Our general aim is to construct models relevant to some of the conjectures discussed in Chapter 30, and eventually models that would prove some of them. From a logical point of view this should be more accessible for Razborov's Conjecture 30.2.1 than for the other conjectures as it concerns a function of the highest complexity. This translates into a higher quantifier complexity of the associated formalized statement and thus offers, in principle, more chances to manipulate things to our advantage.

The model we shall construct in this chapter, the local witness model, will yield the following statement. We shall formulate it now a little informally; the formal version in Section 31.4 will quantify the parameters involved.

Theorem 31.0.2(informal)Let A be an m × n 0–1 matrix that is a (log m,n⅓) design in the sense of Nisan and Wigderson [84], and n < m. Let f be a Boolean function in n⅓ variables that is a hard bit of a one-way permutation. Assume R is an infinite NP-set.

Then it is consistent with Cook's theory PV (and, in fact, with the true universal theory Th∀(LPV) in the language of PV) that

This gives a form of consistency of Razborov's conjecture (in fact, of a stronger statement) and also of Rudich's demi-bit conjecture. We shall discuss this in Section 31.4.

The local witness model K(Fb)

We shall continue using the notation from Section 29.4. In particular, A ∈ Mn will be an m × n0–1 matrix with ℓ 1s per row, in row i the 1s are in columns from Ji ⊆ [n]. We shall fix ℓ ≔ n⅓ and we shall assume that A is (log m, ℓ)-design.