Abstract Syntax for Variable Binders: An Overview

A large variety of computing systems, such as compilers, interpreters, static analyzers, and theorem provers, need to manipulate syntactic objects like programs, types, formulas, and proofs. A common characteristic of these syntactic objects is that they contain variable binders, such as quantifiers, formal parameters, and blocks. It is a common observation that representing such binders using only first-order expressions is problematic since the notions of bound variable names, free and bound occurrences, equality up to alpha-conversion, substitution, etc., are not addressed naturally by the structure of first-order terms (labeled trees). This overview describes a higher-level and more declarative approach to representing syntax within such computational systems. In particular, we shall focus on a representation of syntax called higher-order abstract syntax and on a more primitive version of that representation called λ-tree syntax.