Different ways definitions of $\Sigma_i \text{SAT}$

Question

Some research paper define $\Sigma_i \text{SAT}$ in this way, $$\Sigma_i \text{QBF}=\left\{\enspace\substack{\exists x_1\forall x_2\exists x_3\cdots Q_ix_i\varphi(x_1\cdots x_i) \text{ where } \varphi(x_1\cdots x_i) \text{ is an unquantified boolean formula. } \\}\right\}$$

$\Sigma_i \text{SAT}=\{\text{true } \Sigma_i \text{QBF}\}.$

Another way to define(followed from Barak, Arora book),

$$\Sigma_i \text{SAT}=\left\{\enspace\substack{\phi(u_1\cdots u_i)|\phi(x_1\cdots x_n) \\ \text{ is a boolean formular with a partition of $x_1\cdots x_n$ into $u_1\cdots u_i$ such that } \\ \exists u_1\forall u_2\exists u_3\cdots Q_iu_i[\phi(u_1\cdots u_i)=1] \\}\right\}$$

For example $u_1$ could be $x_1x_2x_3$ and $u_2=x_2x_6$ etc.

can you explain with one example why and how both ways of definition of $\Sigma_i \text{SAT}$ is same or equivalent?

Answer 1

Your two definitions use $x_i$ to represent different things. In the first, $x_i$ represents a sequence of booleans ($x_i \in \{0,1\}^{n_i}$), whereas in the latter $x_i$ represents a single boolean ($x_i \in \{0,1\}$). The $x_i$ in the first definition correspond to the $u_i$ in the second definition.

The first definition defines $\Sigma_i \mathrm{SAT}$ as a set of quantified formulas $\exists x_1 \forall x_2 \dots Q_i x_i \phi(x_1 \dots x_i)$ whereas the second definition defines it as a set of unquantified formulas $\phi(u_1 \dots u_i)$. The difference is mostly cosmetic: you can transform one into the other by just adding or removing the alternating quantifiers. However, to add the quantifiers to an unquantified formula, you need to know the length of the $u_i$, which means the information about the partition $u_1 \dots u_i$ should be stored somewhere in the problem instance. Your second definition doesn't make that completely clear.