Determining the Levels of Some Special Complexity Classes of Sets in the Kleene Arithmetical Hierarchy

Abstract

We shall determine their levels of some special classes of sets of strings such as $\{X\subseteq\Sigma^* : P[X]\neq NP[X]\}$ in the Kleene arithmetical hierarchy on subclasses of $\mathscr{P}(\Sigma^*)$. We shall show that such several classes are proper $\Pi_2^0$, that is, they are $\Pi_2^0$ but not $\Sigma_2^0$.