The Flower Calculus

Author Pablo Donato



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Pablo Donato
  • LIX, École Polytechnique, Palaiseau, France

Acknowledgements

I want to thank Luc Chabassier for writing the Lua script that was used to generate all the flower drawings in this document, and Benjamin Werner for useful feedback on a first draft of this paper. Lastly, I thank Tito for teaching me some invaluable formatting tricks.

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Pablo Donato. The Flower Calculus. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FSCD.2024.5

Abstract

We introduce the flower calculus, a deep inference proof system for intuitionistic first-order logic inspired by Peirce’s existential graphs. It works as a rewriting system over inductive objects called "flowers", that enjoy both a graphical interpretation as topological diagrams, and a textual presentation as nested sequents akin to coherent formulas. Importantly, the calculus dispenses completely with the traditional notion of symbolic connective, operating solely on nested flowers containing atomic predicates. We prove both the soundness of the full calculus and the completeness of an analytic fragment with respect to Kripke semantics. This provides to our knowledge the first analyticity result for a proof system based on existential graphs, adapting semantic cut-elimination techniques to a deep inference setting. Furthermore, the kernel of rules targetted by completeness is fully invertible, a desirable property for both automated and interactive proof search.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
  • Theory of computation → Constructive mathematics
Keywords
  • deep inference
  • graphical calculi
  • existential graphs
  • intuitionistic logic
  • Kripke semantics
  • cut-elimination

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