Higher-Order Uni cation via Combinators

We present an algorithm for uniication in the simply typed lambda calculus which enumerates complete sets of uniiers using a nitely branching search space. In fact, the types of terms may contain type-variables, so that a solution may involve type-substitution as well as term-substitution. the problem is rst translated into the problem of uniication with respect to extensional equality in combinatory logic, and the algorithm is deened in terms of transformations on systems of combinatory terms. These transformations are based on a new method (itself based on systems) for deciding extensional equality between typed combinatory logic terms.