 
Summary: Limitations of the Upward Separation Technique
Eric Allender
Department of Computer Science
Rutgers University
New Brunswick, N.J. 08903
April, 1990
A preliminary version of this paper was presented at the 16th International Colloquium on Automata,
Languages, and Programming [3].
Supported in part by National Science Foundation Research Initiation Grant number CCR8810467.
1
SUMMARY
The upward separation technique was developed by Hartmanis, who used it to show
that E=NE iff there is no sparse set in NPP [15]. This paper shows some inherent
limitations of the technique. The main result of this paper is the construction of an
oracle relative to which there are extremely sparse sets in NPP, but NEE = EE; this
is in contradiction to a result claimed in [14, 16]. Thus, although the upward separation
technique is useful in relating the existence of sets of polynomial (and greater) density in
NPP to the NTIME(T(n)) = DTIME(T(n)) problem, the existence of sets of very low
