$k$-SAT completeness proof when $k$ is linear in number of variables
Your problem is actually in P. First of all, you can assume that no clause contains both a variable and its negation, since such clauses are always satisfied. Suppose that the instance contains $m$ clauses. Then a random assignment will falsify your formula with probability $m/2^{n/2}$. If $m &lt; 2^{n/2}$, then this means that your formula is satisfiable. Otherwise, the input is of size at least $2^{n/2}$, and so you can go over all $2^n$ truth assignments in polynomial time.