Graphic lambda calculus

Graphic lambda calculus consists of a class of graphs endowed with moves between them. It might be considered a visual language in the sense of Erwig [1]. The name comes from the fact that it can be used for representing terms and reductions from untyped lambda calculus. Its main move is called the graphic beta move for its relation to the beta reduction in lambda calculus. However, the graphic beta move can be applied outside the “sector” of untyped lambda calculus, and the graphic lambda calculus can be used for other purposes than that of visually representing lambda calculus. For other visual, diagrammatic representations of lambda calculus see the VEX language [2], or Keenan’s website [3]. The motivation for introducing graphic lambda calculus comes from the study of emergent algebras. In fact, my goal is to eventually build a logic system that can be used for the formalization of certain “computations” in emergent algebras. The system can then be applied for a discrete differential calculus that exists for metric spaces with dilations, comprising Riemannian manifolds and sub-Riemannian spaces with very low regularity. Emergent algebras are a generalization of quandles; namely, an emergent algebra is a family of idempotent right quasigroups indexed by the elements of an Abelian group, while quandles are self-distributive idempotent right quasigroups. Tangle diagrams decorated by quandles or racks are a well-known tool in knot theory [4, 5]. In Kauffman [6] knot diagrams are used for representing combinatory logic, thus forming a graphical notation for untyped lambda cal-