Summary: A Clari cation Concerning the L Hierarchy
Department of Computer Science, Rutgers University
P.O. Box 1179, Piscataway, NJ 08855-1179,USA
October 29, 1997
In AO96 , it is stated without proof that if NC1L = AC0L, then the
L hierarchy collapses to some level. This note provides a proof of that claim.
The reader is referred to AO96 for all background, de nitions, and motivation.
All of the circuit complexity classes mentioned in this note are logtime-uniform
BIS90 , unless otherwise speci ed.
1 The Proof
The proof consists of establishing the following fact.
Fact 1.1 The set of languages NC1DET has a complete set under logspace
This fact is su cient to establish the claim made in AO96 . To see this,
assume that NC1DET = AC0DET. By Claim 1.1, there is a complete set
for AC0DET. Since AC0DET is equal to LLL::L
, this complete set is in