Syntactically and Semantically Regular Languages of λ-Terms Coincide Through Logical Relations

Authors Vincent Moreau , Lê Thành Dũng (Tito) Nguyễn

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Author Details

Vincent Moreau
  • IRIF & Université Paris Cité & Inria Paris, France
Lê Thành Dũng (Tito) Nguyễn
  • Laboratoire de l'informatique du parallélisme (LIP), École normale supérieure de Lyon, France


We would like to thank Amina Doumane, Sam van Gool, Paul-André Melliès and Sylvain Salvati for in-depth discussions that significantly helped us refine our ideas. We are also grateful to Sam and Paul-André for proof-reading drafts of this paper, and to the ReFL discussion group for hosting a reading group on logical relations and normalization by evaluation. The first author would like to thank the Felicissimo family for their support during the writing process of this article.

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Vincent Moreau and Lê Thành Dũng (Tito) Nguyễn. Syntactically and Semantically Regular Languages of λ-Terms Coincide Through Logical Relations. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 40:1-40:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


A fundamental theme in automata theory is regular languages of words and trees, and their many equivalent definitions. Salvati has proposed a generalization to regular languages of simply typed λ-terms, defined using denotational semantics in finite sets. We provide here some evidence for its robustness. First, we give an equivalent syntactic characterization that naturally extends the seminal work of Hillebrand and Kanellakis connecting regular languages of words and syntactic λ-definability. Second, we show that any finitary extensional model of the simply typed λ-calculus, when used in Salvati’s definition, recognizes exactly the same class of languages of λ-terms as the category of finite sets does. The proofs of these two results rely on logical relations and can be seen as instances of a more general construction of a categorical nature, inspired by previous categorical accounts of logical relations using the gluing construction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Denotational semantics
  • Theory of computation → Regular languages
  • regular languages
  • simple types
  • denotational semantics
  • logical relations


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