# 算法设计|COMP3027 Algorithm Design代写

$$\exists x_{1} \forall x_{2} \ldots \exists x_{n-2} \forall x_{n-1} \exists x_{n} \Phi\left(x_{1}, \ldots, x_{n}\right) ?$$
That is, we wish to know whether there is a choice for $x_{1}$, so that for both choices of $x_{2}$, there is a choice for $x_{3}$, and so on, so that $\Phi$ is satisfied. We will refer to this decision problem as Quantified 3-SAT (or, briefly, QSAT).
The original 3 -SAT problem, by way of comparison, simply asked
$$\exists x_{1} \exists x_{2} \cdots \exists x_{n-2} \exists x_{n-1} \exists x_{n} \Phi\left(x_{1}, \ldots, x_{n}\right) ?$$

In other words, in 3-SAT it was sufficient to look for a single setting of the Boolean variables.

Here’s an example to illustrate the kind of reasoning that underlies an instance of QSAT. Suppose that we have the formula
$$\Phi\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1} \vee x_{2} \vee x_{3}\right) \wedge\left(x_{1} \vee x_{2} \vee \overline{x_{3}}\right) \wedge\left(\overline{x_{1}} \vee x_{2} \vee x_{3}\right) \wedge\left(\overline{x_{1}} \vee \overline{x_{2}} \vee \overline{x_{3}}\right)$$
$$\exists x_{1} \forall x_{2} \exists x_{3} \Phi\left(x_{1}, x_{2}, x_{3}\right) ?$$

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