Reducing the complexity of reductions
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
| Hi-index | 0.00 |
We show that all sets complete for NC1 under AC0 reductions are isomorphic under AC0-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 many-one reductions, the sets complete for C under NC0 reductions are all isomorphic under AC0-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 AC0 and NC0 reducibility coincide. This 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 NC0-complete.