A theory of linear typings as flows on 3-valent graphs

TL;DR

This paper develops a novel analogy between lambda calculus typings and graph flows, using embedded 3-valent graphs and algebraic structures, connecting logic, topology, and graph theory.

Contribution

It introduces a new flow-based framework for linear lambda typings on embedded graphs, linking them to classical flow theory and proof-net structures.

Findings

Characterization of when local flows lift to global flows in lambda terms

Development of a flow rewriting theory with topological interpretations

Introduction of polarized flows connecting to proof-nets and bidirectional typing

Abstract

Building on recently established enumerative connections between lambda calculus and the theory of embedded graphs (or "maps"), this paper develops an analogy between typing (of lambda terms) and coloring (of maps). Our starting point is the classical notion of an abelian group-valued "flow" on an abstract graph (Tutte, 1954). Typing a linear lambda term may be naturally seen as constructing a flow (on an embedded 3-valent graph with boundary) valued in a more general algebraic structure consisting of a preordered set equipped with an "implication" operation and unit satisfying composition, identity, and unit laws. Interesting questions and results from the theory of flows (such as the existence of nowhere-zero flows) may then be re-examined from the standpoint of lambda calculus and logic. For example, we give a characterization of when the local flow relations (across vertices) may be categorically lifted to a global flow relation (across the boundary), proving that this holds just in case the underlying map has the orientation of a lambda term. We also develop a basic theory of rewriting of flows that suggests topological meanings for classical completeness results in combinatory logic, and introduce a polarized notion of flow, which draws connections to the theory of proof-nets in linear logic and to bidirectional typing.