It follows immediately from Proposition 3.1 that \(\Sigma^P_1 = \textbf{NP}\) and \(\Pi^P_1 = \textbf{coNP}\). It is also evident from the foregoing definitions that we have the containment relations \(\Sigma^P_n \subseteq \Delta^P_{n+1} \subseteq \Sigma^P_{n+1}\) and \(\Pi^P_n \subseteq \Delta^P_{n+1} \subseteq \Pi^P_{n+1}\) depicted in Figure 4. It is also not difficult to show that \(\textbf{PH} \subseteq \textbf{PSPACE}\). These relationships are similar to those which obtain between the analogously defined \(\Sigma^0_n\)- and \(\Pi^0_n\)-sets in the Arithmetic Hierarchy studied in computability theory (see, e.g., Rogers 1987). But whereas it can be shown by a standard diagonal argument that the arithmetic hierarchy does not collapse, the following are also currently unresolved questions of fundamental importance: