On the Metric Nature of (Differential) Logical Relations
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911476530282496 |
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| author | Lago, Ugo Dal Hoshino, Naohiko Pistone, Paolo |
| author_facet | Lago, Ugo Dal Hoshino, Naohiko Pistone, Paolo |
| contents | Differential logical relations are methods to measure distances between higher-order programs where distances between functional programs are themselves \emph{functions}, relating errors in inputs with errors in outputs. This way, differential logical relations provide a more fine-grained and contextual information of program distances. This paper aims to clarify the metric nature of differential logical relations. We introduce the notion of quasi-quasi-metrics and observe that the cartesian closed category of quasi-quasi-metric spaces reflects the construction of differential logical relations in the literature. The cartesian closed structure induces a fundamental lemma, which can be seen as a compositional reasoning principle for program distances. Furthermore, we investigate the quasi-quasi-metric spaces arising from the interpretation of types, and we prove that they satisfy variants of the strong transitivity condition and indistancy condition, as well as a weak form of the symmetry condition. In the last part of this paper, we introduce a notion of differential prelogical relations arising as a quantitative counterpart of the framework of prelogical relations. Roughly speaking, differential prelogical relations are quasi-quasi-metrics on the collection of programs. The poset of differential prelogical relations has the finest differential prelogical relation presented as a formal quantitative equational theory, while the poset lacks a coarsest differential prelogical relation. The absence of a coarsest differential prelogical relation contrasts with the situations of typed lambda calculi, where the contextual equivalences serve as the coarsest program equivalences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01317 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Metric Nature of (Differential) Logical Relations Lago, Ugo Dal Hoshino, Naohiko Pistone, Paolo Logic in Computer Science Differential logical relations are methods to measure distances between higher-order programs where distances between functional programs are themselves \emph{functions}, relating errors in inputs with errors in outputs. This way, differential logical relations provide a more fine-grained and contextual information of program distances. This paper aims to clarify the metric nature of differential logical relations. We introduce the notion of quasi-quasi-metrics and observe that the cartesian closed category of quasi-quasi-metric spaces reflects the construction of differential logical relations in the literature. The cartesian closed structure induces a fundamental lemma, which can be seen as a compositional reasoning principle for program distances. Furthermore, we investigate the quasi-quasi-metric spaces arising from the interpretation of types, and we prove that they satisfy variants of the strong transitivity condition and indistancy condition, as well as a weak form of the symmetry condition. In the last part of this paper, we introduce a notion of differential prelogical relations arising as a quantitative counterpart of the framework of prelogical relations. Roughly speaking, differential prelogical relations are quasi-quasi-metrics on the collection of programs. The poset of differential prelogical relations has the finest differential prelogical relation presented as a formal quantitative equational theory, while the poset lacks a coarsest differential prelogical relation. The absence of a coarsest differential prelogical relation contrasts with the situations of typed lambda calculi, where the contextual equivalences serve as the coarsest program equivalences. |
| title | On the Metric Nature of (Differential) Logical Relations |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2603.01317 |