Implicit Automata in Typed λ-Calculi I: Aperiodicity in a Non-Commutative Logic

Authors Lê Thành Dũng Nguyễn , Pierre Pradic

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Lê Thành Dũng Nguyễn
  • Laboratoire d'informatique de Paris Nord, Villetaneuse, France
  • Laboratoire Cogitamus (
Pierre Pradic
  • Department of Computer Science, University of Oxford, UK


The initial trigger for this line of work was twofold: the serendipitous rediscovery of Hillebrand and Kanellakis’s theorem by Damiano Mazza, Thomas Seiller and the first author, and Mikołaj Bojańczyk’s suggestion to the second author to look into connections between transducers and linear logic. Célia Borlido proposed looking at star-free languages at the Topology, Algebra and Categories in Logic 2019 summer school. During its long period of gestation, this work benefited from discussions with many other people: Pierre Clairambault, Amina Doumane, Marie Fortin, Jérémy Ledent, Paolo Pistone, Lorenzo Tortora de Falco, Noam Zeilberger and others that we may have forgotten to mention. We also thank the anonymous reviewers for their constructive feedback, especially for their highly relevant pointers to the literature.

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Lê Thành Dũng Nguyễn and Pierre Pradic. Implicit Automata in Typed λ-Calculi I: Aperiodicity in a Non-Commutative Logic. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 135:1-135:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We give a characterization of star-free languages in a λ-calculus with support for non-commutative affine types (in the sense of linear logic), via the algebraic characterization of the former using aperiodic monoids. When the type system is made commutative, we show that we get regular languages instead. A key ingredient in our approach – that it shares with higher-order model checking – is the use of Church encodings for inputs and outputs. Our result is, to our knowledge, the first use of non-commutativity in implicit computational complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic language theory
  • Theory of computation → Linear logic
  • Church encodings
  • ordered linear types
  • star-free languages


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