The Lambda Calculus is Quantifiable
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909394599411712 |
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| author | Maestracci, Valentin Pistone, Paolo |
| author_facet | Maestracci, Valentin Pistone, Paolo |
| contents | In this paper we introduce several quantitative methods for the lambda-calculus based on partial metrics, a well-studied variant of standard metric spaces that have been used to metrize non-Hausdorff topologies, like those arising from Scott domains. First, we study quantitative variants, based on program distances, of sensible equational theories for the $λ$-calculus, like those arising from Böhm trees and from the contextual preorder. Then, we introduce applicative distances capturing higher-order Scott topologies, including reflexive objects like the $D_\infty$ model. Finally, we provide a quantitative insight on the well-known connection between the Böhm tree of a $λ$-term and its Taylor expansion, by showing that the latter can be presented as an isometric transformation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11809 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Lambda Calculus is Quantifiable Maestracci, Valentin Pistone, Paolo Logic in Computer Science Programming Languages Logic F.4.1; F.3.2 In this paper we introduce several quantitative methods for the lambda-calculus based on partial metrics, a well-studied variant of standard metric spaces that have been used to metrize non-Hausdorff topologies, like those arising from Scott domains. First, we study quantitative variants, based on program distances, of sensible equational theories for the $λ$-calculus, like those arising from Böhm trees and from the contextual preorder. Then, we introduce applicative distances capturing higher-order Scott topologies, including reflexive objects like the $D_\infty$ model. Finally, we provide a quantitative insight on the well-known connection between the Böhm tree of a $λ$-term and its Taylor expansion, by showing that the latter can be presented as an isometric transformation. |
| title | The Lambda Calculus is Quantifiable |
| topic | Logic in Computer Science Programming Languages Logic F.4.1; F.3.2 |
| url | https://arxiv.org/abs/2411.11809 |