Abstract

We show that all sets complete for NC¹ under AC 0 reductions are isomorphic under AC 0 -computable isomorphisms. Although our proof does not generalize directly to other complexity classes, we do show that, for all complexity classes C closed under NC¹-computable manyone reductions, the sets complete for C under NC 0 reductions are all isomorphic under AC 0 -computable isomorphisms. Our result showing that the complete degree for NC¹ collapses to an isomorphism type follows from a theorem showing that in NC¹, the complete degrees for AC 0 and NC 0 reducibility coincide.

Citations