An interpretation of system F through bar recursion

• Published in: 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
• Main Author: Blot, Valentin
• Other Authors: Department of Computer Science (Queen Mary University of London), Queen Mary University of London (QMUL)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2017
• Subjects: [INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO]; Iceland
• Online Access: https://hal.archives-ouvertes.fr/hal-01766883; https://hal.archives-ouvertes.fr/hal-01766883/document; https://hal.archives-ouvertes.fr/hal-01766883/file/SystFbarrec.pdf; https://doi.org/10.1109/LICS.2017.8005066

International audience There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from these two interpretations have a fundamentally different computational behavior and their relationship is not well understood. We make a step towards a comparison by defining the first translation of system F into a simply-typed total language with a variant of bar recursion. This translation relies on a realizability interpretation of second-order arithmetic. Due to Gödel's incompleteness theorem there is no proof of termination of system F within second-order arithmetic. However, for each individual term of system F there is a proof in second-order arithmetic that it terminates, with its realizability interpretation providing a bound on the number of reduction steps to reach a normal form. Using this bound, we compute the normal form through primitive recursion. Moreover, since the normalization proof of system F proceeds by induction on typing derivations, the translation is compositional. The flexibility of our method opens the possibility of getting a more direct translation that will provide an alternative approach to the study of polymorphism, namely through bar recursion.