Consider the construction of "A Pseudorandom Generator from any One-way Function" [HILL99] in Hastad et al.
One way to proof pseudorandomness of this construction is by contradiction. That is, an assumed PRG distinguisher is transformed into a OWF inverter (usually through an intermediate notion called pseudoentropy generator). There is both a uniform and a non-uniform version, considering uniform resp. non-uniform distinguishers/inverters. However, in all proofs of the uniform version that I have seen the PRG distinguisher and the OWF inverter are randomized.
Is it (im-)possible to convert a deterministic (uniform) PRG distinguisher into a deterministic (uniform) OWF inverter?