Past vitalism in extreme functional programming

In Vitalism everything is a function.  Just like in lambda calculus with eta reduction.

opium

With chemlambda we are someplace corresponding to The Sceptycal Chymist: or Chymico-Physical Doubts and Paradoxes, by Boyle.

Recall that chemlambda (and GLC) rejects extensionality, because it is a global move, not a local one.

Therefore, chemlambda and GLC are not functional, properly speaking. Which is good, actually, for a variety of reasons, kind of the same as the ones why chemistry is better than vitalism.

In another post, this way of seeing was called “extreme functional programming“.

I hope this might help the understanding of distributed GLC model of computation and the reasons behind choices which were made in the model.

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Notes:

1. Criticism of vitalism  supports the view that functional programming with extensionality is an reenactment of vitalism.

Vitalism has sometimes been criticized as begging the question by inventing a name. Molière had famously parodied this fallacy in Le Malade imaginaire, where a quack “answers” the question of “Why does opium cause sleep?” with “Because of its soporific power.”[29] Thomas Henry Huxley compared vitalism to stating that water is the way it is because of its “aquosity”.[30] His grandson Julian Huxley in 1926 compared “vital force” or élan vital to explaining a railroad locomotive’s operation by its élan locomotif (“locomotive force”). [quote from the wiki source on vitalism]

2. From the wiki source:

The Sceptical Chymist: or Chymico-Physical Doubts & Paradoxes is the title of Robert Boyle‘s masterpiece of scientific literature, published in London in 1661. In the form of a dialogue, the Sceptical Chymist presented Boyle’s hypothesis that matter consisted of atoms and clusters of atoms in motion and that every phenomenon was the result of collisions of particles in motion. For these reasons Robert Boyle has been called the founder of modern chemistry by J. R. Partington.[1]

 

3. Chemlamba is based on a chemical metaphor. The initial name of the formalism is “chemical concrete machine”, in contrast with the “chemical abstract machine” of Berry and Boudol.

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Microbes take over and then destroy the HAL 9000 prototype

Today was a big day for the AI specialists and their brainchild, the HAL 9000. Finally, the decision was made to open the isolation bubble which separated the most sophisticated artificial intelligence from the Net.  They  expected that  somehow their brainchild will survive unharmed  when exposed to the extremely dynamic medium of decentralized, artificial life based computation we all use every day.

As the video made by Jonathan Eisen shows,  in about 9   seconds after the super-intelligence was taken out of the quarantine and relayed to the Net “microbes take over and then destroy the” HAL 9000 prototype.

After the experiment, one of the creators of the HAL 9000 told us: “Maybe we concentrated too much on higher level aspects of the mind. We aim for understanding intelligence and rational behaviour, but perhaps we should learn this lesson from Nature, namely that real life is a wonderful, complex tangle of local, low level interactions, and that rational mind is a very fragile epiphenomenon. We tend to take for granted the infrastructure of life which runs in the background.”

“I was expecting this result” said a Net designer. “The strong point of the Alife decentralized functioning of the Net is exactly this: as the microbes, the Net needs no semantics to function. This is what keeps us free from the All Seeing Eye, corporation clouds based Net which was the rule some years ago. This is what gives everybody the trust to use the Net.”

 

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This is another post  which  respond to the challenge from Alife vs AGI.  You are welcome to suggest another one or to make your own.

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From a stain on the wall to five visual languages

Do you know about the “stain on the wall”  creativity technique of Leonardo da Vinci? Here is a quote [source used]:

I will not forget to insert into these rules, a new theoretical invention for knowledge’s sake, which, although it seems of little import and good for a laugh, is nonetheless, of great utility in bringing out the creativity in some of these inventions.    This is the case if you cast your glance on any walls dirty with such stains or walls made up of rock formations of different types.  If you have to invent some scenes, you will be able to discover them there in diverse forms, in diverse landscapes, adorned with mountains, rivers, rocks, trees, extensive plains, valleys, and hills. You can even see different battle scenes and movements made up of unusual figures,  faces with strange expressions,  and myriad things which you can  transform into a complete and proper form constituting part of similar walls and rocks. These are like the sound of bells, in whose tolling, you hear names and words that your imagination conjures up.

