Formal systems in computer science frequently involve specifications of computations over syntactic structures such as λ-terms, π-calculus expressions, first-order formulas, types, and proofs. This book is concerned, in part, with using higher-order logic to express such specifications. Properties are often associated with expressions by formal systems via syntax-based inference rules. Examples of such descriptions include presentations of typing and operational semantics. Logic programming, with its orientation around rule-based specifications, provides a natural framework for encoding and animating these kinds of descriptions. Variable binding is integral to most syntactic expressions, and its presence typically translates into side conditions accompanying inference rules. While many of the concepts related to binding, such as variable renaming, substitution, and scoping, are logically well understood, their treatment at a programming level is surprisingly difficult. We show here that a programming language based on a simply typed version of higher-order logic provides an elegant approach to performing computations over structures embodying binding.

The agenda just described has a prerequisite: We must be able to make sense of a higher-order logic as a programming language. This is a nontrivial task that defines a second theme that permeates this book. Usual developments of logic programming are oriented around formulas in clausal form with resolution as the sole inference rule. Sometimes a semantics-based presentation is also used, expanding typically into the idea of minimal (Herbrand) models.