Untyped Lambda-Calculus with Input-Output

We introduce an untyped λ-calculus with input-output, based on Gordon's continuation-passing model of input-output. This calculus is intended to allow the classification of possibly infinite input-output behaviors, such as those required for servers or distributed systems. We define two terms to be operationally approximate iff they have similar behaviors in any context. We then define a notion of applicative approximation and show that it coincides with operational approximation for these new behaviors. Last, we consider the theory of pure λ-terms under this notion of operational equivalence.

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