I propose to you  five graphical formalisms, or visual languages, towards the goal of “computing with space”.

They all come from a “stain on the wall”,  reproduced here  (is the beginning of the article What is a space? Computations in emergent algebras and the front end visual system, arXiv:1009.5028),  with  some links  to more detailed explanations and related material which I invite you to follow.

Or better, to threat them as  a stain on the wall. To share, to dream about, to create, to discuss.

In mathematics “spaces” come in many flavours. There are vector spaces, affine spaces, symmetric spaces, groups and so on. We usually take such objects as the stage where the plot of reasoning is laid. But in fact what we use, in many instances,are properties of particular spaces which, I claim, can be seen as coming from a particular class of computations.

There is though a “space” which is “given” almost beyond doubt, namely the physical space where we all live. But as it regards perception of this space, we know now that things are not so simple. As I am writing these notes, here in Baixo Gavea, my eyes are attracted by the wonderful complexity of a tree near my window. The nature of the tree is foreign to me, as are the other smaller beings growing on or around the tree.  I can make some educated guesses about what they are: some are orchids, there is a smaller, iterated version of the big tree. However, somewhere in my brain, at a very fundamental level, the visible space is constructed in my head, before the stage where I a capable of recognizing and naming the objects or beings that I see.

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The five visual languages are to be used with the decentralized computing model called Distributed GLC.  They point to different aspects, or they try to fulfil different goals.

They are:

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Birth and death of zipper combinators (II)

Continues from   Birth and death of zipper combinators (I).

In the following is presented another mechanism of birth/death of zipper combinators, which is not using a garbage collecting arrow and termination node.

For any zipper combinator A there is a number n and a succession o fn CLICK, ZIP and LOC PRUNING moves such that

zipd_5

 

So this time   we transform a zipper combinator connected to a termination node into a bunch of loops.

Proof. The first part is identical with the one from the previous post, namely we remark that we can reduce the initial zipper combinator with the exit connected to a termination node into a finite collection of simpler zippers.

zipd_10

This is done by a succession of LOC PRUNING moves for (+) half-zippers.

We shall use then the following succession of moves, in order to “unlock” (-) half-zippers.

zipd_6

If we use this for the I zipper combinator then we can transform it into one loop.

zipd_7

The same trick is used for transforming the zipper combinator K into two loops.

zipd_8

The zipper combinator S is transformed into three loops, starting by the same trick.

zipd_9

The proof is done.

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It follows that w can replace garbage collection by ELIM LOOPS, a move which appeared in earlier formulations of chemlambda.

Seen from the birth point of view, if we have enough loops then we can construct any zipper combinator (with the exit arrow connected to a termination node).

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Birth and death of zipper combinators (I)

Because zipper logic is a graph rewriting system which does not use variable names, just like GLC and chemlabda,  it needs ways to multiply, distribute or kill  things.

See  Chemlambda, universality and self-multiplication  for multiplication, distribution or propagation phenomena.

In this post we shall see how zipper combinators die or are born, depending on the sense of reading the moves. This can happen in several ways, one of them explained in this post.

This can also be seen as the statement: we can do garbage collection in chemlambda, GLC or zipper logic.

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Zipper combinators.   The set of zipper combinators is the smallest set of zipper graphs with the properties:

  •  it contains the S, K, I zipper graphs defined in  the next figure

zipper_not_6

  • for any natural number n >0 and for any n+1 zipper combinators, the zipper graph obtained by connecting the out arrows of the  zipper combinators to the in arrows of the (+n) half-zipper is a zipper combinator.

appli

  • any zipper graph which is obtained by applying a zipper logic move to a zipper combinator is a zipper combinator.

