1/{\polylog n})n$. This lower bound qualitatively matches the known quasi-polynomial-size bounded-depth Frege proofs ...

We give a general transformation that turns polynomial-size Frege proofs into subexponential-size AC⁰-Frege proofs. This indicates that proving truly exponential lower bounds for AC⁰-Frege is hard, as it is a long-standing open problem to prove ...

We give a general transformation which turns polynomialsize Frege proofs to subexponential-size AC⁰-Frege proofs. This indicates that proving exponential lower bounds for AC⁰-Frege is hard, since it is a longstanding open problem to prove super-...