Implicit Resolution

Let Ω be a set of unsatisfiable clauses, an implicit resolution refutation of Ω is a circuit β with a resolution proof α of the statement “β describes a correct tree-like resolution refutation of Ω”. We show that such a system is p-equivalent to Extended Frege. More generally, let τ be a tautology, a [P,Q]-proof of τ is a pair (α, β) s.t. α is a P -proof of the statement “β is a circuit describing a correct Q-proof of τ”. We prove that [EF,P ] ≤p [R,P ] for an arbitrary Cook-Reckhow proof system P .