A Simplified Suspension Calculus and its Relationship to Other Explicit Substitution Calculi

TL;DR

This paper simplifies and rationalizes the suspension calculus for explicit substitutions in lambda calculus, making it more practical and theoretically robust, and compares it with other systems to understand their properties.

Contribution

The paper introduces a simplified, rationalized version of the suspension calculus that supports logical analysis and has strong theoretical properties, and it classifies various explicit substitution calculi.

Findings

Simplified encoding of substitution composition.

The modified calculus supports logical analyses and type assignments.

The calculus exhibits strong termination and confluence properties.

Abstract

This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this calculus provides a cumbersome encoding of substitution composition, an operation that is important to the efficient realization of reduction. This encoding is simplified here, resulting in a treatment that is easy to use directly in applications. The rationalization consists of the elimination of a practically inconsequential flexibility in the unravelling of substitutions that has the inadvertent side effect of losing contextual information in terms; the modified calculus now has a structure that naturally supports logical analyses, such as ones related to the assignment of types, over lambda terms. The overall calculus is shown to have pleasing theoretical properties such as a strongly terminating sub-calculus for substitution and confluence even in the presence of term meta variables that are accorded a grafting interpretation. Another contribution of the paper is the identification of a broad set of properties that are desirable for explicit substitution calculi to support and a classification of a variety of proposed systems based on these. The suspension calculus is used as a tool in this study. In particular, mappings are described between it and the other calculi towards understanding the characteristics of the latter.