The authors hold that relationships between deterministic and non-deterministic complexity classes are unsettled. To help settle them, the authors adopt the polynomial time-bounded Turing machines and oracles approach used by others. In particular, the authors extend the work reported in 1983 by Homer and Maass [1] to show that for every oracle such as A, B, and C, and any arbitrary real k > 0,

(1) NP ^A ≠ EXP_k^A

(2) P^A ⊊ NP^A ⊊ EXP_k^A

(3) EXP_k^B ⊊ NP^B

(4) NP^C and EXP_k^C are incomparable under inclusion.

In developing these conclusions, the authors offer four theorems, one corollary, one proposition, and six lemmas. The proofs generally depend on set constructions.