Discrepancies between APL-TOE (¬\) and other theories
G.A. Langlet
Some hypotheses of APL-TOE [LanA] disagree with generally-admitted
postulates of neural-circuit or molecular-computing theories, so that, as a
complement to the papers submitted to APL94, a few points, which needed
clarification, are reviewed here.
(Quotations are italicised and embedded between “…”)
"Neurons ARE continuous
dynamical systems, and neuron models must be able to describe smooth, continuous
quantities such as graded transmitter release and time-averaged pulse intensity " [Hopf]
p. 626.
Neurons are made of molecules in which interacting entities are
electrons; an average, between an electron and a "no electron", has
no physical meaning. Never forget that,
even in classical mechanics, averaging two opposite forces which are not
applied to the same point, would lead to ignore the rotating action of a couple
(or the effect of a dipole in magnetism); moreover, it is dangerous to
postulate, especially for "micro-forces" which correspond to
"micro-actions", that two opposite actions are necessarily
simultaneous.
"The input currents of all
synaptic channels are simply additive" [Hopf] p. 627.
They cannot be additive, as individual-electron sequences, except modulo
2, and with timing : successive applications of
¬ within an information processor, e.g. a chain of alternate single and
double chemical bonds, which would perform a "¬\ like" algorithm, again, cannot be simultaneous; hence the general
disagreement :
"We refer to the dynamics
as "classical" because we are ignoring propagation time delays and the quantal nature
of quantum potentials, in analogy to classical mechanics. "
[Hopf] p. 633, in ref.12.
"Energy functions, for
minimization, appropriate to each problem, have to be constructed
" (a general
postulate).
Minimization, seen in modulo 2 integer algebra, simplifies : because of
the property of either 0 or 1, raised to any positive power p, to be...
idempotent, non-linearities disappear; then, a least-square method becomes a least-any-power
method, which will not depend on the choice of any exponent p (such as 2) : the
¬\ algorithm,
when its iterates are seen as a whole, tries to equalize parity-neighboring; it
is, per se, an ideal, then general, optimizer.
One should bear in mind that the modulo 2 integer matrix product, expressed in
APL as
¬.^ can also be written
¬.˜ because
¸×¾ in Z/2Z is also
¸˜¾. The cognitive transform of any
binary signal B which
is defined as the catenation of the successive iterates of
¯1†B„¬\B can also be obtained as a matrix product (convolution) of
B by a Sierpinski conforming matrix (a mirrored geniton); due to the fact
that the Sierpinski matrix is the most perfect self-similar fractal one can
imagine, with a sparsity* (the ratio of 0-terms) which drastically increases as
a function of its size, the
˜
component of the matrix product always selects the minimum item between a
sparse row or column of the fractal matrix, and the corresponding item of
sequence
B. Cognitive- or helix-transforming rearranges information, optimizes it,
and compresses it when it is periodical.
All this is a consequence of ¬\ being the Law electrons like to play with, as explained in one of the
papers [LanE].
"[In switching devices], switches
recognize minimal patterns of ones and zeros, like the patterns recognized by the familiar AND, OR and NOR gates (called
"logic gates" because they
conform to mathematical logic operations). "[Conr]p.56
If AND, OR and NOR gates are "familiar", they are also able to
pollute information. With the NAND gate
(also discussed in some essential papers, by Forrest L. Carter, on cellular
automata and "Molecular Device Computers") and the ones expressed in
APL by symbols < < > > , these Boolean gates (14 out of 16 possible
ones altogether) are not necessary in biological models which have to exhibit
"self-organization" : Boolean gates, with the exception of ¬ known as XOR outside APL, and
of its opposite NXOR (alias EQUAL =), are all, at least partly, Gödelian, i.e.
undecidable : they may add entropy (noise) to information they process, because
it is impossible, in the general case, to reconstruct the original information
from information having being submitted to such gates; the less familiar XOR
and the "unknown-outside-APL" XOR_SCAN are not-polluting.
*Note of the redactor: the author means to say scarcity.
Idiom
¬\ self-encrypts any information into itself as a key, is reversible, (its
in-verse is simply the Gray-code function, in fact a useless function, because
of the rules of periodicity that govern
¬\ when iterated as a multi-soliton wave, for finite information sequences,
even a whole human genome).
