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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

## Authors

## Colleges, School and Institutes

## 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.

## Details

Original language | English |
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Title of host publication | LICS '18 Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science |

Publication status | Published - 9 Jul 2018 |

Event | Thirty-Third Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) - Oxford, United Kingdom Duration: 9 Jul 2018 → 12 Jul 2018 |

### Conference

Conference | Thirty-Third Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) |
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Country | United Kingdom |

City | Oxford |

Period | 9/07/18 → 12/07/18 |

## Keywords

- lambda calculus and combinatory logic, graph theory, nowhere-zero flows, linear logic, skew-closed categories