Pseudo-finite hard instances for a student-teacher game with a Nisan-Wigderson generator

TL;DR

This paper establishes lower bounds on the failure of polynomial-size circuit students in a game involving a Nisan-Wigderson generator, demonstrating the existence of pseudo-finite hard instances with implications for proof complexity.

Contribution

It introduces a lower bound on the number of inputs any polynomial-size circuit student must fail on in a game with a Nisan-Wigderson generator, and constructs a pseudo-finite hard instance set.

Findings

Polynomial-size circuit students cannot succeed on all inputs for certain hard instances.

Existence of a pseudo-finite set of hard instances where all uniform students fail.

Applications in proof complexity via a non-standard model of true arithmetic.

Abstract

For an NP intersect coNP function g of the Nisan-Wigderson type and a string b outside its range we consider a two player game on a common input a to the function. One player, a computationally limited Student, tries to find a bit of g(a) that differs from the corresponding bit of b. He can query a computationally unlimited Teacher for the witnesses of the values of constantly many bits of g(a). The Student computes the queries from a and from Teacher's answers to his previous queries. It was proved by Krajicek (2011) that if g is based on a hard bit of a one-way permutation then no Student computed by a polynomial size circuit can succeed on all a. In this paper we give a lower bound on the number of inputs a any such Student must fail on. Using that we show that there is a pseudo-finite set of hard instances on which all uniform students must fail. The hard-core set is defined in a non-standard model of true arithmetic and has applications in a forcing construction relevant to proof complexity.