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The Minimum Oracle Circuit Size Problem

The Minimum Oracle Circuit Size Problem

Author Name: 

Eric Allender, Dhiraj Holden, and Valentine Kabanets

Publication Type: 
Journal Publications
Journal/Volume: 
Computational Complexity
Publication Date: 
February, 2016
Abstract: 
We consider variants of the Minimum Circuit Size Problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MCSPQBF is known to be complete for PSPACE under ZPP reductions.  We show that it is not complete under logspace reductions, and indeed
it is not even hard for TC0 under uniform AC0 reductions. We obtain a variety of consequences that follow if oracle versions of MCSP are hard for various complexity classes under different types of reductions.  We also prove analogous results for the problem of determining the resource-bounded Kolmogorov complexity of strings, for certain types of Kolmogorov complexity measures.