On Parity and Near-Testability: $P^{A} \neq NT^{A}$ With Probability 1

Abstract

The class of near-testable sets, NT, was defined by Goldsmith, Joseph, and Young. They noted that $P \subseteq NT \subseteq PSPACE$, and asked whether P=NT. This note shows that NT shares the same $m$-degree as the parity-based complexity class $\bigoplus P$ (i.e., $NT\equiv^{p}_{m} \oplus P$) and uses this to prove that relative to a random oracle $A, P^{A} \neq NT^{A}$ with probability one. Indeed, with probability one, $NT^{A} - (NP^{A} \bigcup coNP^{A}) \neq 0$.