# 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:

# The three Moirai, continued

UPDATE:  The problem of connecting two gates, as explained in this post, is equivalent with the oriented Reidemeister move R2c, itself equivalent with R3a, for the untyped lambda calculus crossing macro. Therefore we cannot, in graphic lambda calculus, without the dual of the graphic beta move, at least, solve the problem of gate connections.

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In the post “Ancient Turing machines (I): the three Moirai” I explained how Clotho, Atropos and Lachesis may build together a Lisp-like based Turing machine, in terms of the graphic lambda calculus.

Clotho creates new thread by inserting FAN-OUT gates (the move CREA), Atropos cuts the thread (the move GARB) and Lachesis performs graphic beta reduction. Either they have an infinite reservoir of loops, or Lachesis may also use the Reidemeister move R1a. (We discussed about oriented Reidemeister moves in the post “Generating set of Reidemeister moves for graphic lambda crossings”  ; the names of the moves are those from the paper    by Michael Polyak  “Minimal generating sets of Reidemeister moves“, only that I use the letter “R” from “Reidemeister” instead of “$\Omega$” used by Polyak.)

There is something missing, though, namely how to connect gates, once created. I shall explain this further. After that I shall finish with a reminder of the real goal of these posts, essentially mentioned in my comment of the last post.

Recall that the three Moirai know how to create the lambda abstraction gate, the application gate, the FAN-OUT gate and the termination gate, now the question is how they connect two gates, once they have them. In the next figure is given a solution for this.

So, the problem is this: we have two threads, marked 1-2 and 3-4, we want to obtain a thread from 1 to 4. For this we add a loop and Lachesis performs a graphic beta move (alternatively, without adding a loop, Lachesis does a R1a move). Lachesis continues by doing a second graphic beta move, as indicated in the figure. Finally, she performs a number of beta moves  equivalent with the oriented Reidemeister move R2c (see the  mentioned Polyak’s paper for notation). I have not counted how many moves are needed for R2c , but the number can be inferred from the generation of the move R2c from the moves R1a, R1b, R2a, R3a.

Now the construction is finished, let us leave the Moirai to do their job.
Finally, I shall recall my real goal, which I have never forgot. The real goal is to pass from understanding of the power of this lambda calculus sector of graphic lambda calculus to the real deal, called “computing with space”, namely to understand space from a computational perspective, not as a given receptacle, but as a small list of procedures along with some impossible to verify assertions (like that we may rescale indefinitely space), see “emergent algebras”, which can always be eliminated a posteriori, by a kind of finitization procedure.

# Ancient Turing machines (I): the three Moirai

This is a first post about interpreting the Turing machine in ancient terms (I have at least another interpretation in mind, which I shall explain later).

It’s your choice to interpret it as a tongue-in-cheek or verbatim. Here are the facts.

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Go to the tutorial “Introduction to graphic lambda calculus” if you want to understand the graphic conventions and the moves.

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1. The three Moirai, cite from their wiki page:

In Greek mythology, the Moirai (Ancient Greek: Μοῖραι, “apportioners”, Latinized as Moerae)—often known in English as the Fates—were the white-robed incarnations of destiny (Roman equivalent: Parcae, euphemistically the “sparing ones”, or Fata; also equivalent to the Germanic Norns). Their number became fixed at three: Clotho (spinner), Lachesis (allotter) and Atropos (unturnable). […]

• Clotho ( /ˈklθ/, Greek Κλωθώ [klɔːˈtʰɔː] – “spinner”) spun the thread of life from her distaff onto her spindle. Her Roman equivalent was Nona, (the ‘Ninth’), who was originally a goddess called upon in the ninth month of pregnancy.
• Lachesis ( /ˈlækɨsɪs/, Greek Λάχεσις [ˈlakʰesis] – “allotter” or drawer of lots) measured the thread of life allotted to each person with her measuring rod. Her Roman equivalent was Decima (the ‘Tenth’).
• Atropos ( /ˈætrəpɒs/, Greek Ἄτροπος [ˈatropos] – “inexorable” or “inevitable”, literally “unturning”,[16] sometimes called Aisa) was the cutter of the thread of life. She chose the manner of each person’s death; and when their time was come, she cut their life-thread with “her abhorred shears”.[17] Her Roman equivalent was Morta (‘Death’).

