On Small-depth Frege Proofs for PHP

Johan Håstad

doi:10.46298/theoretics.25.27

Home/Archive/arXiv

arXiv · 2401.15683

On Small-depth Frege Proofs for PHP

Johan Håstad

View original source ↗Read source document ↗Export metadata ↓

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 $n^{\Omega (1/d)}$. 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.

Explore related subjects

cs.CC

Keep this discovery

BibTeX RIS

Johan Håstad. 2024-01-28. On Small-depth Frege Proofs for PHP. https://doi.org/10.46298/theoretics.25.27

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