![]() A series of successful purges that results in no remaining clauses constructs a satisfying truth assignment in the process. If neither purge succeeds, the formula is not satisfiable. If it succeeds, continue with the remaining clauses. Choose an arbitrary element $x_i$ in the remaining clauses and set its value to True, and execute the purge algorithm. ![]() If the purge does not fail, any resulting clauses have exactly two literals. If during the purge process all variables from a clause are excluded via simplification, the purge fails. Call this process of eliminating one-literal clauses a purge. Consider $\phi=(x_1) \land (x_1 \lor x_2) \land ( x_3 \lor \overline \lor x_2)$ can be simplified to $(x_2)$, resulting in more one-clause literals that can further simplify $\phi$.(From Lewis&Papadimitriou, section 6.3) 2-SAT considers Boolean formulas that are the conjunction of $m$ clauses, where each clause is of the form $(x_i)$ or $(x_i \lor x_j)$.MAX-SAT is defined as follows: given a Boolean formula $\phi$ of $F$ clauses and an integer $K$, is there a truth assignment of the variables in $\phi$ that satisfy at least $K \leq F$ clauses? Show via a reduction from SAT that MAX-SAT is NP-complete.Show that NP is closed under union, concatenation, and Kleene star.Show that P is closed under union, complement and concatenation. ![]()
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