On Pseudorandomness with respect to Deterministic Observes

In the theory of pseudorandomness, potential (uniform) observers are modeled as probabilistic polynomial-time machines. In fact many of the central results in that theory are proven via probabilistic polynomial-time reductions. In this paper we show that analogous deterministic reductions are unlikely to hold. We conclude that randomness of the observer is essential to the theory of pseudorandomness. What we actually prove is that the hypotheses of two central theorems (in the theory of pseudorandomness) hold unconditionally when stated with respect to deterministic polynomial-time algorithms. Thus, if these theorems were true for deterministic observers, then their conclusions would hold unconditionally, which we consider unlikely. For example, it would imply (unconditionally) that any unary language in BPP is in P. The results are proven using diagonalization and pairwise independent sample spaces.