Emergent rewrites in knot theory and logic
[video]     [js slides]
Marius Buliga (IMAR)
I explain in what sense new graph rewrite systems emerge from given ones, with two examples:

From sub-riemannian geometry to emergent algebras
A riemannian manifold (X,g) is a length metric space (X,d) by Hopf-Rinow thm.
Problem 1: recover (X,g) from (X,d).
but (1996, M. Gromov) asks for a solution of
Problem 2: recover sub-riemannian (X,D,g) from (X,d).
Sub-riemannian spaces are weird! (except when riemannian)
Sub-riemannian spaces (techniques) are useful!
Drawing conventions
Emergent algebras
Used in: A characterization of sub-riemannian spaces as length dilatation structures constructed via coherent projections a solution of the problem of intrinsic characterization of sub-riemannian manifolds posed by M. Gromov, 1996.
introduced as algebras in arXiv:0907.1520, as a λ calculus in arXiv:1807.02058.
Drawing conventions 2
Tangles
Chora
Conical groups
groups with:


an action by automorphisms


a ⋅ (x * y) = (a ⋅ x) * (a ⋅ y)


which are uniformly contractive:


ε → 0, ε ⋅ x → e, uniformly wrt x in compact set
Examples
Tangles
Rescale!
Commutative emergent algebras
The operation * is commutative iff any of the following:
Let's denote the 3 ports of a dilation node as:

port 1: "from", port 2: "see", port 3: "as"



The operation * is commutative iff all the 6 permutations of ports are also dilations, with coefficients from the anharmonic group: Pure See!
Lambda calculus
(1936, A. Church) Untyped λ calculus is a term rewrite system
Terms:
Term rewrite rule:
 
(1936, A. Church) Pure λ calculus is a term rewrite system
Term rewrite rule:
(1971, C.P. Wadsworth, 1990, J. Lamping) graph rewrite system
Emergent lambda
Aplication can be seen as:



(all the 6 permutations of ports are also dilations, with coefficients from the anharmonic group: Pure See!)
Abstraction can be seen as:



(all the 6 permutations of ports are also dilations, with coefficients from the anharmonic group: Pure See!)
β rewrite from shuffle
only with dilation nodes:

the same with chemlambda v2 nodes:

THANK YOU!

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