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

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.