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

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