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DOI: 10.4230/LIPIcs.FSCD.2016.5

Strong Normalization for the Parameter-Free Polymorphic Lambda Calculus Based on the Omega-Rule.

Authors: Ryota Akiyoshi and Kazushige Terui

Published in: LIPIcs, Volume 52, 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)

Abstract

Following Aehlig, we consider a hierarchy F^p= { F^p_n }_{n in Nat} of parameter-free subsystems of System F, where each F^p_n corresponds to ID_n, the theory of n-times iterated inductive definitions (thus our F^p_n corresponds to the n+1th system of Aehlig). We here present two proofs of strong normalization for F^p_n, which are directly formalizable with inductive definitions. The first one, based on the Joachimski-Matthes method, can be fully formalized in ID_n+1. This provides a tight upper bound on the complexity of the normalization theorem for System F^p_n. The second one, based on the Godel-Tait method, can be locally formalized in ID_n. This provides a direct proof to the known result that the representable functions in F^p_n are provably total in ID_n. In both cases, Buchholz' Omega-rule plays a central role.

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Ryota Akiyoshi and Kazushige Terui. Strong Normalization for the Parameter-Free Polymorphic Lambda Calculus Based on the Omega-Rule.. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{akiyoshi_et_al:LIPIcs.FSCD.2016.5,
  author =	{Akiyoshi, Ryota and Terui, Kazushige},
  title =	{{Strong Normalization for the Parameter-Free Polymorphic Lambda Calculus Based on the Omega-Rule.}},
  booktitle =	{1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)},
  pages =	{5:1--5:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-010-1},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{52},
  editor =	{Kesner, Delia and Pientka, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2016.5},
  URN =		{urn:nbn:de:0030-drops-59718},
  doi =		{10.4230/LIPIcs.FSCD.2016.5},
  annote =	{Keywords: Polymorphic Lambda Calculus, Strong Normalization, Computability Predicate, Infinitary Proof Theory}
}

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Strong Normalization for the Parameter-Free Polymorphic Lambda Calculus Based on the Omega-Rule.

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