Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 124 / 3 / ii / Solution

This is the Karp–Lipton theorem. It is enough to place ${\Pi }_2^{\mathbf{P}}$ inside ${\Sigma }_2^{\mathbf{P}}$. Let $L\in {\Pi }_2^{\mathbf{P}}$, so for a polynomial-time predicate $R$ and polynomially bounded strings,The NP search problem that receives $(x,y)$ and seeks such a $z$ has, by part (i), a polynomial-size circuit family producing a valid witness whenever one exists. For each input length there is therefore a polynomial-size circuit $C$ such that, for every relevant $x,y$, existence of a witness implies $R(x,y,C(x,y))$.

Consequentlywhere the existentially guessed circuit description has polynomial length and evaluation of $C$ is polynomial time. This is a ${\Sigma }_2^{\mathbf{P}}$ description. Hence ${\Pi }_2^{\mathbf{P}}\subseteq {\Sigma }_2^{\mathbf{P}}$; complementation gives the reverse inclusion, and merging adjacent equal quantifier blocks collapses every higher level. Thus the polynomial hierarchy satisfies