The levels of the polynomial hierarchy increase: $\Sigma_j^p \subseteq \Sigma_k^p$ and $\Pi_j^p \subseteq \Pi_k^p$ for $j \le k$, and each of $\Sigma_k^p$ and $\Pi_k^p$ is contained in both $\Sigma_{k+1}^p$ and $\Pi_{k+1}^p$, for $k \ge 1$. Every level is contained in PH. From level 1 up, a definition is padded with a vacuous quantifier block; the step from level 0 goes through the complete problem HORN-SAT.