polynomial hierarchy is a hierarchy

The polynomial hierarchy is a hierarchy. Specifically:

$${\mathrm{\Sigma }}_i^p\cup {\mathrm{\Pi }}_i^p\subseteq {\mathrm{\Delta }}_{i+1}^p\subseteq {\mathrm{\Sigma }}_{i+1}^p\cap {\mathrm{\Pi }}_{i+1}^p.$$

Proof

To see that ${\mathrm{\Sigma }}_i^p\cup {\mathrm{\Pi }}_i^p\subseteq {\mathrm{\Delta }}_{i+1}^p={𝒫}^{{\mathrm{\Sigma }}_i^p}$,observe that the machine which checks its input against its oracle and accepts or rejects when the oracle accepts or rejects (respectively) is easily in $𝒫$, as is the machine which rejects or accepts when the oracle accepts or rejects (respectively). These easily emulate ${\mathrm{\Sigma }}_i^p$ and ${\mathrm{\Pi }}_i^p$ respectively.

Since $𝒫\subseteq 𝒩𝒫$, it is clear that${\mathrm{\Delta }}_i^p\subseteq {\mathrm{\Sigma }}_i^p$. Since ${𝒫}^{𝒞}$ isclosedunder complementation for anycomplexity class$𝒞$ (the associatedmachines are deterministic and always halt, so the complementary machine justreverses which states are accepting), if$L\in {𝒫}^{{\mathrm{\Sigma }}_i^p}\subseteq {\mathrm{\Sigma }}_i^p$ then so is $\overline{L}$,and therefore $L\in {\mathrm{\Pi }}_i^p$.

Unlike the arithmetical hierarchy, the polynomial hierarchy is not known to be proper. Indeed, if $𝒫=𝒩𝒫$ then $𝒫=𝒫\mathscr{H}$, so a proof that the hierarchy is proper would be quite significant.