Computing with space: a tangle formalism for chora and difference

@article{Buliga2011ComputingWS,
  title={Computing with space: a tangle formalism for chora and difference},
  author={Marius Buliga},
  journal={ArXiv},
  year={2011},
  volume={abs/1103.6007}
}
What is space computing,simulation, or understanding? Converging from several sources, this seems to be something more primitive than what is meant nowadays by computation, something that was along with us since antiquity (the word "choros", "chora", denotes "space" or "place" and is seemingly the most mysterious notion from Plato, described in Timaeus 48e - 53c) which has to do with cybernetics and with the understanding of the front end visual system. It may have some unexpected applications… 

Tales Told by Coloured Tangles

An expository account of tangle machines motivated by the problem of describing `covariance intersection' fusion of Gaussian estimators in networks and two examples in which tangles tell stories of adiabatic quantum computations are given.

Tangle machines

  • Daniel MoskovichA. Carmi
  • Business
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2015
This section describes various capacities associated with machines, which provide a measure of how ‘good’ a computation is, and investigates discrepancies between its input and output streams.

Tangled Interactive Proofs

The utility of the formalism to distribute an interactive proof between multiple interacting verifiers is demonstrated by a network of nonadaptive 3-bit query PCP verifiers whose resulting soundness improves over the best known result for a single verifier of this class.

The em-convex rewrite system

We introduce and study em (or "emergent"), a lambda calculus style rewrite system inspired from dilations structures in metric geometry. Then we add a new axiom (convex) and explore its consequences.

Computing with Colored Tangles

It is proved that the model of computation is Turing complete and with bounded resources that it can decide any language in complexity class IP, sometimes with better performance parameters than corresponding classical protocols.

On graphic lambda calculus and the dual of the graphic beta move

This is a short description of graphic lambda calculus, with special emphasis on a duality suggested by the two different appearances of knot diagrams, in lambda calculus and emergent algebra sectors

Graphic Lambda Calculus

We introduce and study graphic lambda calculus, a visual language which can be used for representing untyped lambda calculus, but it can also be used for computations in emergent algebras or for

Local and global moves on locally planar trivalent graphs, lambda calculus and $\lambda$-Scale

It is shown that the lambda calculus contains $\lambda$-Scale calculus, therefore in particular untyped lambda calculus, and the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.

$λ$-Scale, a lambda calculus for spaces with dilations

$\lambda$-Scale is an enrichment of lambda calculus which is adapted to emergent algebras. It can be used therefore in metric spaces with dilations.

Graphic lambda calculus and knot diagrams

The sector covering knot diagrams is explored, which are constructed as macros over the graphic lambda calculus, which has sectors corresponding to untyped lambda calculus and emergent algebras.

References

SHOWING 1-10 OF 29 REFERENCES

More than discrete or continuous: a bird's view

I try to give mathematical evidence to the following equivalence, which is based on ideas from Plato (Timaeus): reality emerges from a more primitive, non-geometrical, reality in the same way as the brain construct the image of reality, starting from intensive properties.

A NON-ARISTOTELIAN SYSTEM AND ITS NECESSITY FOR RIGOUR IN MATHEMATICS AND PHYSICS

A very extensive literature shows that the problems of ‘infinity’ pervade human psycho-logical reactions, starting from the lowest stage of human development up to the present and that without some

Local operations : the embodiment of geometry

It turns out that such very general principles constrain the possible sampling structures greatly, and that the resulting default structure is not unlike the actual structure of the primate front end visual system and also not like the structures as they have "evolved" in computer vision.

Conservative logic

Conservative logic shows that it is ideally possible to build sequential circuits with zero internal power dissipation and proves that universal computing capabilities are compatible with the reversibility and conservation constraints.

What is a space? Computations in emergent algebras and the front end visual system

With the help of link diagrams with decorated crossings, I explain computations in emergent algebras, introduced in arXiv:0907.1520, as the kind of computations done in the front end visual system.

The brain a geometry engine

SummaryAccording to Kant, spacetime is a form of the mind. If so, the brain must be a geometry engine. This idea is taken seriously, and consequently the implementation of space and time in terms of

An Algorithmic Definition of the Axial Map

It is proposed that the analytical power of the axial map in empirical studies derives from the efficient representation of key properties of the spatial configuration that it captures.

From insect vision to robot vision

The understanding of some invertebrate sensory-motor systems has now reached a level able to provide valuable design hints and this approach brings into prominence the mutual constraints in the designs of a sensory and a motor system, in both living and non-living ambulatory creatures.

Emergent algebras as generalizations of differentiable algebras, with applications

We propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq). Algebraically, irqs are related with

RACKS AND LINKS IN CODIMENSION TWO

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a