Normalization Without Syntax

Authors Willem B. Heijltjes, Dominic J. D. Hughes, Lutz Straßburger

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Willem B. Heijltjes
  • Department of Computer Science, University of Bath, UK
Dominic J. D. Hughes
  • Logic Group, University of California Berkeley, CA, USA
Lutz Straßburger
  • Equipe Partout, Inria Saclay, Palaiseau, France


Dominic Hughes would like to thank Wes Holliday, his host at U.C. Berkeley. We thank the referees for their diligence.

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Willem B. Heijltjes, Dominic J. D. Hughes, and Lutz Straßburger. Normalization Without Syntax. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We present normalization for intuitionistic combinatorial proofs (ICPs) and relate it to the simply-typed lambda-calculus. We prove confluence and strong normalization. Combinatorial proofs, or "proofs without syntax", form a graphical semantics of proof in various logics that is canonical yet complexity-aware: they are a polynomial-sized representation of sequent proofs that factors out exactly the non-duplicating permutations. Our approach to normalization aligns with these characteristics: it is canonical (free of permutations) and generic (readily applied to other logics). Our reduction mechanism is a canonical representation of reduction in sequent calculus with closed cuts (no abstraction is allowed below a cut), and relates to closed reduction in lambda-calculus and supercombinators. While we will use ICPs concretely, the notion of reduction is completely abstract, and can be specialized to give a reduction mechanism for any representation of typed normal forms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Lambda calculus
  • combinatorial proofs
  • intuitionistic logic
  • lambda-calculus
  • Curry-Howard
  • proof nets


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