Author Topic: Computation Using Knots  (Read 3383 times)

knot

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Computation Using Knots
« on: January 09, 2013, 04:10:45 AM »
Anybody know anything about computation using knots?

First, one can build things like abacuses.

Second, and more interestingly, I imagine that it would be possible to encode problems into knots and, using characteristics of the knot, find the answer. For example, maybe there is a way to encode some small class of true-false logical statements into knots, and pull on two ends to find the answer: if the knot was equivalent to the unknot, the answer would be "true".

I wonder whether there are any problems that can be encoded that are less trivially isomorphic than "Is this knot equivalent to the unknot?"

This subject, though vague and unlikely to give good methods of computation, though probably tedious and dull, is interesting to me.

X1

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Re: Computation Using Knots
« Reply #1 on: January 09, 2013, 05:28:52 AM »
   So, you are about to write the sequence of Leibniz s "Monadology", and invent a universal knotting machine able to compute, and, why knot, to think, and to replicate itself, using, as elements, the elementary knot transformations that leave the knot topologically invariant ( the Reidemeister moves )( or retain any other invariant property of the knot ). ! ! !  Oh, my knotGod ! This IS an abstract idea, indeed ! Von Neumann would be proud of it !...
   William Thomson ( Lord Kelvin) tried to explain the atoms by knots, but it turned out that God had not utilized his idea. We can imagine intelligent beings at another Galaxy, that think using knot-brains made by tangles of one-atom-thin chains, transforming into zillions of curvilinear shapes, that is, thoughts... Even beyond this, may be their knot-brains will be made by quantum particles, able to perform quantum computations. (1) If a classic brain, made by classic objects, can be so irrational and stupid as the brain of humans, we can imagine how irrational and stupid will be a quantum brain !  :)
   In this Forum I believe we are still a little more modest. We would be happy if somebody would help us enumerate all the bends made by two interlocked overhand knots, for example... We are not experts on mathematical Knots or on Artificial intelligence. and I doubt if we are experts even on practical knots ( we still do not know which Sheet bend is stronger ! ), or on plain intelligence ( we can not agree on anything ! ). :)
     
   
1. Computing with quantum knots. by G.P.Collins   
marcuslab.harvard.edu/otherpapers/SciamTQC.pdf

roo

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Re: Computation Using Knots
« Reply #2 on: January 09, 2013, 06:19:02 AM »
Anybody know anything about computation using knots?

First, one can build things like abacuses.

Second, and more interestingly, I imagine that it would be possible to encode problems into knots and, using characteristics of the knot, find the answer. For example, maybe there is a way to encode some small class of true-false logical statements into knots, and pull on two ends to find the answer: if the knot was equivalent to the unknot, the answer would be "true".

I wonder whether there are any problems that can be encoded that are less trivially isomorphic than "Is this knot equivalent to the unknot?"

This subject, though vague and unlikely to give good methods of computation, though probably tedious and dull, is interesting to me.

With a flat strip and an overhand knot you can approximate a pentagon and its associated angles.  An octogon can also be made per ABoK 2590 or a hexagon per ABoK 2962. I don't know if you had anything like that in mind.   These are pretty limited cases, and don't extend very much into general computation.

Most computational systems, be it electronic, pneumatic, or mechanical, usually need to be able to work quickly.  I'm not sure how that could be pulled off with knots except in some forced, superfluous part of a mechanical computer.
« Last Edit: January 09, 2013, 06:24:10 AM by roo »
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struktor

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Re: Computation Using Knots
« Reply #3 on: January 09, 2013, 07:37:59 PM »
The Egyptians knew that a triangle with sides 3, 4, and 5 make a 90 degres angle.


Rosary

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kd8eeh

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Re: Computation Using Knots
« Reply #4 on: January 10, 2013, 01:29:23 AM »
Actually, this could likely be done.  What we can do is represent all the logic gates with knots that slip selectively, or colapse.  Then, you pull on some ropes and you are left with certain output ropes pulled on.  Of course, the device would not be practical, but i see no reason that constructing logic gates is impossible.  It may likely require a pegboard on the back, or something of that nature, so that friction doesn't skrew things up.

The obvious question follows; Can anyone construct a nand gate or an xor gate that uses ropes as input?  You may be able to replicate a transistor with many slipped knots, and that could help.

