We prove space hierarchy and separation results for randomized and

other semantic models of computation with advice. 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 are the following. Let $s(n)$ be any

space-constructible function that is $Omega(log n)$ and such that

$s(a n) = O(s(n))$ for all constants $a$, and let $s'(n)$ be any

function that is $omega(s(n))$.

- 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 randomized 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.

The condition that $s(a n)=O(s(n))$ is a technical condition

satisfied by typical space bounds that are at most linear. We also

obtain weaker results that apply to generic semantic models of

computation.