One head is better than two: a polynomial restriction for propositional definite Horn forgetting

Logical forgetting is NP-complete even in the simple case of propositional Horn formulae, and may exponentially increase their size. A way to forget is to replace each variable to forget with the body of each clause whose head is the variable. It takes polynomial time in the single-head case: each variable is at most the head of a clause. Some formulae are not single-head but can be made so to simplify forgetting. They are single-head equivalent. The first contribution of this article is the study of a semantical characterization of single-head equivalence. Two necessary conditions are given. They are sufficient when the formula is inequivalent: it makes two sets of variables equivalent only if they are also equivalent to their intersection. All acyclic formulae are inequivalent. The second contribution of this article is an incomplete algorithm for turning a formula single-head. In case of success, forgetting becomes possible in polynomial time and produces a polynomial-size formula, none of which is otherwise guaranteed. The algorithm is complete on inequivalent formulae.