A Gentzen-Style Monadic Translation of Gödel’s System T

Author Chuangjie Xu

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Chuangjie Xu
  • Ludwig-Maximilians-Universität München, Germany


The author is grateful to Martín Escardó, Hajime Ishihara, Nils Köpp, Paulo Oliva, Thomas Powell, Benno van den Berg, Helmut Schwichtenberg for many fruitful discussions and the anonymous reviewers for their valuable suggestions and comments.

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Chuangjie Xu. A Gentzen-Style Monadic Translation of Gödel’s System T. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We introduce a syntactic translation of Gödel’s System 𝖳 parametrized by a weak notion of a monad, and prove a corresponding fundamental theorem of logical relation. Our translation structurally corresponds to Gentzen’s negative translation of classical logic. By instantiating the monad and the logical relation, we reveal the well-known properties and structures of 𝖳-definable functionals including majorizability, continuity and bar recursion. Our development has been formalized in the Agda proof assistant.

Subject Classification

ACM Subject Classification
  • Theory of computation → Program analysis
  • Theory of computation → Type theory
  • Theory of computation → Proof theory
  • monadic translation
  • Gödel’s System T
  • logical relation
  • negative translation
  • majorizability
  • continuity
  • bar recursion
  • Agda


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