Approaches to Polymorphism in Classical Sequent Calculus

  • Alexander J. Summers
  • Steffen van Bakel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3924)


\(\mathcal X\) is a relatively new calculus, invented to give a Curry-Howard correspondence with Classical Implicative Sequent Calculus. It is already known to provide a very expressive language; embeddings have been defined of the λ-calculus, Bloo and Rose’s λ x, Parigot’s λμ and Curien and Herbelin’s \({\overline{\lambda}\mu\tilde{\mu}}\). We investigate various notions of polymorphism in the context of the \(\mathcal X\)-calculus. In particular, we examine the first class polymorphism of System F, and the shallow polymorphism of ML. We define analogous systems based on the \(\mathcal X\)-calculus, and show that these are suitable for embedding the original calculi.

In the case of shallow polymorphism we obtain a more general calculus than ML, while retaining its useful properties. A type-assignment algorithm is defined for this system, which generalises Milner’s \({\cal W}\).


Classical Logic Atomic Type Reduction Rule Natural Deduction Sequent Calculus 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexander J. Summers
    • 1
  • Steffen van Bakel
    • 1
  1. 1.Department of ComputingImperial College LondonLondonU.K.

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