- Authors: J. Kinne and D. van Melkebeek.

- Reference:
*Computational Complexity*, 19: 423-475, 2010.
2009.

- Earlier version:

#### Abstract

We prove space hierarchy and separation results for randomized and other
semantic models of computation with advice where a machine is only
required to behave appropriately when given the correct advice sequence.
Previous works on hierarchy and separation theorems
for such models focused on time as the resource. We obtain tighter
results with space as the resource.
Our main theorems deal with space-bounded randomized machines that
always halt, and read as follows.
Let *s*(*n*) be any space-constructible monotone function that is
Ω(log *n*) and let
*s*'(*n*) be any function such that
*s*'(*n*) = ω(*s*(*n*+*as*(*n*)))
for all constants *a*.

- There exists a language computable by
*two-sided* error randomized
machines using *s*'(*n*) space and one bit of advice that is not
computable by *two-sided* error machines using *s*(*n*)
space and min(*s*(*n*),*n*) bits of advice.

- There exists a language computable by
*zero-sided* error randomized
machines in space *s*'(*n*) with one bit of advice that is not
computable by *one-sided* error randomized machines using
*s*(*n*) space and min(*s*(*n*),*n*) bits of advice.

If, in addition, *s*(*n*) = O(*n*) then the condition on
*s*' above can be relaxed to
*s*'(*n*) = ω(*s*(*n*+1)).
This yields tight space hierarchies for
typical space bounds *s*(*n*) that are at most linear.
We also obtain results that apply to generic semantic models of
computation.