2. Let’s interpret their activity as something equivalent to a Turing machine. I shall use untyped lambda calculus, which has the same computational power as Turing machines. Better, I choose to work with graphic lambda calculus (tag archive , first paper), which has a sector equivalent with untyped lambda calculus.

The challenge is to arrive to generate all graphs in $GRAPH$by using the three Moirai, specifically by formalizing their activity in terms of graphic lambda.

The following figure contains this, let’s contemplate it and then pass to explanations.

CLOTHO   is creating the thread, namely the new move called “CREA” (from “creation”):

Basically she introduces a FAN-OUT gate into the thread. In order to make this gate to function as FAN-OUT, she also needs  from the graphic lambda calculus the moves CO-COMM (which allows her to permute the outputs) and CO-ASSOC (which allows her to not care about the order of application of a cascade of FAN_OUT gates).

ATROPOS cuts the thread, namely she is performing a move which I shall call “GARB” (from “garbage”), which is a new move introduced in graphic lambda calculus:

She picks from the moves of graphic lambda calculus LOCAL PRUNING and ELIMINATION OF LOOPS, which are kind of her style.

LACHESIS  is doing only one move, the graphic beta, described here (and see the paper) as a braiding move, when seen in knot diagrams macro. (She might actually be able to do also the oriented Reidemeister 1a move, see further.)

This is a graphic form of $\beta$ reduction, so you may say that LACHESIS  is performing something akin to $\beta$ reduction.

3. How does it work? The Moirai have a thread to start from. Their first goal is to produce the gates. They can easily  have two gates, one appearing after GARB, the other appearing after CREA. They still need the application gate (corresponding to the application operation in lambda calculus) and the lambda abstraction gate.

They also need to have enough threads to play with. Here are two ways of getting them. The first one is using only GARB and CREA moves. The dashed green curves represent the input and the output of their activities. The dashed red curves indicate where the moves are applied.

Another way of producing two threads from one, more specifically producing a new thread and also keeping the old one, uses also LOCAL PRUNING:

If the Moirai have only one thread and no loop, then we have to add to LACHESIS’s competences the three Reidemeister moves, or at least the Reidemeister 1a move:

Then LACHESIS may use her graphic beta move in order to get a thread and a loop.  ATROPOS has to refrain to use her ELIMINATION OF LOOPS for later!

Now the three Moirai are ready to produce the application and lambda abstraction gates. CLOTHO and LACHESIS  start with two threads (which they already have), in order to get to an intermediary step.

From here, with some help from ATROPOS, they  get a lambda gate and an application gate.

From here the Moirai have to be very clever and patient in order to construct the graphs which correspond to the lambda calculus terms needed for something equivalent of a Turing machine. They have to be clever because they want to construct graphs in $GRAPH$ from the lambda calculus sector, and for this they have to cleverly use loops in order to satisfy, at the end, the global conditions which graphs from the lambda calculus sector satisfy (that is, basically, the condition that whatever exits from the right hand side exit of a lambda gate, has to either end in garbage, or to continue until it enters by the input of the said lambda gate).

Their work could be made easier if they learn a bit of LISP and they follow the indications of  this paper.

That’s it.

We are left with three, very vague questions:

1. Could it be that the Moirai take some shorcuts through the maze of constructing a Turing machine and instead, thread our fates in an equivalent (or more general?) way, but using less sophisticated building blocks?

2. As they spun the destiny of the Universe, they do it in a computable fashion?

3. Could the Moirai build Moirai? (I find this hard to believe, by looking at the GLOBAL CONDITIONS they have to achieve by pure wisdom.)

