Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem

We show that all sets that are complete for NP under non-uniform AC0 reductions are isomorphic under non-uniform AC0-computable isomorphisms. Furthermore, these sets remain NP-complete even under non-uniform NC0 reductions. More generally, we show two theorems that hold for any complexity class C closed under uniform NC1-computable many-one reductions. Gap: The sets that are complete for C under AC0 and NC0 reducibility coincide. Isomorphism: The sets complete for C under AC0 reductions are all isomorphic under isomorphisms computable and invertible by AC0 circuits of depth three. Our Gap Theorem does not hold for strongly uniform reductions: we show that there are Dlogtime-uniform AC0-complete sets for NC1 that are not Dlogtime-uniform NC0complete.

• Type: Journal
• Year: 1998
• Where: JCSS
• Authors: Manindra Agrawal, Eric Allender, Steven Rudich
• Keywords: JCSS 1998, Non-uniform Ac0 Reductions, Non-uniform Nc0 Reductions, Reductions