Question
An "empty implication" is of the form " x", i.e., an implication indicating that variable x must be set to True. Below is a variant of the greedy satisfiability algorithm for Horn formulas. It can be implemented to run in linear time (i.e. in time proportional to the number of literals in the formula) using simple data-structures. Fast Greedy Horn: Set every variable in the formula to False. Set W = {u : v appears on the right of an empty implication}. While W is not empty, do steps 4, 5: Remove a variable u from W, and set it to True. For each clause c where u appears on the left-hand side, remove v from the left side of c. If no literals remain on the left side of c (i.e., c becomes an empty implication), add the variable on the right side of c into W (if it is not already in W). Check if the current True/False assignments make the negative clauses true. If yes, output that the formula is satisfied. If no, output that the formula is not satisfiable. Explain why this algorithm is correct. You may simply show how it is an implementation of the greedy satisfiability algorithm that we already learned. That is, show that this algorithm makes decisions (of which variables to set True) that would have been made in the original algorithm (which we already know is correct). Or you can give a proof of correctness using the invariant approach, similar to the way the greedy satisfiability algorithm in the book was proved correct in problem 1.
Explanation / Answer
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