# Right angles everywhere (II), about the gnomon

In this post I shall write about the gnomon. According to wikipedia,

The gnomon is the part of a sundial that casts the shadow. Gnomon (γνώμων) is an ancient Greek word meaning “indicator”, “one who discerns,” or “that which reveals.”

In the next figure are collected the minimal ingredients needed for understanding the gnomon: the sun, a vertical shape and its horizontal shadow.

That is the minimal model of the ancient greek visual universe: sun, a man and its shadow on the beach. It is a speculation, but to me, a gnomon seems to be a visual atom.

Pythagoreans extracted from this minimal visual universe the pattern and used it for giving an explanation for the human vision, described by the next figure.

Here the sun is replaced by the eye (of a god, initially, but the pattern might apply to a mortal also), the light rays emanated by the sun are assimilated with the lines  of vision (from here the misconception that the ancient greeks really believed that the eyes shoot rays which illuminate the field of vision) and the indivisible pair man-shadow becomes the L-shape of a gnomon.  An atom of vision.

Here comes a second level of understanding the gnomon, also of pythagoreic flavor. I cite again from the wiki page:

Hero defined a gnomon as that which, added to an entity (number or shape), makes a new entity similar to the starting entity.

This justifies the Euclid’ picture of the gnomon, as a generator of self-similarity:

(image taken from the wiki page on gnomon)
So maybe the word “atom” is less appropriate than “generator”. In conclusion, according to ancient greeks, a gnomon (be it a triple sun-man-shadow or a pair eye – elementary L-shape) is the generator of the visual perception, via the mechanism of self-similarity.

In their architecture, they tried to make this obvious, readable.  Because it’s scalable (due to the relation with self-similarity), the architectural solution of constructing with gnomons  invaded the world.

# Right angles everywhere (I)

Look at almost any building in the contemporary city, it’s constructed from right angles, assembled into rectangles, assembled into boxes. We expect, in fact,  a room to have a rectangular floor, with vertical walls meeting in right angles. Exceptions are either due to architectural fancies or to historical constraints or mistakes.

When a kid draws a house, it looks like a rectangle, with the  triangle of the roof on top.

Is this normal? Where does this obsession of the right angle comes from?

The answer is that behind any right angle is hidden a gnomon. We build like this because we  are Pythagoras children, living by the rules and categories of our cultural ancestors, the ancient greeks.

Let’s see:
(I) In ancient times,  or in  places far from the greeks  (and babylonians), other architectural forms are preferred, like the  roundhouse. Here’s a Scottish broch (image taken from this wiki page)

and here’s a Buddhist stupa (image taken from the wiki page)

Another ancient building form is the step pyramid , like the Great Ziggurat of Ur (image taken from the last wiki page)

or the egyptian pyramids, or any other famous  pyramid in the world (there are plenty of them, in very different cultural frames).

Here is a Sardinian Nuraghe

Conclusion: round, conical, pyramidal is the rule, there are no right angles there!

Until the greeks: here’s the Parthenon

It is made of gnomons, here’s one (from the wiki page)

# The gnomon in the greek theater of vision, I

In the post Theatron as an eye I proposed the Greek Theater, or Theatron (as opposed to the “theater in a box”, or Cartesian Theater, see further) as a good model for   vision.

Any model of vision should avoid the homunculus fallacy. What looks less understood is that any good model of vision should avoid the scenic space fallacy. The Cartesian Theater argument against the existence of the homunculus is not, by construction, an argument against the scenic space. Or, in the Cartesian Theater, homunculus and scenic space come to existence in a pair. As a conclusion, it seems that there could not be a model of vision which avoids the homunculus but is not avoiding the scenic space. This observation is confirmed by facts: there is no good, rigorous  model of vision up to date, because all proposed models rely on the a priori existence of a scenic space. There is, on the contrary, a great quantity of experimental data and theoretical partial models which show just how complex the problem of vision is. But, essentially, from a mathematician viewpoint, it is not known how to even formulate the problem of vision.

