Short Propositional Refutations for Dense Random 3CNF Formulas

TL;DR

This paper shows that standard propositional proof systems, specifically TC^0-Frege proofs, can efficiently refute random 3CNF formulas with a high clause-to-variable ratio, using spectral techniques and formal reasoning.

Contribution

It introduces polynomial-size TC^0-Frege refutations for random 3CNF formulas with Ω(n^{1.4}) clauses, leveraging spectral methods and formal proof translations.

Findings

Polynomial-size TC^0-Frege refutations for random 3CNF with Ω(n^{1.4}) clauses.

Spectral techniques enable reasoning about eigenvectors in propositional proof systems.

Formal systems of arithmetic facilitate the proof of spectral correctness in propositional logic.

Abstract

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notably are the exponential-size resolution refutation lower bounds for random 3CNF formulas with $\Omega(n^{1.5-\epsilon}) $ clauses [Chvatal and Szemeredi (1988), Ben-Sasson and Wigderson (2001)]. On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with $\Omega(n^{2}/\log n) $ clauses, shown by Beame et al. (2002). In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are $TC^0$ formulas (i.e., $TC^0$-Frege proofs) for random 3CNF formulas with $ n $ variables and $ \Omega(n^{1.4}) $ clauses. The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek (2006). Since the soundness of these witnesses is verified using spectral techniques, we develop an appropriate way to reason about eigenvectors in propositional systems. To carry out the full argument we work inside weak formal systems of arithmetic and use a general translation scheme to propositional proofs.