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

2026-04-30

AI Summary

This work investigates lower bounds on the refutation length of CNF formulas in tree-like semantic proof systems under row-length restrictions. By constructing a specific family of CNF formulas and combining probabilistic methods, combinatorial constructions, and proof complexity analysis, the study establishes the first superpolynomial lower bounds for refutation length in tree-like Frege and higher-order threshold proof systems when the row length lies between polynomial and exponential. The results demonstrate that, under the given row-length constraints, almost all formulas in the constructed family require superpolynomial-length refutations across a broad range of tree-like semantic proof systems, significantly extending the scope of existing lower-bound theories in proof complexity.

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 $\exp\bigl(s(n)\bigr)$.

Problem

• proof complexity
• tree-like Frege
• lower bounds
• bounded line size
• semantic proof systems

Innovation

• tree-like Frege
• superpolynomial lower bounds
• bounded line size
• semantic proof systems
• threshold proof systems