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

12/16/2022

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by   Mateusz Pyzik, et al.

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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 Λ_$, 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 Λ_$ 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 Λμ_v, λ_$, Λ_$ and λ all correspond equationally.