The duality of computation

We present the μ -calculus, a syntax for λ-calculus + control operators exhibiting symmetries such as program/context and call-by-name/call-by-value. This calculus is derived from implicational Gentzen's sequent calculus LK, a key classical logical system in proof theory. Under the Curry-Howard correspondence between proofs and programs, we can see LK, or more precisely a formulation called LKμ , as a syntax-directed system of simple types for μ -calculus. For μ -calculus, choosing a call-by-name or call-by-value discipline for reduction amounts to choosing one of the two possible symmetric orientations of a critical pair. Our analysis leads us to revisit the question of what is a natural syntax for call-by-value functional computation. We define a translation of λμ-calculus into μ -calculus and two dual translations back to λ-calculus, and we recover known CPS translations by composing these translations.

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