In the influent paper “The brain a geometry engine”  J. Koenderink proposes that (at least a part of) the visual mechanism is doing a kind of massively parallel computation, by using an embodiment of the geometry of jet spaces (the euclidean infinitesimal geometry of a smooth manifold)  of the scenic space. Jean Petitot continues along this idea, by proposing a neurogeometry of vision based essentially on the sub-riemannian geometry of those jet spaces. This an active mathematical area of research, see for example “Antropomorphic image reconstruction via hypoelliptic diffusion“, by Ugo Boscain et al.

Sub-riemannian geometry is one of my favorite mathematical subjects, because it  is just a  particular model of a metric space with dilations.  Such spaces are somehow fundamental for the problem of vision, I think. Why? because there is behind them a purely relational formalism, called “emergent algebra“, which allow to understand “understanding space” in a purely relational way. Thus I hope emergent algebras could be used in order to formulate the problem of vision as the problem of computing with space, which in turn could be used for getting a good model of vision.

To my surprise, some time ago I have found that this  very complex subject has a respectable age, starting with Pythagora  and Plato!  This is how I arrived to write this blog, as an effort to disseminate what I progressively understand.

This brings me back to the theater and, finally, to gnomon. I cite from previous wiki link:

Hero defined a gnomon as that which, added to an entity (number or shape), makes a new entity similar to the starting entity.

In the greek theater, a gnomon sits in the center of the orchestra (which is the circular place where things happen in the greek thater, later replaced by the scene in the theater in a box). Why?

# Theatron as an eye

I want to understand what “computing with space” might be. By making  a parallel with the usual computation, there are three ingredients which need to be identified: what are the computing with space equivalents of

1. the universal computing gate (in usual computing this is the transistor)

2. the universal machine (in usual computing this is the Turing machine)

3. what is the universal machine doing by using its arrangement of universal computing gates (in usual computing this is the algorithm).

I think that (3) is (an abstraction of) the activity of map making, or space exploration. The result of this activity is coded by a dilation structure, but I have no idea HOW such a result is achieved. Once obtained though, a mathematical model of the space is the consequence of  a priori assumptions (that we can repeat in principle indefinitely the map making operations) which lead to the emergent algebraic and differential structure of the space.

The universal gate (1), I think, is the dilation gate, or the map-territory relation.

Today I want to pave the way to the discovery of the universal machine (2). This is related to my previous posts The Cartesian Theater: philosophy of mind versus aerography and Towards aerography, or how space is shaped to comply with the perceptions of the homunculus.

My take is that the Greek Theater, or Theatron (as opposed to the “theater in a box”, or Cartesian Theater) is a good model for an universal machine.

For today, I just want to point to the similarities between the theatron and the eye.

The following picture represents the main parts of the theatron (the ancient greek meaning of “theatron” is “place of seeing). In black are written the names of the theatron parts and in red you see the names of the corresponding parts of the eye, according to the proposed similarity.

Let me proceed with the meaning of these words:

– Analemmata means the pedestal of a sundial (related with analemma and analemmatic sundial; basically a theatron is an analemmatic sundial, with the chorus as the gnomon). I suggest to parallel this with the choroid of the eye.

– Diazomata (diazoma means “belt”), proposed to be similar with the retina.

Prohedria (front seating) is a privilege to sit in the first few rows at the bottom of the viewing area. Similar with the fovea (small pit), responsible for sharp central vision.

Skene (tent), the stage building, meant to HIDE the workings  of the actors which are not part of the show, as well as the masks and other materials. When a character dies, it happens behind the skene. Eventually, the skene killed the chorus and  became the stage. The eye equivalent  of this is the iris.

Parodos (para – besides, counter, and ode – song) entrance of the chorus. Eye equivalent is the crystalline lens.

– Orchestra, the ancient greek stage, is the place where the chorus acts, the center of the greek theater. Here we pass to abstraction: the eye correspondent is the visual field.