# Gnomonic cubes: a simultaneous view of the extended graphic beta move

Recall that the extended  beta move is equivalent with the pair  of  moves :

where the first move is the graphic beta move and the second move is the dual of the beta move, where duality is (still loosely) defined by the following diagram:

In this post I want to show you that it is possible to view simultaneously these two moves. For  that I need to introduce the gnomonic cube. (Gnomons appeared several times in this blog, in expected or unexpected places, consult the dedicated tag “gnomon“).

From the wiki page about the gnomon,     we see that

A three dimensional gnomon is commonly used in CAD and computer graphics as an aid to positioning objects in the virtual world. By convention, the X axis direction is colored red, the Y axis green and the Z axis blue.

(image taken from the mentioned wiki page, then converted to jpg)

A gnomonic cube is then just a cube with colored faces. I shall color the faces of the gnomonic cube with symbols of the gates from graphic lambda calculus! Here is the construction:

So, to each gate is associated a color, for drawing conveniences. In the upper part of the picture is described how the faces of the cube are decorated. (Notice the double appearance of the $\Upsilon$ gate, the one used as a FAN-OUT.)  In the lower part of the picture are given 4 different views of the gnomonic cube. Each face of the cube is associated with a color. Each color is associated with a gate.

Here comes the simultaneous view of the pair of moves which form, together, the extended beta move.

In this picture is described a kind of a 3D move, namely the pair of gnomonic cubes connected with the blue lines can be replaced by the pair of red lines, and conversely.

If you project to the UP face of the dotted big cube then you get the graphic beta move. The UP view is the viewpoint from lambda calculus (metaphorically speaking).

If you project to the RIGHT face then you get the dual of the graphic beta move. The RIGHT view is  the viewpoint from emergent algebras (…).

Instead of 4 gates (or 5 if we count $\varepsilon^{-1}$ as different than $\varepsilon$), there is only one: the gnomonic cube. Nice!

# Dual of the graphic beta move implies some Reidemeister moves

For the extended beta move see the dedicated tag.  See also the tutorial “Introduction to graphic lambda calculus“.

The extended beta move is still in beta version. In this post I want to explore some consequences of the dual of the graphic beta move.

I proved earlier the the extended beta move is equivalent with the pair (graphic beta move, dual of the graphic beta move), let me recall this pair in the following picture.

Here are some particular applications of the dual of the graphic beta move. The first is this:

But this is equivalent with the emergent algebra move R1a (via a CO-COMM move).  Likewise, another application of of the dual of the graphic beta move is this:

which is the same as   the emergent algebra move R2a.

The third application is this one:

and it was discussed in the post “Second thoughts on the dual of the extended beta move“, go there to see if this move may be interpreted as a pruning move (or an elimination of loops).

Finally, there is also this:

which was not mentioned before. It suggests that

behaves like the generic point in algebraic geometry.

# Second thoughts on the dual of the extended beta move

After the night I found another way to look at the dual of the extended beta move. Recall that in the previous post was proved that the dual of the extended beta move is equivalent with this mystery move.

The graph from the left hand side can be expressed by using the (emergent algebra) crossing macros:

This form of the graph makes obvious where to apply the extended beta move. Let’s see what we get.

Therefore it looks like the mystery move is a combination of the extended beta move and elimination of loops. There is no need to replace the termination gate by a trivalent graph, as suggested  in the previous post, although that is an idea worthy of further exploration.

# Dual of the extended beta move. Termination as implosion or blow-out

Thanks Stephen for this comment, it made me think about the dual of the extended beta move and about the elimination of the termination gate.

The extended beta move has been introduced here. It is not, for the moment, part of the tutorial for graphic lambda calculus, but it will be soon.

Meanwhile, let me explore further the duality suggested in the post “Lambda scale in graphic lambda calculus and diagram crossings“.

The duality, although unclear as a precise mathematical object, is described by these three correspondences:

The last correspondence should be understood like it appears in the left hand sides of the “dual” moves from the “Graphic beta move extended …” which I reproduce here:

OK, so what is the dual of the extended beta move, then?  First, let me remind you the extended beta move:

The dual should be this:

If we look at the graph from the right hand side, then we see we can apply a beta move, like this:

Therefore the “dual extended beta” move is equivalent with the curious looking move:

That’s kind of a pruning move, but it is not on the list. Should it be?

We reason like this: by using the extended beta move and the Reidemeister 2 move (for emergent algebras) we arrive at

We may repeat indefinitely this succesion of moves, or, alternatively, we may use the extended beta move with an arbitrary $\mu$  from the group $\Gamma$. The intuition behind this is that the gate $\bar{\varepsilon}$ is a dilation gate, which has an input coming from the fan-out gate and the other connected to the output of the gate, therefore, by circulating along this loop, we apply an endless number of dilations of coefficient $\varepsilon$. At the limit (if such a concept makes sense at this level of generality), either the things blow to infinity (if $\varepsilon > 1$) or they implode to the value given by the fan-out gate(if $\varepsilon < 1$), or they circulate forever in a loop (if $\varepsilon = 1$) . In all cases the graph with only one input and no outputs, which we see in the previous figure, behaves exactly like the termination gate! Hence the mystery move (??) is simply a local pruning move.

Therefore, we may as well eliminate the termination gate and replace it in all the formalism by the gate from the previous figure.

By this replacement the “dual extended beta move” is true, as a consequence of the usual graphic beta move. Moreover, if we do this replacement, then we shall have pure trivalent graphs in the formalism because the ugly univalent termination gate is replaced by a trivalent graph!

