Principal Types as Lambda Nets

Authors Pietro Di Gianantonio , Marina Lenisa

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Pietro Di Gianantonio
  • University of Udine, Italy
Marina Lenisa
  • University of Udine, Italy


We thank Beniamino Accattoli, Paolo Coppola, Furio Honsell, Ivan Scagnetto, and Gabriele Vanoni for helpful discussions on the subject. We are indebted to the anonymous referees for their valuable comments on the present work.

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Pietro Di Gianantonio and Marina Lenisa. Principal Types as Lambda Nets. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We show that there are connections between principal type schemata, cut-free λ-nets, and normal forms of the λ-calculus, and hence there are correspondences between the normalisation algorithms of the above structures, i.e. unification of principal types, cut-elimination of λ-nets, and normalisation of λ-terms. Once the above correspondences have been established, properties of the typing system, such as typability, subject reduction, and inhabitation, can be derived from properties of λ-nets, and vice-versa. We illustrate the above pattern on a specific type assignment system, we study principal types for this system, and we show that they correspond to λ-nets with a non-standard notion of cut-elimination. Properties of the type system are then derived from results on λ-nets.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lambda calculus
  • Theory of computation → Type structures
  • Theory of computation → Linear logic
  • Lambda calculus
  • Principal types
  • Linear logic
  • Lambda nets
  • Normalization
  • Cut elimination


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