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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 14:10] vkuncak |
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- | ====== Polynomial Algorithm for Horn Clauses ====== | + | ====== Boolean Constraint Propagation and Polynomial Algorithm for Horn Clauses ====== |
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. | ||
+ | |||
+ | Such clause is either of the form $\{p\}$ where $p \in V$ is a variable, or $\{\lnot p\}$ for $p \in V$, or of the form $\{\lnot p_1, \ldots, \lnot p_n, q\}$ for $n \ge 1$, that is $p_1 \land \ldots \land p_n \rightarrow q$. | ||
+ | |||
+ | 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, do //boolean constraint propagation//: |
- | * 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 $p$ | ||
- | * remove $\lnot p$ from all literals | + | * remove $\lnot p$ from all clauses |
- | * if there is an empty clause, set is not satisfiable | + | * if there is an empty clause, the set is unsatisfiable |
- | * if no contradiction found, the set is satisfiable | + | * if no empty clause found after repeating the above, the set is satisfiable |
+ | |||
+ | Boolean constraint propagation is a sound inference rule. If we obtain contradiction, the set is therefore unsatisfiable. | ||
+ | |||
+ | 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. | ||
+ | |||
+ | This algorithm does polynomial amount of work for each propositional variable, so it is polynomial. | ||
+ | |||
+ | Conclusion: the difficulty are clauses with at least two positive literals, they require case analysis. | ||