What do you think?

# Lachesis, computation and desire to explore (Ancient Turing machines III)

This post continues the previous:

In the fiction of the Moirai performing as an ancient Turing machine, destiny  is akin to computation:

– it might be anyhow, like the Turing machine which could do anything,

– but once threaded by Clotho, Atropos and Lachesis, destiny is fixed (as in greek tragedies), like the outcome of a computation is fixed once the program and the data are given (by the gods  holding the keyboard).

Where is the part of exploration here? Where is the free will to leave the destiny’s path and take a walk on the field, towards that distant glitter from the mountain, far away…

Exploration is not computation, like (trying to understand something and) formulating problems is not solving problems.

Let’s get pragmatic and suppose that Lachesis is capable of doing the extended beta move instead of the more limited graphic beta move. Is this enough for allowing exploration into one’s destiny? Yes, if we interpret exploration (as done before) as being governed by the emergent algebra gate $\varepsilon$.

In the previous Moirai posts I showed how the three fates may construct any graph representing a lambda calculus term, starting from an initial loop. What the Moirai could not do, was to construct a $\varepsilon$ gate. Now, with the extended beta move available, here is how they could do it:

# Graphic beta move extended, to explore

This post continues the previous: “Lambda scale in graphic lambda calculus and diagram crossings“.

Suppose, just suppose for the moment that there is an extension of the graphic beta move, suggested by the previous post. I shall call this move “extended beta move”, or “(ext $\beta$)”, although a better name would be “scaled beta move”.

The extension to consider has the following form:

What a move! Who could have guessed such a move, involving four gates, without thinking about diagram crossings AND about lambda calculus?

Now, after expressing my enthusiasm, let’s see what this move can do.

First of all, per the previous post, the extended beta move implies the graphic beta move (by using (ext2) and LOCAL PRUNING).  Recall that in terms of the crossing macro involving only lambda calculus gates, the graphic beta move looks like this:

which is the same as this:

The extended beta move implies also a new move, expressed via the emergent algebra crossings, namely  this:

(See the last post: by the graphic beta move, which is implied by the extended one, the emergent algebra crossing from the RHS of the figure transforms into the graph from the RHS of the first figure, which now, by the extended beta move, transforms into the graphs from the LHS of the last figure.)

Let us give a name to this last move, call it “epsilon beta move”.

Now, fact: the usual graphic beta move together with the epsilon beta move are equivalent with the extended beta move.

Proof: we have to prove only one implication, because we have already proved the other. For this, let us express the graph from the RHS of the first figure with the help of the crossing macros. It looks like this:

Now we see that by applying first a epsilon beta move and then an usual graphic beta move, we recover the extended beta move.

Anyway, the extended beta move is left for further exploration.

# Lambda scale in graphic lambda calculus and diagram crossings

Background:

1. The lambda epsilon calculus was introduced in  arXiv:1205.0139 [cs.LO]    “$\lambda$-Scale, a lambda calculus for spaces with dilations”. Further, graphic lambda calculus was introduced in arXiv:1207.0332 [cs.LO]  “Local and global moves on locally planar trivalent graphs, lambda calculus and $\lambda$-Scale”.

2.   In the post “3D crossings in emergent algebras” I noticed a very intriguing resemblance between crossings in emergent algebras (as encoded in graphic lambda) and crossings macros in graphic lambda. More specifically, here is  the encoding of a crossing in emergent algebras:

which is very much like the crossing macro

Likewise, here is the other crossing in emergent algebras:

and its “twin” crossing macro

Their behaviour differs only with respect to the Reidemeister 3 move, which holds only in self-distributive (or “linear”) emergent algebras.

Question:   Are these crossing macros related in graphic lambda calculus?

Answer: Yes, through an idea introduced in the $\lambda$-Scale paper (which is actually the first proposal of a calculus for emergent algebra, later transformed into “graphic lambda calculus”).

Here is the explanation. In $\lambda$-Scale there are two operations, namely the $\lambda$ abstraction and a $\varepsilon$ operation, one for every coefficient $\varepsilon$ in a commutative group.  The application operation from lambda calculus comes as a composite of these two fundamental operations. Moreover, the $\varepsilon$ operation IS NOT THE SAME AS the operation from emergent algebras. The emergent algebra operation is also defined as a composite of the two fundamental operations of $\lambda$-Scale (see the paper, definition 2.2).

Later, in graphic lambda calculus, I renounced at the $\varepsilon$ operation of $\lambda$-Scale, replacing it with the operation from emergent algebras. To be clear, I used implicitly proposition 3.2 from the mentioned paper (which has a slightly misleading figure attached), which gives a description of the fundamental $\varepsilon$ operation of $\lambda$-Scale calculus in terms of the abstraction operation, the application operation  and the emergent algebra operation.

This turns out to be the key of the explanation I am writing about.

I shall define now a macro in graphic lambda calculus (inspired by the said proposition 3.2) which corresponds to the fundamental $\varepsilon$ operation from $\lambda$-Scale calculus. Here is it:

Let us play with this macro, by using it in a graph almost similar with the one where the graphic beta move applies:

Let us consider the particular case $\varepsilon = 1$. By using the (ext2) move and then local pruning we get the following graph:

I shall now apply the graphic beta move for the general $\varepsilon$, like this:

which is very nice, because we obtain the crossing macro of emergent algebras.

Finally, in the case when $\varepsilon = 1$, by an (ext2) move, followed by a local pruning pruning, we may transform the macro crossing from emergent algebra into this:

which ends the explanation.