Decidable structures between Church-style and Curry-style

Authors Ken-etsu Fujita, Aleksy Schubert

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Ken-etsu Fujita
Aleksy Schubert

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Ken-etsu Fujita and Aleksy Schubert. Decidable structures between Church-style and Curry-style. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 190-205, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


It is well-known that the type-checking and type-inference problems are undecidable for second order lambda-calculus in Curry-style, although those for Church-style are decidable. What causes the differences in decidability and undecidability on the problems? We examine crucial conditions on terms for the (un)decidability property from the viewpoint of partially typed terms, and what kinds of type annotations are essential for (un)decidability of type-related problems. It is revealed that there exists an intermediate structure of second order lambda-terms, called a style of hole-application, between Church-style and Curry-style, such that the type-related problems are decidable under the structure. We also extend this idea to the omega-order polymorphic calculus F-omega, and show that the type-checking and type-inference problems then become undecidable.
  • 2nd-order lambda-calculus
  • type-checking
  • type-inference
  • Church-style and Curry-style


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