Normalisation Control in Deep Inference via Atomic Flows

TL;DR

This paper introduces atomic flows, a geometric graph-based formalism that simplifies the analysis of normalization in deep inference, leading to a general normalization theorem encompassing cut elimination.

Contribution

The paper presents atomic flows as a syntax-independent, geometric tool for normalisation in deep inference, advancing the theoretical understanding of proof normalization.

Findings

Proves a new general normalization theorem for propositional logic

Atomic flows are independent of logical inference rules

Provides a more intuitive geometric formalism for normalization

Abstract

We introduce `atomic flows': they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax.