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sav08:polynomial_algorithm_for_horn_clauses [2008/03/12 01:32] vkuncak created |
sav08:polynomial_algorithm_for_horn_clauses [2008/03/12 01:41] vkuncak |
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| A Horn clause is a clause that has at most one positive literal. | A Horn clause is a clause that has at most one positive literal. | ||
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| + | Such clause is either of the form $\{p\}$ where $p \in V$ is a variable, or of the form $\{\lnot p_1, \ldots, \lnot p_n, q$ for $n \ge 1$, that is $p_1 \land \ldots \land p_n \rightarrow q$. | ||
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| + | The following algorithm eliminates clauses of the form $\{p\}$, keeping only clauses that have at least one assumption. | ||
| To check satisfiability of a set of Horn clauses: | To check satisfiability of a set of Horn clauses: | ||
| - | * set initially all variables to false | ||
| * while the set contains a clause of the form $\{p\}$ where $p$ is a propositional variable: | * while the set contains a clause of the form $\{p\}$ where $p$ is a propositional variable: | ||
| * erase all clauses that contain literal $p$ | * erase all clauses that contain literal $p$ | ||
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| * if there is an empty clause, set is not satisfiable | * if there is an empty clause, set is not satisfiable | ||
| * if no contradiction found, the set is satisfiable | * if no contradiction found, the set is satisfiable | ||
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| + | On $\{p\}$ we conclude that $p$ must be true and derive valid consequences of this fact. If we obtain contradiction, the set of clearly unsatisfiable. | ||
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| + | Moreover, if loop terminates and there are no empty clauses, then every clause contains a negative literal. The assignment that sets all remaining variables to //false// is a satisfying assignment. | ||
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| + | This algorithm does polynomial amount of work for each propositional variable, so it is polynomial. | ||
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| + | Conclusion: the difficulty are clauses with at least two positive literals, they require case analysis. | ||