# 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 where is a variable, or for , or of the form for , that is .

The following algorithm eliminates clauses of the form , 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 where is a propositional variable, do boolean constraint propagation:
• erase all clauses that contain literal
• remove 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.