A Simpler Undecidability Proof for System F Inhabitation

Authors Andrej Dudenhefner, Jakob Rehof

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Andrej Dudenhefner
  • Technical University of Dortmund, Dortmund, Germany
Jakob Rehof
  • Technical University of Dortmund, Dortmund, Germany


We would like to thank Paweł Urzyczyn for sharing his insights on second order propositional logic provability, which helped to develop the presented results.

Cite AsGet BibTex

Andrej Dudenhefner and Jakob Rehof. A Simpler Undecidability Proof for System F Inhabitation. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 2:1-2:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Provability in the intuitionistic second-order propositional logic (resp. inhabitation in the polymorphic lambda-calculus) was shown by Löb to be undecidable in 1976. Since the original proof is heavily condensed, Arts in collaboration with Dekkers provided a fully unfolded argument in 1992 spanning approximately fifty pages. Later in 1997, Urzyczyn developed a different, syntax oriented proof. Each of the above approaches embeds (an undecidable fragment of) first-order predicate logic into second-order propositional logic. In this work, we develop a simpler undecidability proof by reduction from solvability of Diophantine equations (is there an integer solution to P(x_1, ..., x_n) = 0 where P is a polynomial with integer coefficients?). Compared to the previous approaches, the given reduction is more accessible for formalization and more comprehensible for didactic purposes. Additionally, we formalize soundness and completeness of the reduction in the Coq proof assistant under the banner of "type theory inside type theory".

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • System F
  • Lambda Calculus
  • Inhabitation
  • Propositional Logic
  • Provability
  • Undecidability
  • Coq
  • Formalization


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