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A Clari cation Concerning the L Hierarchy Eric Allender
 

Summary: A Clari cation Concerning the L Hierarchy
Eric Allender
Department of Computer Science, Rutgers University
P.O. Box 1179, Piscataway, NJ 08855-1179,USA
e-mail: allender@cs.rutgers.edu
October 29, 1997
Abstract
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
many-one reducibility.
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

  

Source: Allender, Eric - Department of Computer Science, Rutgers University

 

Collections: Computer Technologies and Information Sciences