Question

Let $\mathbf{CNFk}=\left\{\right\}$ is a satisfiable cnf-formula where each variable appears in at most k places}.

a. Show that$\mathbf{CNF}\mathbf{2}?\mathbf{P}$ .

b. Show that$\mathbf{CNF}\mathbf{3} \mathbf{is}\mathbf{NP}$-complete.

Short answer

It is clear that$\varphi$is satisfiable if and only if $r_y(⟨\varphi ⟩)$is satisfiable. The $r_y$ is a reduced polynomial time in terms of the number of variable in$\varphi$ from$CW\_\left\{3\right\}isNP-$ complete.

Step-by-step solution

Step 1:Chemical molecule of the equation

Now have to show that $CM{K}_2\in P_{-}$.

Let$T_y$be the polynomial tine decider for${\text{CH}}_{2^{-}}$

$T_p$can be described as $\{\left\{T\right\}wS$:

$T_3=$on input$⟨\varphi ⟩=$

Converting variable into polynomial

According to CNF rules,

Consider the first choose of$\varphi$$\varphi$. If it is of the from $x$, and there is$x in \varphi$ , reject

Solve CNF where c occurs in every choose, where negation of c does not appear-

Every time $T_1$processes each variable and reaches either accept or reject Because of this the number of$\varphi$ might decrease by 1 or Hence running time of$T_p$ becomes polynomial tine in terms of the number of variables.

So,$C{H}_2\in P.$

Thus,$GW\_\left\{3\right\}isNP-$ complete