Why Are Proofs Relevant in Proof-Relevant Models?
Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that...
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| Vydáno v: | Proceedings of ACM on programming languages Ročník 7; číslo POPL; s. 218 - 248 |
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09.01.2023
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| Abstract | Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that they can be presented as type systems as well. We prove that all the models living in this class satisfy an Approximation Theorem stating that the interpretation of a program corresponds to the filtered colimit of the denotations of its approximants. As in the relational case, the quantitative nature of our models allows to prove this property via a simple induction, rather than using impredicative techniques. Unlike relational models, our 2-dimensional graph models are also proof-relevant in the sense that the interpretation of a λ-term does not contain only its typings, but the whole type derivations. The additional information carried by a type derivation permits to reconstruct an approximant having the same type in the same environment. From this, we obtain the characterization of the theory induced by the categorified graph models as a simple corollary of the Approximation Theorem: two λ-terms have isomorphic interpretations exactly when their B'ohm trees coincide. |
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| AbstractList | Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that they can be presented as type systems as well. We prove that all the models living in this class satisfy an Approximation Theorem stating that the interpretation of a program corresponds to the filtered colimit of the denotations of its approximants. As in the relational case, the quantitative nature of our models allows to prove this property via a simple induction, rather than using impredicative techniques. Unlike relational models, our 2-dimensional graph models are also proof-relevant in the sense that the interpretation of a λ-term does not contain only its typings, but the whole type derivations. The additional information carried by a type derivation permits to reconstruct an approximant having the same type in the same environment. From this, we obtain the characterization of the theory induced by the categorified graph models as a simple corollary of the Approximation Theorem: two λ-terms have isomorphic interpretations exactly when their B'ohm trees coincide. |
| ArticleNumber | 8 |
| Author | Kerinec, Axel Manzonetto, Giulio Olimpieri, Federico |
| Author_xml | – sequence: 1 givenname: Axel orcidid: 0000-0003-0920-8847 surname: Kerinec fullname: Kerinec, Axel email: kerinec@lipn.univ-paris13.fr organization: Université Sorbonne Paris Nord, France / LIPN, France / CNRS, France – sequence: 2 givenname: Giulio orcidid: 0000-0003-1448-9014 surname: Manzonetto fullname: Manzonetto, Giulio email: manzonetto@univ-paris13.fr organization: Université Sorbonne Paris Nord, France / LIPN, France / CNRS, France – sequence: 3 givenname: Federico orcidid: 0000-0003-1485-5360 surname: Olimpieri fullname: Olimpieri, Federico email: f.olimpieri@leeds.ac.uk organization: University of Leeds, UK |
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(e_1_2_1_4_1) 1977; 1132 Saville Philip (e_1_2_1_58_1) 2007 Ehrhard Thomas (e_1_2_1_20_1) Jacobs Bart (e_1_2_1_40_1) 1997; 62 e_1_2_1_41_1 e_1_2_1_24_1 e_1_2_1_62_1 e_1_2_1_64_1 e_1_2_1_28_1 e_1_2_1_49_1 e_1_2_1_26_1 e_1_2_1_47_1 Kelly Max (e_1_2_1_45_1) 2005 Roberto (e_1_2_1_2_1) 1998 de Carvalho Daniel (e_1_2_1_17_1) 2009 e_1_2_1_31_1 e_1_2_1_54_1 e_1_2_1_8_1 e_1_2_1_56_1 e_1_2_1_50_1 e_1_2_1_10_1 e_1_2_1_33_1 e_1_2_1_52_1 e_1_2_1_16_1 e_1_2_1_39_1 Seely Robert A. G. (e_1_2_1_60_1) 1987 e_1_2_1_14_1 e_1_2_1_37_1 e_1_2_1_18_1 Karazeris Panagis (e_1_2_1_43_1) 2001 e_1_2_1_65_1 Bucciarelli Antonio (e_1_2_1_12_1) 2014; 354 e_1_2_1_23_1 e_1_2_1_46_1 e_1_2_1_61_1 e_1_2_1_21_1 e_1_2_1_44_1 e_1_2_1_63_1 e_1_2_1_27_1 e_1_2_1_25_1 e_1_2_1_48_1 e_1_2_1_29_1 Hyland J. Martin E. (e_1_2_1_35_1) 1976 Ehrhard Thomas (e_1_2_1_22_1) 2006; 197 Joyal André (e_1_2_1_42_1) e_1_2_1_30_1 e_1_2_1_55_1 e_1_2_1_5_1 Benabou J. (e_1_2_1_7_1) Barendregt Hendrik Pieter (e_1_2_1_6_1) e_1_2_1_57_1 e_1_2_1_3_1 e_1_2_1_13_1 e_1_2_1_34_1 e_1_2_1_51_1 e_1_2_1_1_1 e_1_2_1_11_1 e_1_2_1_32_1 e_1_2_1_53_1 e_1_2_1_38_1 e_1_2_1_15_1 e_1_2_1_36_1 e_1_2_1_59_1 e_1_2_1_9_1 e_1_2_1_19_1 |
| References_xml | – reference: Thomas Ehrhard and Laurent Regnier. 2006. Böhm Trees, Krivine’s Machine and the Taylor Expansion of Lambda-Terms. In CiE (Lecture Notes in Computer Science, Vol. 3988). Springer, 186–197. – reference: Giulio Manzonetto and Domenico Ruoppolo. 2014. Relational Graph Models, Taylor Expansion and Extensionality. In Proceedings of the 30th Conference on the Mathematical Foundations of Programming Semantics, MFPS 2014, Ithaca, NY, USA, June 12-15, 2014, Bart Jacobs, Alexandra Silva, and Sam Staton (Eds.) (Electronic Notes in Theoretical Computer Science, Vol. 308). Elsevier, 245–272. https://doi.org/10.1016/j.entcs.2014.10.014 10.1016/j.entcs.2014.10.014 – reference: G. Maxwell Kelly. 1980. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. 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