Boolean Constraint Propagation and Polynomial Algorithm for Horn Clauses

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 negative literal.

To check satisfiability of a set of Horn clauses:

  • while the set contains a clause of the form $\{p\}$ where $p$ is a propositional variable, do boolean constraint propagation:
    • erase all clauses that contain literal $p$
    • remove $\lnot p$ from all clauses
    • if there is an empty clause, the set is unsatisfiable
  • 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.