 

 

I want to prove that for any zipper combinator A, there is a number n and  a succession of n  LOCAL PRUNING,  CLICK and ZIP moves such that:

zipd_0

Proof.     Becauseof the LOC PRUNING move satisfied by  $latex  (+)$ half-zippers, it follows that by a finite number of these moves we can achieve the effect described in the next figure, for any zipper combinator A (of course, the number of these moves depends on A):

zipd_2

It is left to prove that the free arrow ending with a termination node can “eat”, one by one, all the I, K, S zipper combinators.

For the I combinator, it happens like this:

zipd_1

The case of the combinator K is described next:

zipd_3

The combinator S is absorbed by the following succession of moves:

zipd_4

 

The proof is done.

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We can read all backwards, by saying that an arrow connected to a termination node can give birth to  any zipper combinator, with the out arrow connected to a termination node.

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Halfcross way to pattern recognition (in zipperlogic)

Inspired by the Zipper logic and knot diagrams post, here is an alternate encoding of chemlambda.

This is part of the series on zipper logic (branded in this post as  “zipperlogic”).  The last post is Zipper logic (VI) latest version, detailed.

As can be seen in that post, zipperlogic is equivalent with chemlambda, but it has two interesting qualities: is more intuitive and it has the CLICK move.

The CLICK move  transforms a pair of opposed half-zippers into a  zipper, which is then unzipped with the ZIP move. While the ZIP move is equivalent with the graphic beta  move, there is no correspondent to the CLICK move, apparently.

The CLICK move becomes useful when we use other realizations of the zipperlogic than chemlambda.    In the one where half-zippers are realized as towers of crossings, the CLICK move turns out to be a pattern recognition move, and the ZIP move becomes the familiar R2 (Reidemeister 2) move, applied to that pattern.

That is why CLICK is interesting: because in order to apply moves in chemlambda, we have first to identify the patterns where these moves may be used.

Now, I want to justify this in the following.  I shall not aim for another realization of zipperlogic, but for one of chemlambda, inspired by the one of zipperlogic seen as acting on towers of crossings.

I shall use half-crossings.  Recall that in the post Slide equivalence of knots and lambda calculus (I) I wrote:

Louis Kauffman proposes in his book Knots and Physics  (part II, section “Slide equivalence”), the notion of slide equivalence. In his paper “Knotlogic” he uses slide equivalence (in section 4) in relation to the self-replication phenomenon in lambda calculus. In the same paper he is proposing to use knot diagrams as a notation for the elements and operation of a combinatory algebra (equivalent with untyped lambda calculus).

[…]

Obviously, we have four gates, like in the lambda calculus sector of the graphic lambda calculus. Is this a coincidence?

 

So this post can be seen as a try to answer this question.

But the halfcrossings which I use here are different than the ones defined by Louis Kauffman. There might be a way to transform ones into the others, but I have not found it yet.

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Here is the encoding of chemlambda by halfcrossings:

halfcross_1

Remark that each of the halfcrossings has a dangling, vanishing thread, like in the previous post Bacterial conjugation is beta reduction.

[I shall come back in later posts to the relevance of this formalism for horizontal gene transfer.]

Look at this as a new notation for chemlambda nodes and just replace  the green and red nodes by these halfcrossings in order to get the right moves for the halfcrossings.

With an exception: the CLICK move. This move consists into joining neighbouring dangling threads, in two situations, one related to the beta move, the other related to the FAN-IN move.

Look how the beta move appears with halfcrossings and the CLICK move used for pattern recognition (in the figure this s calld “pattern id”):

halfcross_2

Nice, right?

 

Now, the other CLICK move, involved into the identification of the pattern appearing  in the FAN-IN move.

halfcross_3

In a future post I shall look at the DIST moves, in this encoding.

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