The Suspension Calculus and its Relationship to Other Explicit Treatments of Substitution in Lambda Calculi

The intrinsic treatment of binding in the lambda calculus makes it an ideal data structure for representing syntactic objects with binding such as formulas, proofs, types, and programs. Supporting such a data structure in an implementation is made difficult by the complexity of the substitution operation relative to lambda terms. To remedy this, researchers have suggested representing the meta level substitution operation explicitly in a refined treatment of the lambda calculus. The benefit of an explicit representation is that it allows for a fine-grained control over the substitution process, leading also to the ability to intermingle substitution with other operations on lambda terms. This insight has lead to the development of various explicit substitution calculi and to their exploitation in new algorithms for operations such as higher-order unification. Considerable care is needed, however, in designing explicit substitution calculi since within them the usually implicit operations related to substitution can interact in unexpected ways with notions of reduction from standard treatments of the lambda calculus. This thesis describes a particular realization of explicit substitutions known as the suspension calculus and shows that it has many properties that are useful in a computational setting. One significant property is the ability to combine substitutions. An earlier version of the suspension calculus has such an ability, but the complexity of the machinery realizing it in a complete form has deterred its direct use in implementations. To overcome this drawback a derived version of the calculus had been developed and used in practice. Unfortunately, the derived calculus sacrifices generality and loses a property that is important for new approaches to unification. This thesis redresses this situation by presenting a modified form of the substitution combination mechanism that retains the generality and the computational properties of the original calculus while being simple enough to use directly in implementations. These modifications also rationalize the structure of the calculus, making it possible to easily superimpose additional logical structure over it. We illustrate this capability by showing how typing in the lambda calculus can be treated in the resulting framework and by presenting a natural translation into the λσ-calculus, another well-known treatment of explicit substitutions. Another contribution of this thesis is a survey of the realm of explicit substitution calculi. In particular, we describe the computational properties that are desired in this setting and then characterize various calculi based on how well they capture these. We utilize the simplified suspension calculus in this process. In particular, we describe translations between the other popular calculi and the suspension calculus towards understanding and contrasting their relative capabilities. Finally, we discuss an elusive property of explicit substitution calculi known as preservation of strong normalization and discuss why there is hope that the suspension calculus possesses this property.