Computing with space: done!

The project of giving a meaning to “computing” part of “Computing with space” is done, via the $\lambda$-Scale calculus and its graphic lambda calculus (still in preview mode).

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UPDATE (09.01.2013): There is now a web tutorial about graphic lambda calculus on this blog.  At some point a continuation of “Computing with space …” will follow, with explicit use of this calculus, as well as applications which were mentioned only briefly, like why the diagram explaining the “emergence” of the Reidemeister III move gives a discretized notion of scalar curvature for a metric space with dilations.

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Explanations.  In the “Computing with space…” paper I claimed that:

1. – there is a “computing” part hidden behind the idea of emergent algebras

2. – which is analogous  with the hypothetical computation taking place in the front-end visual system.

The 1. part is done, essentially. The graphic version of $\lambda$-Scale is in fact very powerful, because it contains as sectors:

– lambda calculus

– (discrete, abstract) differential calculus

– the formalism of tangle diagrams.

These “sectors” appear as subsets $S$ of graphs in $GRAPH$ (see the preview paper for definitions), for which the condition $G \in S$ is global, together with  respective selections of  local or global graphic moves (from those available on $GRAPH$) which transform elements of $S$ into elements of $S$.

For example, for lambda calculus the relevant set is $\lambda$-GRAPH and the moves are (ASSOC) and  the graphic $\beta$ move (actually, in this way we obtain a formalism a bit nicer than lambda calculus; in order to obtain exactly lambda calculus we have to add the stupid global FAN-OUT and global pruning moves).

For differential calculus we need to restrict to graphs like those in $\lambda$-GRAPH, but also admitting dilation gates. We may directly go to $\lambda$-Scale, which contains lambda calculus (made weaker by adding the (ext) rules, corresponding to $\eta$-conversion) and differential calculus (via emergent algebras). The moves are (ASSOC), graphic $\beta$ move, (R1), (R2), (ext1), (ext2) and, if we want a dumber version,  some global FAN-OUT and pruning moves.

For tangle diagrams see the post Four symbols and wait for the final version of the graphic calculus paper.

SO  now, I declare part 1. CLOSED. It amounts to patiently writing all details, which is an interesting activity by itself.

Part 2. is open, albeit now I have much more hope to give a graph model for the front-end of the visual system, which is not relying on assumptions about the geometric structure of the space, linear algebra, tasks and other niceties of the existing models.

UPDATE  02.07.2012. I put on arxiv the graphical formalism paper, it should appear on 03.07.2012. I left outside of the paper a big chunk of very intriguing facts about various possible crossing definitions, for another paper.

Four symbols

Here is a diagram involving the four symbols in the graphic lambda-Scale:

– At the left are used drawing conventions from the preview paper “Local and global moves on planary trivalent graphs, lambda calculus and lambda-Scale“.

– At the right are used drawing conventions from “Computing with space…” page 21.

We can pass from a drawing to another by a graphic beta move.

To me, this looks like

$e^{i \pi} = -1$

a formula which involves four symbols too.

Gromov on entropy and Souriau

Gromov just posted on his page the paper In a Search for a Structure, Part 1: On Entropy. June 19, 2012.   With much interest I started to read it and my first impression is that I have seen something like this before (but I might be wrong, please excuse me if so) in the HUGE last paper (book)

Grammaire de la nature (version du 8 juillet 2007)

by Jean-Marie Souriau, the inventor of symplectic geometry and geometric quantization, among others.

Specifically, I refer to  the way Souriau treats probabilities in “CLE 8: Calcul des Hasards”, p. 209.

The book is a must-read! It is a summum of Souriau mathematical view of Nature, specifically concerning symmetry, entropy, relativity and quantum mechanics.

Gromov, with his enormous geometrical knowledge (different than Souriau’ though)  points to sofic groups, this  I need a lot of time to understand.

UPDATE: I am starting to understand the sofic group notion of Gromov and learning to appreciate it, it’s related to constructions with approximate groups, apparently.

Two halves of beta, two halves of chora

In this post I want to emphasize a strange similarity between the beta rule in lambda calculus and the chora construction (i.e. encircling a tangle diagram).

Motivation? Even if now clearer, I am still not completely satisfied by the degree of interaction between lambda calculus and emergent algebras, in the proposed lambda-Scale calculus. I am not sure if this is because lambda-Scale is yet not explored, or because there exist a more streamlined version of lambda calculus as a macro over the emergent algebras.

