  It is a widely believed, though unproven, conjecture that the capability of
postselection increases the language recognition power of both probabilistic
and quantum polynomial-time computers. It is also unknown whether
polynomial-time quantum machines with postselection are more powerful than
their probabilistic counterparts with the same resource restrictions. We
approach these problems by imposing additional constraints on the resources to
be used by the computer, and are able to prove for the first time that
postselection does augment the computational power of both classical and
quantum computers, and that quantum does outperform probabilistic in this
context, under simultaneous time and space bounds in a certain range. We also
look at postselected versions of space-bounded classes, as well as those
corresponding to error-free and one-sided error recognition, and provide
classical characterizations. It is shown that $\mathsf{NL}$ would equal
$\mathsf{RL}$ if the randomized machines had the postselection capability.
