The paper presents a fresh perspective on the connection between call-by-value (CbV) evaluation in the λ-calculus and proof theory. It shows that the most basic presentation of Gentzen's sequent calculus for minimal intuitionistic logic, dubbed the "vanilla" sequent calculus, can be seen as a logical interpretation of CbV evaluation.
The key insights are:
The ineliminable cuts in the CbV reading of natural deduction correspond exactly to occurrences of the left rule (⇒l) for implication in the vanilla sequent calculus. This disentangles the overloaded role of the cut rule in CbV natural deduction.
The author introduces "vanilla λ-terms" that decorate vanilla sequent proofs, without the application construct specific to natural deduction. This avoids the need for additional concepts like stoup or separate judgments for values.
The author defines a notion of "cut elimination at a distance" for the vanilla λ-calculus, which avoids the burden of commuting conversion rules found in many CbV calculi with explicit substitutions.
The central result is that the vanilla λ-calculus and the value substitution calculus (a well-known formalism for CbV) are mutually termination-preserving simulations. This shows that the vanilla sequent calculus provides a neat logical foundation for CbV evaluation.
The paper also proves strong normalization for typed vanilla λ-terms, and discusses how the vanilla sequent calculus gives a first-class status to sub-term sharing in CbV normal forms.
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by Beniamino Ac... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2409.19722.pdfDeeper Inquiries