2011/01/20 by Sebastian Müller, Müller, Sebastian, Iddo Tzameret +1
Computer Science · #03F20 #03F30 #68Q15 #68Q17 #Advanced Algebra and Logic #Computational Complexity (cs.CC) #Cryptography and Data Security #F.2.2 #F.4.1 #FOS: Computer and information sciences #I.2.3 #Logic, Reasoning, and Knowledge #acm:03F20 #acm:03F30 #acm:68Q15 #acm:68Q17 #cs.CC #msc:03F20 #msc:03F30 #msc:68Q15 #msc:68Q17
paper · pdf · doi:10.48550/arxiv.1101.3970
62 pages; improved introduction and abstract, and a changed title. Fixed some typos
openalex publication_date 2011/01/20 · arxiv created 2011/06/03 · arxiv updated 2011/06/06 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
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 Ω(n1.5-ε) 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 Ω(n2/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 TC0 formulas (i.e., TC0-Frege proofs) for random 3CNF formulas with n variables and Ω(n1.4) 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.