Superpolynomial Length Lower Bounds for Tree-Like Semantic Proof Systems with Bounded Line Size

TL;DR

This paper establishes superpolynomial lower bounds on the length of tree-like semantic proof systems with bounded line size, demonstrating inherent complexity for certain explicit CNF formulas.

Contribution

It introduces explicit formulas requiring super-polynomial proof length in tree-like semantic proof systems with bounded line size, advancing understanding of proof complexity.

Findings

Superpolynomial lower bounds for tree-like Frege refutations with bounded line size.

Lower bounds extend to tree-like degree-d threshold systems with d up to n^{1-ε}.

Results apply to semantic proof systems with bounded number of lines.

Abstract

We prove superpolynomial length lower bounds for the semantic tree-like Frege refutation system with bounded line size. Concretely, for any function $n^{2-\varepsilon} \leq s(n) \leq 2^{n^{1-\varepsilon}}$ we exhibit an explicit family $\mathcal{A}$ of $n$-variate CNF formulas $A$, each of size $|A| \le s(n)^{1+\varepsilon}$, such that if $A$ is chosen uniformly from $\mathcal{A}$, then asymptotically almost surely any tree-like Frege refutation of $A$ in line-size $s(n)$ is of length super-polynomial in $|A|$. Our lower bounds apply also to tree-like degree-$d$ threshold systems, for $d \approx \log\bigl(s(n)\bigr)$, that is, for $d$ up to $n^{1-\varepsilon}$. More generally, our lower bounds apply to the semantic version of these systems and to any semantic tree-like proof system where the number of distinct lines is bounded by $\exp\bigl(s(n)\bigr)$.