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

Jan Krajíček

doi:10.2168/lmcs-8(3:9)2012

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arXiv · 1207.0393

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

Jan Krajíček

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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.

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Jan Krajíček. 2012-07-02. Pseudo-finite hard instances for a student-teacher game with a Nisan-Wigderson generator. https://doi.org/10.2168/lmcs-8(3:9)2012

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