The Minimum Oracle Circuit Size Problem
Eric Allender, Dhiraj Holden, and Valentine Kabanets
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 MCSP^{QBF} 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 TC^{0} under uniform AC^{0} 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.
Versions
- journal version in Computational Complexity (to appear)
- extended abstract in Proceedings of the Thirty-Second Symposium on Theoretical Aspects of Computer Science (STACS'15), pages 21-33, 2015.
- preliminary full version in Electronic Colloquium on Computational Complexity ECCC-TR14-176.