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- W4301758049 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." @default.
- W4301758049 created "2022-10-05" @default.
- W4301758049 creator A5008964203 @default.
- W4301758049 date "2018-04-27" @default.
- W4301758049 modified "2023-10-01" @default.
- W4301758049 title "A theory of linear typings as flows on 3-valent graphs" @default.
- W4301758049 doi "https://doi.org/10.48550/arxiv.1804.10540" @default.
- W4301758049 hasPublicationYear "2018" @default.
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