Polynomial hierarchy levels

All languages are subsets of $\{0,1{\}}^{\ast }$. Put ${\mathrm{\Sigma }}_0^p={\mathrm{\Pi }}_0^p=P$ as in The class P. For a fixed integer $k\ge 1$, $L\in {\mathrm{\Sigma }}_k^p$ means that a deterministic polynomial-time predicate $R$ and polynomially bounded, polynomial-time computable lengths $p_1,\dots ,p_k$ satisfy $x\in L⟺\mathrm{\exists }{y}_1\in \{0,1{\}}^{p_1(\mathrm{\mid }x\mathrm{\mid })}\mathrm{\forall }{y}_2\in \{0,1{\}}^{p_2(\mathrm{\mid }x\mathrm{\mid })}\cdots Q_k{y}_k\in \{0,1{\}}^{p_k(\mathrm{\mid }x\mathrm{\mid })}R(x,y_1,\dots ,y_k)\mathrm{.}$ Here $Q_j$ is existential for odd $j$ and universal for even $j$. Starting universally defines ${\mathrm{\Pi }}_k^p$. Write $\mathrm{P}\mathrm{H}={\bigcup }_{k\ge 0}{\mathrm{\Sigma }}_k^p$. The level and all polynomials are fixed for a language, not part of the input. Length zero allows the single empty string. Variable-length witnesses can be encoded with a length field and padding; every padded string decodes to a bounded-length string, so universal blocks introduce no invalid-witness exception.