Reducing the complexity of reductions
Manindra Agrawal, Eric Allender, Russell Impagliazzo, Toniann Pitassi, Steven Rudich · 1997
We prove that the Berman-Hartmanis isomorphism conjecturers true under ACO reductions.More generafly, we show three theorems that hold for any comdexitv class C closed under (uniform) TCO-commtable man~-one" reductions.Isomorp'hism:The sets c~mplete for Cunder ACO reductions are afl isomorphic under isomorphisms computable and invertible by ACO circuits of depth three.Ga : p The sets that are complete for C under ACO and NC reducibility coincide.Stop Gap: The sets that are complete for C under ACO[mod 2] and ACO 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.)