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

lmcs:788 - Logical Methods in Computer Science, August 13, 2012, Volume 8, Issue 3
Pseudo-finite hard instances for a student-teacher game with a Nisan-Wigderson generator

Authors: Krajíček, Jan

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.


Source : oai:arXiv.org:1207.0393
DOI : 10.2168/LMCS-8(3:9)2012
Volume: Volume 8, Issue 3
Published on: August 13, 2012
Submitted on: February 22, 2012
Keywords: Computer Science - Computational Complexity,Mathematics - Logic,F.2.2,F.4.1


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