A theory of linear typings as flows on 3-valent graphs
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 &quo...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
27.04.2018
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| Subjects | |
| Online Access | Get full text |
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| Summary: | 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. |
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| DOI: | 10.48550/arxiv.1804.10540 |