Llambda

Llambda is a hypothetical formal framework in theoretical computer science that extends the lambda calculus with a linear type discipline to model resource usage in computation. The name signals the combination of linearity concepts with lambda abstraction, and the framework is used to reason about how often a value is used and how memory is managed during evaluation.

Terms and types in Llambda are built from variables, lambda abstractions, and applications, similar to standard lambda calculus. The type system introduces linear function types, written A ⊸ B, where an argument of type A must be consumed exactly once, and unrestricted types, written !A, which permit duplication or erasure. A term is well-typed if its free variables are used in accordance with their types.

Evaluation follows a form of beta reduction with linear constraints: (λx. t) u reduces to t[x := u], but only when the substitution respects linear usage. Linear variables cannot be duplicated or discarded unless they are wrapped in the ! modality, which lifts a linear type to an unrestricted one. This yields a substructural form of evaluation that explicitly tracks resource usage.

Llambda sits conceptually between the standard lambda calculus and linear logic, offering a lens on how function application interacts with data usage and memory management. It serves as a theoretical tool for exploring resource-sensitive computation and has influenced teaching and research in programming language theory.

Example terms illustrate the idea: λx. x is a linear identity function, using its argument exactly once. More complex terms may require the ! modality to permit duplication, depending on the chosen presentation of the rules.