Linear Second-Order Uni cation

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