Also, I am working again on the paper put on preview (version 05.06.2012) about planar trivalent graphs ans lambda calculus, after finishing the  course notes on intrinsic sub-riemannian geometry.

So, I let my mind hovering over …

As explained in the draft paper, the beta rule in lambda calculus  is a LOCAL rule, described in this picture (advertised here):

It is made by two halves: the left half contains the lambda abstraction, the right half contains the application operation. In between there is a wire. The rule says that these two halves annihilate somehow and the wire is replaced by a dumb crossing with no information about who’s on top.

Let us contemplate an elementary chora, made also by two halves:

We can associate to this figure a move, which consists in the annihilation of the left (difference gate) and right (sum gate) halves, followed by the replacement of the “wire” by an equivalent crossing

Preview of two papers, thanks for comments

Here are two papers:

Local and global moves on planary trivalent graphs, lambda calculus and lambda-Scale (update 03.07.2012, final version, appears as arXiv:1207.0332)

Sub-riemannian geometry from intrinsic viewpoint    (update 14.06.2012: final version, appears as arxiv:1206.3093)

which are still subject to change.  Nevertheless most of what I am trying to communicate is there. I would appreciate  mathematical comments.

This is an experiment,  to see what happens if I make previews of papers available, like a kind of a blog of papers in the making.

Intrinsic characterizations of riemannian and sub-riemannian spaces (I)

In this post I explain what is the problem of intrinsic characterization of riemannian manifolds, in what sense has been solved in full generality by Nikolaev, then I shall comment on the proof of the Hilbert’s fifth problem by Tao.

In the next post there will be then some comments about Gromov’s problem of giving an intrinsic characterization of sub-riemannian manifolds, in what sense I solved this problem by adding a bit of algebra to it. Finally, I shall return to the characterization of riemannian manifolds, seen as particular sub-riemannian manifolds, and comment on the differences between this characterization and Nikolaev’ one.

1. History of the problem for riemannian manifolds. The problem of giving an intrinsic characterization of riemannian manifolds is a classic and fertile one.

Problem: give a metric description of a Riemannian manifold.

Background: A complete riemannian manifold is a length metric space (or geodesic, or intrinsic metric space) by Hopf-Rinow theorem. The problem asks for the recovery of the manifold structure from the distance function (associated to the length functional).

For 2-dim riemannian manifolds the problem has been solved by A. Wald [Begrundung einer koordinatenlosen Differentialgeometrie der Flachen, Erg. Math. Colloq. 7 (1936), 24-46] (“Begrundung” with umlaut u, “Flachen” with umlaut a, sorry for this).

In 1948 A.D. Alexandrov [Intrinsic geometry of convex surfaces, various editions] introduces its famous curvature (which uses comparison triangles)  and proves that, under mild smoothness conditions  on this curvature, one is capable to recover the differential structure and the metric of the 2-dim riemannian manifold. In 1982 Alexandrov proposes as a conjecture that a characterization of a riemannian manifold (of any dimension) is possible in terms of metric (sectional) curvatures (of the type introduced by Alexandrov) and weak smoothness assumptions formulated in metric way (as for example Holder smoothness). Many other results deserve to be mentioned (by Reshetnyak, for example).

2. Solution of the problem by Nikolaev. In 1998 I.G. Nikolaev [A metric characterization of riemannian spaces, Siberian Adv. Math. , 9 (1999), 1-58] solves the general problem of intrinsic characterization of $C^{m,\alpha}$ riemannian spaces:

every locally compact length metric space $M$, not linear at one of its points,  with $\alpha$ Holder continuous metric sectional curvature of the “generalized tangent bundle” $T^{m}(M)$ (for some $m=1,2,…$, which admits local geodesic extendability, is isometric to a $C^{m+2}$ smooth riemannian manifold..

Therefore:

• he defines a generalized tangent bundle in metric sense
• he defines a notion of sectional curvature
• he asks some metric smoothness of this curvature

and he gets the result.

3. Gleason metrics and Hilbert’s fifth problem. Let us compare this with the formulation of the solution of the Hilbert’s fifth problem by Terence Tao. THe problem is somehow similar, namely recover the differential structure of a Lie group from its algebraic structure. This time the “intrinsic” object is the group operation, not the distance, as previously.

Tao shows that the proof of the solution may be formulated in metric terms. Namely, he introduces a Gleason metric (definition 4 in the linked post), which will turn to be a left invariant riemannian metric on the (topological) group. I shall not insist on this, instead read the post of Tao and also, for the riemannian metric description, read this previous post by me.