Nisan-Wigderson generators in proof systems with forms of interpolation

Proof complexity generators are used to define candidate hard tautologies for strong proof systems like Frege proof system or Extended Frege. They were independently introduced by Krajı́ček [3] and by Alekhnovich, Ben-Sasson, Razborov, and Wigderson [1]. Roughly speaking, the tautologies encode the fact that b / ∈ Rng(g) for an element b outside of the range of a generator g : {0, 1} 7−→ {0, 1}, where m > n, defined by a circuit of size m. If g : {0, 1}t(n)O(1) 7−→ {0, 1}2n sends codes of t(n)-size circuits with n inputs to the truth tables of functions they compute, then the tautologies f / ∈ Rng(g) say that f has no t(n)-size circuits. Denote such a formula by ¬Circuitt(n)(f). The hardness of such tautologies can be interpreted as the hardness of proving circuit lower bounds. This captures an element of self-reference in the P vs NP problem. As Razborov pointed out in [7], to prove the hardness of ¬Circuitt(n)(f) in a proof system, it is sufficient to show that there exists a generator g : {0, 1}0 7−→ {0, 1}2n with arbitrary t0(n) ≤ 2 and such that g is