Profinite lambda-terms and parametricity
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929241370656768 |
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| author | van Gool, Sam Melliès, Paul-André Moreau, Vincent |
| author_facet | van Gool, Sam Melliès, Paul-André Moreau, Vincent |
| contents | Combining ideas coming from Stone duality and Reynolds parametricity, we formulate in a clean and principled way a notion of profinite lambda-term which, we show, generalizes at every type the traditional notion of profinite word coming from automata theory. We start by defining the Stone space of profinite lambda-terms as a projective limit of finite sets of usual lambda-terms, considered modulo a notion of equivalence based on the finite standard model. One main contribution of the paper is to establish that, somewhat surprisingly, the resulting notion of profinite lambda-term coming from Stone duality lives in perfect harmony with the principles of Reynolds parametricity. In addition, we show that the notion of profinite lambda-term is compositional by constructing a cartesian closed category of profinite lambda-terms, and we establish that the embedding from lambda-terms modulo beta-eta-conversion to profinite lambda-terms is faithful using Statman's finite completeness theorem. Finally, we prove that the traditional Church encoding of finite words into lambda-terms can be extended to profinite words, and leads to a homeomorphism between the space of profinite words and the space of profinite lambda-terms of the corresponding Church type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12475 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Profinite lambda-terms and parametricity van Gool, Sam Melliès, Paul-André Moreau, Vincent Logic in Computer Science Combining ideas coming from Stone duality and Reynolds parametricity, we formulate in a clean and principled way a notion of profinite lambda-term which, we show, generalizes at every type the traditional notion of profinite word coming from automata theory. We start by defining the Stone space of profinite lambda-terms as a projective limit of finite sets of usual lambda-terms, considered modulo a notion of equivalence based on the finite standard model. One main contribution of the paper is to establish that, somewhat surprisingly, the resulting notion of profinite lambda-term coming from Stone duality lives in perfect harmony with the principles of Reynolds parametricity. In addition, we show that the notion of profinite lambda-term is compositional by constructing a cartesian closed category of profinite lambda-terms, and we establish that the embedding from lambda-terms modulo beta-eta-conversion to profinite lambda-terms is faithful using Statman's finite completeness theorem. Finally, we prove that the traditional Church encoding of finite words into lambda-terms can be extended to profinite words, and leads to a homeomorphism between the space of profinite words and the space of profinite lambda-terms of the corresponding Church type. |
| title | Profinite lambda-terms and parametricity |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2301.12475 |