Higher-order abstract syntax

In computer science, higher-order abstract syntax (abbreviated HOAS) is a technique for the representation of abstract syntax trees for languages with variable binders. An abstract syntax is abstract because it is represented by mathematical objects that have certain structure by their very nature. For instance, in first-order abstract syntax (FOAS) trees, as commonly used in compilers, the tree structure implies the subexpression relation, meaning that no parentheses are required to disambiguate programs (as they are, in the concrete syntax). HOAS exposes additional structure: the relationship between variables and their binding sites. In FOAS representations, a variable is typically represented with an identifier, with the relation between binding site and use being indicated by using the same identifier. With HOAS, there is no name for the variable; each use of the variable refers directly to the binding site. There are a number of reasons why this technique is useful. First, it makes the binding structure of a program explicit: just as there is no need to explain operator precedence in a FOAS representation, there is no need to have the rules of binding and scope at hand to interpret a HOAS representation. Second, programs that are alpha-equivalent (differing only in the names of bound variables) have identical representations in HOAS, which can make equivalence checking more efficient. One mathematical object that could be used to implement HOAS is a graph where variables are associated with their binding sites via edges. Another popular way to implement HOAS (in, for example, compilers) is with de Bruijn indices. The first programming language which directly supported λ-bindings in syntax was the higher-order logic programming language λProlog. The paper that introduced the term HOAS used λProlog code to illustrate it. Unfortunately, when one transfers the term HOAS from the logic programming to the functional programming setting, that term implies the identification of bindings in syntax with functions over expressions.