Scroll nets

TL;DR

Scroll nets are a new graph-based formalism inspired by Peirce's topological notation, enabling representation and normalization of proofs in propositional logic with potential applications in logic and computation.

Contribution

The paper introduces scroll nets, a novel formalism derived from existential graphs, with a graph-theoretic definition and a detour-elimination procedure, bridging logic and lambda calculus.

Findings

Scroll nets can simulate normalization in simply typed lambda calculus.

They provide a diagrammatic and combinatorial framework for propositional proofs.

A detour-elimination procedure analogous to cut-elimination is developed.

Abstract

We introduce a new formalism for representing proofs in propositional logic called "scroll nets". Its fundamental construct is the "scroll", a topological notation for implication proposed by C. S. Peirce at the end of the 19th century as the basis for his diagrammatic system of existential graphs (EGs). Scroll nets are derived from EGs by following the Curry-Howard methodology of internalizing inference rules inside judgments, just as terms in type theory internalize natural deduction rules. We focus on the intuitionistic implicative fragment of EGs, starting from a natural diagrammatic representation of scroll nets, and then distilling their combinatorial essence into a purely graph-theoretic definition. We also identify a notion of detour, that we use to sketch a detour-elimination procedure akin to cut-elimination. We illustrate how to simulate normalization in the simply typed $\lambda$-calculus, demonstrating both the logical and computational expressivity of our framework.