Year: 2025
Type: article
Abstract: We study Frege proofs for the one-to-one graph Pigeon Hole Principle defined on the $n\times n$ grid where $n$ is odd. We are interested in the case where each formula in the proof is a depth $d$ formula in the basis given by $\land$, $\lor$, and $\neg$. We prove that in this situation the proof needs to be of size exponential in $n^{Ω(1/d)}$. If we restrict the size of each line in the proof to be of size $M$ then the number of lines needed is exponential in $n/(\log M)^{O(d)}$. The main technical component of the proofs is to design a new family of random restrictions and to prove the appropriate switching lemmas. 42 pages. This is the TheoretiCS journal version
Source: TheoretiCS
Author: Johan Håstad
Institution: KTH Royal Institute of Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4416847758
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