In the course of studying the relationship between arithmetic and complexity theory, it is often useful to consider functions in addition to sets. For instance we may consider the class \(\textbf{FP} =_{\text{df}}\Box^P_1\) of functions computable by a deterministic Turing machine in polynomial time. Similarly, we define \(\Box^P_{n+1}\) to be the class of functions computable by a deterministic Turing machine in polynomial time with the use of an oracles for a set in level \(\Sigma^P_n\) of \(\textbf{PH}\). As in the case of \(\textbf{PH}\), it is not known whether the hierarchy \(\Box^P_1 \subseteq \Box^P_2 \subseteq \ldots\) collapses.