Ph containments and polynomial space

Given: the objects and hypotheses in the statement above.

For every $k\ge 0$, ${\mathrm{\Sigma }}_{k+1}^p=N{P}^{{\mathrm{\Sigma }}_k^p},{\mathrm{\Pi }}_{k+1}^p=coN{P}^{{\mathrm{\Sigma }}_k^p},{\mathrm{\Delta }}_{k+1}^p=P^{{\mathrm{\Sigma }}_k^p}\mathrm{.}$ For $k\ge 1$ a fixed complete bounded-alternation QBF language can replace the class oracle. At $k=1$, this gives the usual satisfiability oracle. The quantifier levels also equal polynomial-time alternating computations with at most $k$ blocks of existential/universal choices, beginning with the indicated polarity. With a fixed base oracle $A$, the same oracle characterization holds using access to both $A$ and a language in ${\mathrm{\Sigma }}_k^{p,A}$. (Quantifier and oracle characterizations of ph).

For each fixed $k\ge 0$, both ${\mathrm{\Sigma }}_k^p$ and ${\mathrm{\Pi }}_k^p$ are closed under polynomial-time many-one preimages and polynomially many uniformly indexed conjunctions and disjunctions. Precisely, if the language of pairs $(x,i)$ lies in the class and $0\le i<q(\mathrm{\mid }x\mathrm{\mid })$ for a polynomial-time computable polynomial bound $q$, then both “every such $i$” and “some such $i$” define languages in that class. Complements interchange ${\mathrm{\Sigma }}_k^p$ and ${\mathrm{\Pi }}_k^p$, and each is contained in both classes at level $k+1$. These assertions also hold with one fixed base oracle. (Ph quantifier block closure).

The class PSPACE is the union of all deterministic polynomial-space classes: $\mathrm{P}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}:={\bigcup }_{p\text{ polynomial}}\mathrm{D}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}(p(n))\mathrm{.}$ The class NPSPACE is the union of all nondeterministic polynomial-space classes: $\mathrm{N}\mathrm{P}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}:={\bigcup }_{p\text{ polynomial}}\mathrm{N}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}(p(n))\mathrm{.}$ Equivalently, a language $L$ lies in PSPACE when there is a deterministic multitape Turing machine deciding $L$ using at most $O(n^c)$ space for some constant $c$, and similarly $L$ lies in NPSPACE when there is such a nondeterministic decider. The meanings of $\mathrm{D}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}(s(n))$ and $\mathrm{N}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}(s(n))$ are those of the stated convention. (The classes PSPACE and NPSPACE).