¬\ is "isentropic", compatible with the
way entities with self-organization capabilities may "live", with NO
production of entropy. As an algorithm, ¬\ is the "optimum
optimorum" one can imagine for a perfect reversible computer, performing serial-to-parallel
conversions of information and conversely, producing the equivalent of a FFT
and its inverse as well, solving breathtaking puzzles, the solution of which
would require the exact inversion of huge matrices (such problems lead, with
conventional arithmetic, to intractable large systems of high-order
differential equations that our present-time computers could NOT solve, because
of the obliged truncations of floating-point processors, the best
entropy-producing tools that were - unfortunately - invented and used by Man
during the second half of the 20th century).
About reversibility, see [Fred] and especially Baker [Baker] who
reinforces the properties of
¬\ (though he seems to ignore APL and the idiom); this author starts from
ideal considerations on Newton's method for square-root finding, and about data
sorting (“ and
““ are indeed connected to ¬\ ), and proposes a "reversible programming language" (Y-LISP).
"The world of computing is
divided into two radically different domains.
The classical
world achieves programmability at the expense of evolutionary adaptability and computational efficiency.
The biomolecular world is not programmable..." etc... [Conr] p. 57.
Algorithmic compression, as well as theoretical derivations have first
led to emit a conjecture (1989), then to prove, that ¬\ must be the nucleus of all
algorithms; so, the subtle but arbitrary distinction between programmable
devices and not-programmable ones will vanish : any man-written computer
program which uses, as a scalar function, anything else than ¬ (XOR gates) is not an optimal
version of the algorithm : algorithmic compression has not been completed yet;
conversely, biomolecular structures SHALL use (they have no choice) the main
property of electrons, which indeed corresponds to a ¬\ model, at the quantum level.
A subtle distinction between binary computing on one hand, and tactile
recognition of the shape of a molecule by another molecule like a key in a
lock, on the other hand, also collapses, once useful binary computing has been
proved to be reducible to the use of ¬ and ¬\ alone :
the role that is played by shapes was investigated [Lans] especially to show
that ¬\ can
produce shapes that one indeed observes in Nature (even if the model remains
bi-dimensional); then, the study of "tactile recognition" in
connection with learning and vision, was paradoxically undertaken, thanks to
the universality and the efficiency of the Braille alphabet for the blind [LanB]:
Tactile recognition (first learning how to read with fingertips, then
reading just for pleasure or, recursively, with an education goal, of learning
Latin, music, mathematics or... the Morse alphabet) is performed by blind
people on very simple binary matrices, scanned in parallel along three
rows. When one looks
at blind people's fingers running on the lines of their Braille books, one SEES
¬\ at work, directly.
References
[Baker] Henry G. Baker, "NReversal of fortune -
The Thermodynamics of Garbage Collection", Proc. of Int'l. Workshop on Memory
Management, St
[Conr] Michael Conrad, "The Lure of Molecular Computing", IEEE
Spectrum (Oct. 1986).
[Fred] E.Fredkin & T. Toffoli, "Conservative Logic", MIT
report MIT/LCS/TM-197 (1981) or Int. J. of Theor. Phys. 21, 3/4, (1982) p. 219.
[Hopf] John H. Hopfield & David W. Tank "Computing with Neural Circuits : A Model", Science, Vol. 233, (
[LanA] Gérard A. Langlet "Towards the Ultimate APL-TOE", APL
Quote Quad, Vol. 23, No 1 (July 1992); APL92,
[LanB] Gérard A. Langlet "From the Braille Alphabet for the Blind to
the Genetic Code", submitted to Apl94 (Sept. 1994; B-Antwerp).
[LanE] Gérard A. Langlet "An APL Game for the Electrons",
submitted to APL94.
[LanS] Gérard A. Langlet, "Building the APL Atlas of Natural
Shapes", APL Quote-Quad", Vol.23, No 1 (Aug. 1993); APL93,
CDN-Toronto.
[LanV] Gérard A. Langlet, "The APL Theory of Human Vision",
submitted to APL94.
[Marr] D. Marr; "
Vision", Freeman
& Co, San Francisco (1982).
[Str] Lubert Streyer, "Biochemistry”, Stanford
University, W.H. Freeman & Co.,
SCM/LIT : Service de Chimie Moléculaire, Laboratoire d'informatique Théorique (Service of Molecular Chemistry, Laboratory for Theoretical Computer Science, French AEC).