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

Abstract

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.

Feasibly constructive proofs of succinct weak circuit lower bounds

“…This improves [52, Proposition 4.3] establishing a one round Student-Teacher computation, and, in fact, is a combination of folklore arguments (e.g. [8,22,45] contain similar constructions). Proposition 4.8.…”

“…This is (43) if we can eliminate the first conjunct (45). But since m (k + 1) log n and n n 0 after the substitution (45) is a tautology whose variables are only the auxiliary variables used in the definition of the translation.…”