Is every formula with a unique minimal model equivalent to a horn clause?

Question

I know that every horn clause has an unique minimal model. But what about the other way around? Is every formula that has a unique minimal model always equivalent to a horn clause?

Answer 1

I suppose that you mean "set of Horn clauses", rather than "Horn clause". But even then the assumption does not hold. Consider the formula $\psi = (p \to (q \lor r))$.

• $\psi$ has a unique minimal model, namely $\emptyset$ (the model in which all atoms are false).

• $\psi$ is not equivalent to any set $S$ of Horn clauses: Suppose that it were equivalent to a Horn clause set $S$. Then $\psi \land p$ would be equivalent to the Horn clause set $S' = S \cup \{p\}$. However, $S'$ has two minimal models, namely $\{p,q\}$ and $\{p,r\}$, contradicting the assumption that it is a Horn clause set.