Syntax and Semantics of Abstract Binding Trees

binding trees (abts) are a generalization of abstract syntax trees where operators may express variable binding structure as part of their arities. Originally formulated by Peter Aczel [1], unisorted abts have been deployed successfully as the uniform syntactic framework for several implementations of Constructive Type Theory, including Nuprl [3], MetaPRL [13] and JonPRL [21]. In Practical Foundations for Programming Languages [12], Robert Harper develops the multi-sorted version of abstract binding trees and proposes an extension to include families of operators indexed by symbols, which are, unlike variables, subject to only distinctnesspreserving renaming and not substitution; furthermore, symbols do not appear in the syntax of abts, and are only introduced as parameters to operators. This extension to support symbols is essential for a correct treatment of programming languages with open sums (e.g. ML’s exn type), as well as assignable references. In parallel, M. Fiore and his collaborators have developed the categorical semantics for several variations of second-order algebraic theories [8, 10, 6, 7]; of these, the simply sorted variants are equivalent to Harper’s abstract binding trees augmented with a notion of second-order variable (metavariable). The contribution of this paper is the development of the syntax and semantics of multi-sorted nominal abts, an extension of second order universal algebra to support symbol-indexed families of operators. Additionally, we have developed the categorical semantics for abts formally in Constructive Type Theory using the Agda proof assistant [18]; we have also developed an implementation for abstract binding trees in Standard ML, which we intend to integrate into the JonPRL proof assistant [21]. 1 Categorical Preliminaries Fix a set S of sorts; we will say τ sort when τ ∈ S. In this section, we will develop a theory of sets varying over collections of S-sorted symbols. Let I be the category of finite cardinals and their injective maps; then the comma construction I ↓ S, is the category of contexts of symbols whose objects are finite sets of symbols U and sort-assignments U S s , and whose morphisms are sort-preserving renamings; we will write Υ for a symbol context (U, s). 1 ar X iv :1 60 1. 06 29 8v 1 [ cs .L O ] 2 3 Ja n 20 16 Remark 1.1. To be precise, I ↓ S is an abuse of notation for the comma category |−|I ↓ ∆[S], with |−|I the forgetful functor from I to Set, and ∆[S](∗) , S a constant functor from 1 to Set. Formally, an object of I ↓ S is, then, a triple 〈U, ∗, s〉withU an object in I, ∗ the unique object in 1, and s a function |U|I S; an arrow ρ : 〈U, ∗, s〉 〈V , ∗, t〉 is a commuting triangle like the following: |U|I |V |I