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Curry-Howard for Sequent Calculus at Last!

Author José Espírito Santo

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Keywords
  • co-control
  • co-continuation
  • vector notation
  • let-expression
  • formal sub- stitution
  • context substitution
  • computational lambda-calculus
  • classical lo

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Abstract

This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an equivalent, "external" interpretation, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the formal duality between control and co-control is studied, in the context of classical logic. The duality cannot be observed in the sequent calculus, but rather in a system unifying sequent calculus and natural deduction.

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José Espírito Santo. Curry-Howard for Sequent Calculus at Last!. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 165-179, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.TLCA.2015.165

Author Details

José Espírito Santo

References

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