Reducing the complexity of reductions

Abstract

We prove that the Berman-Hartmanis isomorphism conjecture is true under AC0 reductions. More generally, we show three theorems that hold for any complexity class C closed under (uniform) TC0-computable many-one reductions. Isomorphism: The sets complete for C under AC0 reductions are all isomorphic under isomorphisms computable and invertible by AC0 circuits of depth three. Gap: The sets that are complete for C under AC0 and NC0 reducibility coincide. Stop Gap: The sets that are complete for C under AC0 [mod 2] and AC0 reducibility do not coincide. (These theorems hold both in the non-uniform and P-uniform settings.) To prove the second theorem for P-uniform settings, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.)