FW: A property of recursion
Hi all, I finally got what I needed on another list so I dont need to be
here anymore - have fun with your project. Chris.
-----Original Message-----
From: Chris Lofting [mailto:chrislofting@ozemail.com.au]
Sent: Saturday, 9 April 2005 6:50 PM
To: Society for chaos theory in psychology
Subject: RE: A property of recursion
Just a small update from this afternoon's readings.... thanks to prompts
from those on the "Society for chaos theory in psychology" list. My focus
has been on identifying the source of how the XOR operator is aiding in
extracting parts information from a set of recursively-derived categories.
Val's reference to laws of form prompted me to go back and start a
're-read' - beginning with Kaufman's "Self-reference and recursive forms".
Whilst doing that I realised that Kate had asked me about feedback.
Without combining the two I would not have 'pondered' more over this casual
"yeh we know that" paragraph of Kaufman's
"the concept of feedback is normally associated with a system that can be
decomposed into inputs and outputs. The feedback is 'feeding back' of (a
transform of) the output into the input. This results in circulatory and
recursive patterns in the system. Under appropriate conditions, the
recursion settles down to a steady form of interaction" p59. J. Social.
Biol. Struct. 1987 10, 53-72
so what?
My original focus here has been on understanding of a discovered property of
the XOR function when applied to categories derived from the recursion of a
dichotomy. The REPRESENTATION of those categories has been in the form of
bit patterns, where each bit reflects some 'input' from a level of
recursion. e.g. 001 means 'nothing from level 1, nothing from level 2,
something from level 3' OR 'integrating from level 1, integrating from level
2, differentiating from level 3'.
We could use wave forms with each level being a marker of cps measured as
2^n cps where n is the level from 1 upwards. This would use
constructive/destructive interference patterns to give us a waveform
representing the quality)
so what?
To **IMPLEMENT** the XOR function in the neurology required TWO neurons in
that we need a feedback system to do it.
IOW the XOR function in our brains ensures, guarantees, DEMANDS, recursion
to be a fundamental property of extracting information from the 'complex',
simply due to the fact that to implement it we need feedback (NOT the case
with AND, NOT etc)
With the recursing of an ASYMMETRIC format (axons = FM (pulse), dendrites =
AM (wave), FM/AM = differentiating/integrating dynamics) that allows for the
application of recursion to itself (using the XOR operator). It is this XOR
methodology that (a) ensures the whole is encoded in all parts, and (b)
allows for the extraction of a particular through the use of masking the bit
pattern of category A with the bit pattern of category B,to bring out a
description, by analogy, of the result in the form of qualities of category
C.
What is indicated here is that the AM data at the dendrites is XOR-ed into
parts, pulse data, by the soma and transmitted down the axon in a REVERSE
ordering of general-to-particular to ensure building of data
context-to-text. Given a context so more information is limited using energy
conservation, we just add 'bits' to the top as needed once the bulk has been
transmitted to set the 'ground'.
Zoom up to the hemispheres of the neocortex and we seem to see this same
sort of FM/AM 'dynamic' but now applied to complex thoughts etc due to the
increase in bandwidth as we move from neuron to whole brain.
From another source, to instigate a memory system you need at the
fundamental level the XOR operator. IOW memory is fundamentally recursive in
form (since XOR requires recursion and memory requires XOR it follows that
memory will be mapped in a recursive manner - which is what IDM covers - the
template for meaning etc)
Feedback means using the past to aid in the perception of the current and/or
imagination of the future. In IDM this is mapped to the 'dimension of
precision'. Here we have:
(a) NO feedback, each moment is 'unique'. representable as
...1,1,1,1,1,1,1,1
(b) feedback limited to the last moment, scale is immaterial. ADD the
moments to get, ...1,2,3,4,5,6,7,8 - a beginning of 'sequence' and the AND
operator - linkage.
(c) go back the last TWO moments - ...1,2,3,5,8,13,21,... the fibonacci
sequence
(d) go back the last THREE to get the tribonacci sequence and so on.
(e) given a FINITE bound, as in a 'beginning', then go back to the beginning
and we get ...1,2,4,8,16,32,64...
In image formats - at the level of the fibonacci sequence we have a cicular
spiral pattern. At the level of the binary sequence we have a square spiral
pattern - inbetween are all of the 'variations on a theme'.
BEYOND (e) and we move into complexity/chaos dynamics (Verhulst, Mandelbrot
et al)
The focus on PRECISION means that each of the above sequences serve as
positions for interpretation of reality -some will do it through patterns of
the fibonacci, most of us do it through patterns of the binary (or
'beyond' - i.e. use of imagination where we put in 'extra' energy that
destablises, allows for 'emergence' etc)
The RECURSION of a dichotomy is covered in the binary - what about the
others? Their recursion is of fractal dimension. The range is 1.6^n to 2.0^n
(less than the fibonacci sequence and there is little or no 'growth'. Beyond
2.0 and we get into complexity etc) For high precision we have to use the
realm of the binary, the exclusive OR. The tightness of the spirals is
determined by what number PAIRS are used out the sequence to create the
spiral. (using the fib pattern as an example - 5/8 is wide, 13/21 tighter)
I still have not reached the EXACT, formal, description of what is going on
but it looks like this:
Given:
000, 001, 010, 011, 100, 101, 110, 111
where these are recursed from the 0/1 dichotomy, if I then apply recursion
to EACH I will get the same patterns I get when I XOR the whole set of
qualities with each. I apply recursion to each by 'bit flipping' - we step
through each quality's representation using binary number patterns. Thus to
'flip' through 000 left-to-right we have:
000, 100, 010, 110, 001, 101, 011, 111
IOW the link of recursion to XOR means I will get the same patterns be it
applying recursion to each quality or just XOR-ing all against one. The
DIFFERENCE here is that the use of recursion just gives me a 'parts list'
ordered by analogy (an AND format), whereas the XOR gives me the X-ness of
the particular property A expressed through context B in the form of an
analogy to property C (since the language for a row is finite and it is
supposed to describe 'all there is', so analogy/metaphor is the 'preferred'
form of description).
I have used 'bit flipping' before to describe things but did not equate the
need for recursion to create the XOR operator in our brains and so the
association of 'bit flipping' (recursion) with XOR-ing.
Chris.