https://hal.inria.fr/hal-02390267/document

Introduction to Deep Inference

We define and study a term calculus implementing higher-order node replication. It is used to specify two different (weak) evaluation strategies: call-by-name and fully lazy call-by-need, that are shown to be observationally equivalent by using type theoretical technical tools.

/pdf/the-spirit-of-node-replication-25wi68kfjm.pdf

The Spirit of Node Replication

We present the spinal atomic \(\lambda \)-calculus, a typed \(\lambda \)-calculus with explicit sharing and atomic duplication that achieves spinal full laziness: duplicating only the direct paths between a binder and bound variables is enough for beta reduction to proceed. We show this calculus is the result of a Curry–Howard style interpretation of a deep-inference proof system, and prove that it has natural properties with respect to the \(\lambda \)-calculus: confluence and preservation of strong normalisation.

https://link.springer.com/content/pdf/10.1007%2F978-3-030-45231-5_30.pdf

Spinal Atomic Lambda-Calculus

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David Sherratt

Papers

A lambda-calculus that achieves full laziness with spine duplication