Call-By-Name Is Just Call-By-Value with Delimited Control

Delimited control operator shift0 exhibits versatile capabilities: it can express layered monadic effects, or equivalently, algebraic effects. Little did we know it can express lambda calculus too! We present \Lambda_\ $, a call-by-value lambda calculus extended with shift0 and control delimiter$ $ $with carefully crafted reduction theory, such that the lambda calculus with beta and eta reductions can be isomorphically embedded into$ \Lambda_$ $via a right inverse of a continuation-passing style translation. While call-by-name reductions of lambda calculus can trivially simulate its call-by-value version, we show that addition of shift0 and$ $ $is the golden mean of expressive power that suffices to simulate beta and eta reductions while still admitting a simulation back. As a corollary, calculi$ \Lambda\mu_v $,$ \lambda_$ $,$ \Lambda_$ $and$ \lambda $ all correspond equationally.