There  is a practical application this may have to a mechanical computer: memory.  Storing information on a rope is significantly easier than calculating with them.  I know there was a system developed by the incans to use knots to record information.  A computer could likely use a knotted line and run it through a narrow gap, and when it gets stuck, you have found a knot.  You could easily read and wright to this (wrighting being a bit trickier, but doable mechanically).  Perhaps you could also use slip knots in different directions, and then apply tension to one end, again pulling through the gap.  The system would move in pulses, that could then likely corespond to binary code.

This idea may not be so theoretically bleak as you think.

X1

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Re: Computation Using Knots
« Reply #5 on: January 10, 2013, 02:08:18 AM »
i see no reason that constructing logic gates is impossible.

  Why should a computing machine made by knots be a digital device ? The knots themselves are made by continuous 1D curves in 3D space, so we should think of an analogue computer here, not a digital one.
  In fact, the only way one can surpass the limitations of the very slow motion of any segment of rope, however thin, light or short (in comparison to the speed of an electron or a photon) is to utilize the many degrees of freedom, the many states, the knot-made-computer could offer. So, we should not think of "logic gates", and the corresponding two-value logic, but of something much different. The instruction set can have MANY members, as an invariant knot ( topologically or otherwise ) can be in MANY distinct geometrical shapes/states.
 

kd8eeh

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Re: Computation Using Knots
« Reply #6 on: January 10, 2013, 04:03:37 AM »
This seems impossible, as friction is so unpredictable.  From a theory prospective, the set of all knots is countable, so a digital means expresses perfectly well the amount of information that may be stored.  Perhaps the algorithims for "programing" such a machine may be widely varied by the variance of knots.  Still, i do not see how, with how unpredictable friction is, any truly analog device can be made, at least not in the real world. 

Perhaps i just don't understand your method you intend to use to impliment such a device.  While it is feasable to instead of using binary encoding use base-n encoding for some fixed n, i don't thiink that is what you want.  Perhaps you mean rather to somehow topologically manipulate the knot, and count crossings or something like that.  however, it is very difficult to have a rope count crossings in another rope, so this device would become more of a party trick or a thing to replace an abicus, which itself is easy enough to fassion from rope.

X1

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Re: Computation Using Knots
« Reply #7 on: January 10, 2013, 04:37:22 AM »
the set of all knots is countable

   Why is this so ?  :) Is there a new twist in the generalized continuum hypothesis ?  :)
The set of all knots ( in a dense continuous space) is larger even than what we can imagine - larger than the set of real numbers ( because they themselves are sets of such sets).

   Now, suppose it was, and you could translate everything that a complex (but supposedly countable) large set of analogue states/instructions could encode, starting from a simple reduced small set of digital states/instructions. That would not mean that the later would always be faster than the former, even if the processes/speed of an electronic or photonic computer would be much faster than the processes/speed of a mechanical computer. Starting by pasting already typed long paragraphs, you can sometimes finish the same text faster than typing it letter by letter - however slow is the speed you paste, in comparison to the speed you type.

struktor

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Re: Computation Using Knots
« Reply #8 on: January 13, 2013, 12:48:21 PM »
Modified Stern-Brocot tree (part)     (or modifed Calkin-Wilf tree)
 
           1

          2/1
 
     1            0

   3/2          3/1
 
  1    0       1    0

5/3  5/2  4/3  4/1


System modified Stern-Brocot tree binary-coded


                                 Ai/Bi
 
                  1                                    0

        (Ai+Bi)/Ai                       (Ai+Bi)/Bi




0 - right  ; 1 - left

i - index level

numerator
Ai=A(i-1)+B(i-1)

denominator
if 0 then Bi=B(i-1)
if 1 then Bi=A(i-1)

1    for 2/1
10   for 3/1
11   for 3/2
100  for 4/1
101  for 4/3
110  for 5/2
111  for 5/3

 :) (upside-down) Euclidean Algorithm Using Subtraction Only   :)

Generating coprime pairs
http://igkt.net/sm/index.php?topic=4149.0

Euclidean Algorithm Using Subtraction Only
http://www.naturalnumbers.org/EuclidSubtract.html
« Last Edit: January 13, 2013, 04:57:18 PM by struktor »

X1

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Re: Computation Using Knots
« Reply #9 on: January 18, 2013, 12:40:46 AM »