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.

Improved introduction and abstract and a changed title. Fixed some typos.

Separating different propositional proof systems---that is, demonstrating that one proof system cannot efficiently simulate another proof system---is one of the main goals of proof complexity. Nevertheless, all known separation results between non-abstract proof systems are for specific families of hard tautologies: for what we know, in the average case all (non-abstract) propositional proof systems are no stronger than resolution. In this paper we show that this is not the case by demonstrating 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. By known lower bounds on resolution refutations, this implies an exponential separation of $TC^0$-Frege from resolution in the average case.

The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek [FOCS'06]. 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, use a general translation scheme to propositional proofs and then show how to turn these proofs into random 3CNF refutations.