lambdai

Lambdai is a fictional formalism in the field of theoretical computer science, conceived as a variant of the lambda calculus that uses explicit indexing on variables to manage scope and resource usage. The term combines "lambda" with "i" to indicate indexed variables. In Lambdai, expressions are built from variables x_i, abstractions λx_i.M, and applications M N, where i is an index indicating the binding level or resource annotation. The indices support a canonical representation of binding and help avoid alpha-conversion because substitution can be performed with explicit indices.

Evaluation follows a beta-like rule, adapted to index-aware substitution, where the bound variable’s index is propagated to the substituted term. In typing variants, Lambdai supports simple types and linear types, the latter enforcing that terms are used exactly once, which makes Lambdai suitable for modeling resource-sensitive computations.

Lambdai is closely related to de Bruijn index representations and to the simply typed lambda calculus. It can be translated to and from standard lambda terms with appropriate index handling, and it provides a readable alternative when explicit binding information is desirable. Example terms include λx_1. x_1 for the identity function with a fixed index, or (λx_1. M) N_2 where the indices reflect binding relationships.

Applications and usage are primarily academic, serving as a teaching tool for illustrating indexed binding and resource-aware computation. See also lambda calculus, de Bruijn indices, linear lambda calculus, and typed lambda calculus.