The Flower Calculus
Pablo Donato
TL;DR
Addressing the need for a graphical, intuitionistic-first-order logic proof system, this work introduces the flower calculus, a nested-flowers representation inspired by Peirce's existential graphs. The authors prove soundness of the full calculus and completeness of an analytic fragment with respect to Kripke semantics, adapting cut-elimination techniques to a deep-inference setting. They show the invertibility of a kernel of rules, which benefits automated and interactive proof search, and implement a Coq formalization showing a bidirectional simulation with cut-free sequent calculus alongside a prototype Flower Prover with a public online version.
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.