Although our proof does not generalize directly to other complexity
classes, we do show that, for all complexity classes C closed under
NC^1-computable many-one 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 NC^1 collapses to an
isomorphism type follows from a theorem showing that in NC^1, the complete
degrees for AC^0 and NC^0 reducibility coincide. This theorem does not
hold for strongly uniform reductions: we show that there are
Dlogtime-uniform AC^0-complete sets for NC^1 that are not Dlogtime-uniform
NC^0-complete.
Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1996/96-04.ps.gz
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