Partial polymorphic type inference and higher-order unification

We show that the problem of partial type inference in the nth-order polymorphic &lgr;-calculus is equivalent to nth-order unification. On the one hand, this means that partial type inference in polymorphic &lgr;-calculi of order 2 or higher is undecidable. On the other hand, higher-order unification is often tractable in practice, and our translation entails a very useful algorithm for partial type inference in the &ohgr;-order polymorphic &lgr;-calculus. We present an implementation in &lgr;Prolog in full.