Linear Second-Order Unification

We present a new class of second-order unification problems, which we have called linear. We deal with completely general second-order typed unification problems, but we restrict the set of unifiers under consideration: they instantiate free variables by linear terms, i.e. terms where any λ-abstractions bind one and only one occurrence of a bound variable. Linear second-order unification properly extends context unification studied by Comon and Schmidt-Schauß. We describe a sound and complete procedure for this class of unification problems and we prove termination for three different subcases of them. One of these subcases is obtained requiring Comon’s condition, another corresponds to Schmidt-Schauß’s condition, (both studied previously for the case of context unification, and applied here to a larger class of problems), and the third one is original, namely that free variables occur at most twice.