A Faithful and Quantitative Notion of Distant Reduction for the Lambda-Calculus with Generalized Applications

We introduce a call-by-name lambda-calculus \(\lambda Jn\) with generalized applications which is equipped with distant reduction. This allows to unblock \(\beta\)-redexes without resorting to the standard permutative conversions of generalized applications used in the original \(\Lambda J\)-calculu...

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Published in:arXiv.org
Main Authors: José Espírito Santo, Kesner, Delia, Peyrot, Loïc
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 26.07.2024
Subjects:
Calculus
Normalizing
Reduction
ISSN:2331-8422
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Abstract We introduce a call-by-name lambda-calculus \(\lambda Jn\) with generalized applications which is equipped with distant reduction. This allows to unblock \(\beta\)-redexes without resorting to the standard permutative conversions of generalized applications used in the original \(\Lambda J\)-calculus with generalized applications of Joachimski and Matthes. We show strong normalization of simply-typed terms, and we then fully characterize strong normalization by means of a quantitative (i.e. non-idempotent intersection) typing system. This characterization uses a non-trivial inductive definition of strong normalization --related to others in the literature--, which is based on a weak-head normalizing strategy. We also show that our calculus \(\lambda Jn\) relates to explicit substitution calculi by means of a faithful translation, in the sense that it preserves strong normalization. Moreover, our calculus \(\lambda Jn\) and the original \(\Lambda J\)-calculus determine equivalent notions of strong normalization. As a consequence, \(\lambda J\) inherits a faithful translation into explicit substitutions, and its strong normalization can also be characterized by the quantitative typing system designed for \(\lambda Jn\), despite the fact that quantitative subject reduction fails for permutative conversions.
AbstractList We introduce a call-by-name lambda-calculus \(\lambda Jn\) with generalized applications which is equipped with distant reduction. This allows to unblock \(\beta\)-redexes without resorting to the standard permutative conversions of generalized applications used in the original \(\Lambda J\)-calculus with generalized applications of Joachimski and Matthes. We show strong normalization of simply-typed terms, and we then fully characterize strong normalization by means of a quantitative (i.e. non-idempotent intersection) typing system. This characterization uses a non-trivial inductive definition of strong normalization --related to others in the literature--, which is based on a weak-head normalizing strategy. We also show that our calculus \(\lambda Jn\) relates to explicit substitution calculi by means of a faithful translation, in the sense that it preserves strong normalization. Moreover, our calculus \(\lambda Jn\) and the original \(\Lambda J\)-calculus determine equivalent notions of strong normalization. As a consequence, \(\lambda J\) inherits a faithful translation into explicit substitutions, and its strong normalization can also be characterized by the quantitative typing system designed for \(\lambda Jn\), despite the fact that quantitative subject reduction fails for permutative conversions.
Author José Espírito Santo
Peyrot, Loïc
Kesner, Delia
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  fullname: Peyrot, Loïc
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Title A Faithful and Quantitative Notion of Distant Reduction for the Lambda-Calculus with Generalized Applications
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