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 $\Lambda J$-calculus with...
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| Published in: | Logical methods in computer science Vol. 20, Issue 3 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V
01.01.2024
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| ISSN: | 1860-5974, 1860-5974 |
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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. 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 | Peyrot, Loïc Kesner, Delia Santo, José Espírito |
| Author_xml | – sequence: 1 givenname: José Espírito surname: Santo fullname: Santo, José Espírito – sequence: 2 givenname: Delia surname: Kesner fullname: Kesner, Delia – sequence: 3 givenname: Loïc surname: Peyrot fullname: Peyrot, Loïc |
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applications which is equipped with distant reduction. This allows to unblock... We introduce a call-by-name lambda-calculus $\lambda Jn$ with generalized applications which is equipped with distant reduction. This allows to unblock... |
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| Title | A Faithful and Quantitative Notion of Distant Reduction for the Lambda-Calculus with Generalized Applications |
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