A remark on pseudo proof systems and hard instances of the satisfiability problem

We link two concepts from the literature, namely hard sequences for the satisfiability problem sat and so-called pseudo proof systems proposed for study by Krajíček. Pseudo proof systems are elements of a particular nonstandard model constructed by forcing with random variables. We show that the existence of mad pseudo proof systems is equivalent to the existence of a randomized polynomial time procedure with a highly restrictive use of randomness which produces satisfiable formulas whose satisfying assignments are probably hard to find.