Type Inference of Linear Lambda Calculus with First-Class Continuations

Previous work proposed the linear lambda calculus with non-linear first-class continuations. In the linear lambda calculus with non-linear first-class continuations, a value is handled linearly, meaning it can be used once and only once; however, a continuation can be duplicated many times. From the viewpoint of the CurryHoward isomorphism, the calculus is considered as dual of the usual lambda calculus without first-class continuation. And this paper proposes an inference algorithm and a system for the type inference. The linear lambda calculus with non-linear first-class continuations is more limited than the linear lambda calculus without first-class continuations. Therefore, the paper manages both of the two calculis using the type inference. It introduces a modal variable in the types to substitute the linear modality. During type inference, modal variables are instantiated.