As n increases, deciding membership in TWO PLAYER SAT_n becomes harder, analogous to how determining winning strategies for longer games of Go or chess becomes harder

We then define \(\Sigma^P_{n+1}\) to be the class of problems of the form \(X = \{x \mid \exists y \leq p(\lvert x\rvert) R(x,y)\}\) where \(R(x,y) \in \Pi^P_{n}\) and \(\Pi^P_{n+1}\) to be the class of problems of the form \(X = \{x \mid \forall y \leq q(\lvert x\rvert) S(x,y)\}\) where \(S(x,y) \in \Sigma^P_{n}\) and \(p(n)\) and \(q(n)\) are both polynomials. \(\Delta^P_n\) is the set of problems which are in both \(\Sigma^P_{n}\) and \(\Pi^P_{n}\). 1 that \(\Sigma^P_1 = \textbf{NP}\) and \(\