A Lambda Term Representation Inspired by Linear Ordered Logic

TL;DR

This paper presents a novel lambda term representation inspired by ordered logic that efficiently tracks variable occurrences, reducing space leaks and improving interpreter performance in proof checkers.

Contribution

It introduces a new nameless lambda term representation that encodes variable positions and contexts, enabling exact environment management without traversals.

Findings

Correctness of the new representation is proven.

Experimental evaluation shows performance improvements in a proof checker.

The approach reduces space leaks in interpreters.

Abstract

We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, and application carries the additional information where to cut the variable context into function and argument part. This way, complete information about free variable occurrence is available at each subterm without requiring a traversal, and environments can be kept exact such that they only assign values to variables that actually occur in the associated term. Our approach avoids space leaks in interpreters that build function closures. In this article, we prove correctness of the new representation and present an experimental evaluation of its performance in a proof checker for the Edinburgh Logical Framework. Keywords: representation of binders, explicit substitutions, ordered contexts, space leaks, Logical Framework.