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 $\Sigma_2^0$.

In this paper we prove that the results obtained in [1] for $L_p$-norm are still valid for an arbitrary Luxemburg norm.

The number of the distinct points entered by a random walk once and only once in the first $n$ steps is called the single point range up to time $n$. We consider the random walk on the $d$ dimensional integer lattice. When $d\geq 4$, the author showed a limiting behavior of the variance of the single point range and established the central limit theorem. In this note, we proved the law of the iterated logarithm in the same case.