Abstract: We study the Minimum Circuit Size Problem (MCSP): given the truth-table of a Boolean function f and a number k, does there exist a Boolean circuit of size at most k computing f? This is a fundamental NP problem that is not known to be NP-complete. Previous work has studied consequences of the NP-completeness of MCSP. We extend this work and consider whether MCSP may be complete for NP under more powerful reductions. We also show that NP-completeness of MCSP allows for amplification of circuit complexity. We show the following results.

• If MCSP is NP-complete via many-one reductions, the following circuit complexity amplification result holds: If NP ∩ co-NP requires 2^nΩ(1)-size circuits, then E^NP requires 2^Ω(n)-size circuits.
• If MCSP is NP-complete under truth-table reductions, then EXP ≠ NP ∩ SIZE(2^nε) for some ε > 0 and EXP ≠ ZPP. This result extends to polylog Turing reductions.

• Proceedings of the 35th IARCS Conference on Foundations of Software Technology and Theoretical Computer Science (Bangalore, India, December 15-19, 2015). Volume 45 of Leibniz International Proceedings in Informatics, pages 236-245, 2015.
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