Everybody's Got To Be Somewhere

Abstract

The key to any nameless representation of syntax is how it indicates the variables we choose to use and thus, implicitly, those we discard. Standard de Bruijn representations delay discarding maximally till the leaves of terms where one is chosen from the variables in scope at the expense of the rest. Consequently, introducing new but unused variables requires term traversal. This paper introduces a nameless 'co-de-Bruijn' representation which makes the opposite canonical choice, delaying discarding minimally, as near as possible to the root. It is literate Agda: dependent types make it a practical joy to express and be driven by strong intrinsic invariants which ensure that scope is aggressively whittled down to just the support of each subterm, in which every remaining variable occurs somewhere. The construction is generic, delivering a universe of syntaxes with higher-order metavariables, for which the appropriate notion of substitution is hereditary. The implementation of simultaneous substitution exploits tight scope control to avoid busywork and shift terms without traversal. Surprisingly, it is also intrinsically terminating, by structural recursion alone.

Intrinsically-typed definitional interpreters for linear, session-typed languages

“…The representation of lexical variables var is nothing more than the observation that the context is a singleton. This nameless representation of binders is known as the co-de-Bruijn representation [2,26]. Whereas in a de-Bruijn representation of binding, the choice between variables in scope is delayed until the leaves of the syntax tree, in a co-de-Bruijn representation, the choice is made at the earliest opportunity.…”

“…We now combine separation logic with the technique of Allais et al [3] and McBride [26] to specify the typed syntax of a linearly-typed lambda calculus (LTLC) concisely:…”

“…The term co-de-Bruijn was coined by McBride [26], who exploits this representation in Agda to ensure that variable shifts in wellscoped, non-linear lambda terms do not require term traversals. He shows that hereditary substitutions on these terms become structurally recursive.…”

“…Neither Abel and Kraus [2], nor McBride [26] make the connection with separation logic, or make use of a magic wand-like connective, which was crucial in typing our functional abstractions.…”

Intrinsically typed compilation with nameless labels

“…The co-context of the composition is simply the union of the exports and imports. Our representation is inspired by the nameless co-de-Bruijn representation of lexical binding by McBride [2018], and the nameless representation of linear references by Rouvoet et al [2020a].…”