Extensions and applications of higher-order unification

This dissertation explores the problem of unification in various typed $\lambda$-calculi, developing and proving the correctness and completeness of unification algorithms for various calculi within a single general framework, and then demonstrating the practical importance of these algorithms by means of example applications. We being by presenting our general framework for unification, based on transformations of unification problems. Then, in this framework, we develop a new unification algorithm for a $\lambda$-calculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development involves significant complications not arising Huet's corresponding algorithm for the simply typed $\lambda$-calculus, primarily because it must deal with ill-typed terms. We then extend this algorithm first for dependent product ($\Sigma$) types, and second for implicit polymorphism. In the latter case, the algorithm is incomplete, though still quite useful in practice. The last part of the dissertation provides examples of the usefulness of the algorithms. The general idea is to use a $\lambda$-calculus as a meta-language for representing various other languages (object-languages). The rich structure of a typed $\lambda$-calculus, as opposed to traditional, first-order abstract syntax trees, allows us to express rules, e.g., program transformation and logical inference rules, that are more succinct, more powerful, and easier to reason about. We can then use unification in the meta-language to mechanize application of these rules.