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Revisiting Zucker’s Work on the Correspondence Between Cut-Elimination and Normalisation

  • Christian Urban
Chapter
Part of the Trends in Logic book series (TREN, volume 39)

Abstract

Zucker showed that in the fragment of intuitionistic logic whose formulae are build up from \(\wedge \), \(\supset \) and \(\forall \) only, every reduction sequence in natural deduction corresponds to a reduction sequence in the sequent calculus and vice versa. Unfortunately, the technical machinery in Zucker’s work is rather cumbersome and complicated. One contribution of this chapter is to greatly simplify his arguments. For example he defined a cut-elimination procedure modulo an equivalence relation; our cut-elimination procedure will be a simple term-rewriting system instead. Zucker also showed that the correspondence breaks down when the connectives \(\vee \) or \(\exists \) are included. We shall show that this negative result is not because cut-elimination fails to be strongly normalising for these connectives, as asserted by Zucker, rather it is because certain cut-elimination reductions do not correspond to any normalisation reduction.

Keywords

Intuitionistic logic Sequent calculus Cut-elimination Natural deduction Normalisation 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of InformaticsKing’s College LondonLondonUK

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