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 notable are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Ω(n1.5-ε) clauses (Chvatal and Szemeredi [13], Ben-Sasson and Wigderson [9]). 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. [5]. 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 polynomialsize propositional refutations whose lines are TC0formulas (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 [19]. 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.

Cook and Krajíček have recently obtained the following Karp-Lipton collapse result in bounded arithmetic: if the theory PV proves NP⊆ P/ poly , then the polynomial hierarchy collapses to the Boolean hierarchy, and this collapse is provable in PV . Here we show the converse implication, thus answering an open question posed by Cook and Krajíček. We obtain this result by formalizing in PV a hard/easy argument of Buhrman et al. [2003]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček. In particular, we obtain several optimality results for proof systems using advice. We further show that these optimal systems are equivalent to natural extensions of Frege systems.