Adjoint Natural Deduction

Authors Junyoung Jang , Sophia Roshal , Frank Pfenning , Brigitte Pientka

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Junyoung Jang
  • McGill University, Montreal, Canada
Sophia Roshal
  • Carnegie Mellon University, Pittsburgh, USA
Frank Pfenning
  • Carnegie Mellon University, Pittsburgh, USA
Brigitte Pientka
  • McGill University, Montreal, Canada

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Junyoung Jang, Sophia Roshal, Frank Pfenning, and Brigitte Pientka. Adjoint Natural Deduction. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Adjoint logic is a general approach to combining multiple logics with different structural properties, including linear, affine, strict, and (ordinary) intuitionistic logics, where each proposition has an intrinsic mode of truth. It has been defined in the form of a sequent calculus because the central concept of independence is most clearly understood in this form, and because it permits a proof of cut elimination following standard techniques. In this paper we present a natural deduction formulation of adjoint logic and show how it is related to the sequent calculus. As a consequence, every provable proposition has a verification (sometimes called a long normal form). We also give a computational interpretation of adjoint logic in the form of a functional language and prove properties of computations that derive from the structure of modes, including freedom from garbage (for modes without weakening and contraction), strictness (for modes disallowing weakening), and erasure (based on a preorder between modes). Finally, we present a surprisingly subtle algorithm for type checking.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Linear logic
  • Substructural Logic
  • Type Systems
  • Functional Programming


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