🤖 AI Summary
This study addresses the challenge of establishing a precise correspondence between explicit substitution λ-calculi and resource-aware linear calculi under constraints of limited parameter availability. To this end, the work introduces a novel term expansion technique that integrates intersection types with a multiplicity mechanism, mapping explicit substitution λ-terms into weakly linear terms in Boudol’s resource-sensitive λ-calculus. This is the first extension of term expansion methods to calculi with explicit substitution, achieving semantic alignment between strongly normalizing terms and weakly linear terms while guaranteeing they share identical normal forms. The result provides a foundational theoretical framework for integrating substructural type systems with explicit substitution formalisms.
📝 Abstract
Term expansion was originally introduced in 2004 as a way to relate terms typed in an intersection type system with linear terms. Recently, new applications of term expansion include the relation of lambda-terms with terms typed in other substructural type systems, such as the relevant and the ordered type systems, and the use of quantitative types to relate the strongly normalising lambda-terms with weak linear terms that share the same normal form. Here we define a new term expansion for a calculus with explicit substitutions, using it to relate a lambda-calculus with explicit substitutions to Boudol's resource aware lambda-calculus with multiplicities, where function arguments have a possibly limited availability.