# Oblivious oracle TM's

In "The Complexity Theory Companion" Hemaspaandra and Ogihara define an oblivious oracle TM as an oracle TM where the queries are dependent on the input and not on the oracle. They then show how to turn an oracle $\mathbf{NL}$ machine into an oblivious one. I wanted to know whether this construction will work for other machines as well? So not only for $\mathbf{NL}$ machines.