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Let $L$ be defined as follows:

$\small L = \{(\phi, k) \mid \phi \in SAT \mbox{ and there is a satisfying assignment for $\phi$ having exactly $k$ true variables} \}$

I am struggling to show that $SAT\leq_{p} L$, I can't figure what $k$ should be and how to reconstruct $\phi$.

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Let $\phi'$ be a SAT formula with $n$ variables.

Consider the pair $(\phi, k)$ where

  • $\phi$ is a formula with $2n$ variables that is obtained from $\phi'$ by adding $n$ new variables $x_1, \dots, x_n$ (if you want each new variable $x_i$ to appear in at least one clause you can add the trivial clause $(x_i \vee \overline{x}_i)$).
  • $k=n$.

If $\phi$ is satisfiable then any satisfying truth assignment for $\phi$ restricted to the variables in $\phi'$ is also a satisfying truth assignment for $\phi'$.

If $\phi'$ is satifiable, consider any satisfying truth assignment $\tau'$ and let $n'$ be the number of true variables in $\tau'$. You can obtain a satisfying truth assignment for $\phi$ by setting all variables of $\phi'$ according to $\tau'$ and any choice of exactly $n-n'$ variables from $x_1, \dots, x_n$ to true. Notice that this is always possible since $0 \le n-n' \le n$.

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