An Isomorphism Theorem for Circuit Complexity

We show that all sets complete for NC1 under AC0 reductions are isomorphic under AC0computable 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.1 1A version of this paper will appear in the Proceedings of the Eleventh Annual IEEE Conference on Computational Complexity (formerly Structure in Complexity Theory), 1996.