Kurzweil Rays

Modeling the Singularity is Near

Robert de Marrais

Thothic Technology Partners, LLC

What Ray Kurzweil Says

Kurzweil's astonishingly accurate forecasts of futuristic technologies (predicting the ubiquity of the Internet 10 years before it happened, etc.) are not arrived at by tapping into dream-states or using Ouija boards.  They are, on the contrary, arrived at by a small set fundamental notions, which will inspire our modeling below.

The Law of Accelerating Returns

In the classical LINEAR way of thinking, even exponential growth must taper off, yielding S-curves:  constrained by environmental limits, change is additive.

In the NONLINEAR picture, growth is fundamentally multiplicative.  Here, exponential growth is not cut off by linear constraints; rather, the rate of exponential growth itself accelerates!

The paradigmatic for-instance is Moore's Law:  the price-performance of computation at the onset of the 20th century was doubling every 3 years; when Intel founder Gordon Moore announced his law in 1965, the "doubling rate" was down to about 2 years; now, it's about 1 year -- and shrinking still.

In fact, Moore's law is the "Fifth Paradigm" (but the first to be consciously recognized) guiding the exponential growth of computing.  Locally, each (per the linear bias) may seem governed by an S-curve; but globally, when one starts to exhaust itself, there is a "baton-toss" to another technology, keeping the relay race going.

The "Sixth Paradigm," 3-D molecular computing, is already developing in niche areas, ready to take the baton-toss from integrated circuits, the way the transistor -- whose niche of emergence was found in portable radios -- took the baton-toss from the no-further-shrinkable vacuum-tube (itself succeeding electromechical, then relay technologies).

In fact, Moore's-Law-like paradigmatic "relay races" involving multiple "epochs" (six, in this case) provide the context for understanding the evolution of virtually everything!

Evolution by Indirection

"Evolution works through indirection:  each stage or epoch uses the information-processing methods of the previous epoch to create the next." (SIN, 14)

"...in the third epoch, DNA-guided evolution produced organisms that could detect information with their own sensory organs and process and store that information in their own brains and nervous systems.  These were made possible by second-epoch mechanisms (DNA and epigenetic information of proteins and RNA fragments that control gene expression), which (indirectly) enabled and defined third-epoch information processing mechanisms (the brains and nervous systems of organisms).  The third epoch started with the ability of early animals to recognize patterns, which still accounts for the vast majority of the activity in our brains."  (SIN, 16)

Nonlinearity of Computation Implies Fractality of Design

"A key requirement for a self-organizing system is a nonlinearity:  some means of creating outputs that are not simple weighted sums of the inputs....  Work initiated by Alan Turing on theoretical models of computation around the same time [as early neural-net models] also showed the computation requires a nonlinearity.  A system that simply creates weighted sums of inputs cannot perform the essential requirements of computation."  (SIN, 155)

The simplest local nonlinearity is the threshold where jumps to unanticipated new states occur when some critical value is reached.  The simplest global nonlinearity is based - like Kurzweil's graphics above - on either semi-log or log-log relationships:  the self-similar power laws of fractals providing the figures of stasis for Chaotic attractors analogous to the fixed points of linear mechanics.

“A deterministic fractal is a design in which a single design element (called the ‘initiator’) is replaced with multiple elements (together called the ‘generator’).  In a second iteration of fractal expansion, each element in the generator itself becomes an initiator and is replaced with the elements of the generator (scaled to the smaller size of the second-generation initiators).  This process is repeated many times, with each newly created element of a generator becoming an initiator and being replaced with a new scaled generator.  Each new generation of fractal expansion adds apparent complexity but requires no additional design information....”  (SIN, 46)

“A key aspect of a probabilistic fractal is that it enables the generation of a great deal of apparent complexity, including extensive varying detail, from a relatively small amount of design information.  Biology uses the same principle.  Genes supply the design information, but the detail in an organism is vastly greater than the genetic design information.”  (op. cit.)

As our own research revealed earlier this decade, the Cayley-Dickson redoubling of the Number Line’s Reals to 2D Imaginaries, 4D Quaternions, 8D Octonions, and 16D ZD-harboring Sedenions, necessarily entails the emergence of “meta-fractals”(whose“pixels” map to CPU-indexed dimensions), opening up beginning in 32-D (and requiring endless redoubling to be revealed in their entirety).

Weaving Multiple Threads, Or:  "Surfing is the Mother of Invention"

“As an inventor in the 1970s, I came to realize that my inventions needed to make sense in terms of the enabling technologies and market forces that would exist when the inventions were introduced, as that world would be a very different one from the one in which they were conceived.  I began to develop models of how distinct technologies - electronics, communications, computer processors, memory, magnetic storage, and others - developed and how these changes rippled through markets and ultimately our social institutions.  I realized that most inventions fail not because the R&D department can’t get them to work but because the timing is wrong.  Inventing is a lot like surfing:  you have to anticipate and catch the wave at just the right moment.”  (SIN, 3)

This is the essence of The Kurzweil Method:  trace the “Moore’s Law” evolution (What’s the “doubling window”?  At what rate is that window itself contracting?) of each component technology necessary to your final invention or prognostication.  “Surf is up” at the point where all the pieces can be conjoined - with simulation of those unavailable at one stage providing means of cobbling together the earlier stages’ components in anticipation of that final moment.  [Compare to “superposition of harmonics” in linear approach of Fourier Series and, hence, Quantum Mechanics.]

Like the “crossing of streams” at the climax of the first “Ghost Busters” movie, the “Aha!” moment comes when multiple “Kurzweil Rays” are brought into intersection in the most timely manner.  If this bringing together is an active process, it’s inventing; when, instead, it is a passive analysis of trends indicating pending emergences of new patterns, we call it forecasting.  But the same abstract method guides both.

In our modeling, each such “Ray” is a fractal-harboring display of boxes-within-boxes (where the cells of the fractal proper are those where zero-divisors are absent:  where something concrete, that is, is made manifest).  When the nested boxes are broken out and arrayed along a 3rd axis, each such square comprising the “Ray” harbors one “generation” of CPUs, with the distance between them corresponding to the “doubling window.”

Correlated with fractal emergence in our approach, are “ZD-free” zones - pure copies of the 8-D Octonions, wherein the whole “period-doubling” tactic can be applied afresh.  It is at these sites, called “cowbird’s nests” in our most recent “Complex Systems” monograph, that we can calibrate the “crossing of streams” that guides Kurzweil’s scheme.

As the subtitling of that monograph indicated, the problem motivating it was finding a method for modeling what evolutionary biologists call “bricolage” - a term they in turn borrowed from structural mythologist Claude Lévi-Strauss, which an American audience might well find most readily grasped by a TV-series reference:  it’s “what MacGyver does” with duct tape, pen knife, and whatever stuff is lying around, to solve pressing technologically-surmountable problems.

What Are "Kurzweil Rays"?

“An important meta-idea:  that the power of ideas to transform the world is itself accelerating.”  (SIM, 4)

We need one notion external to Kurzweil’s systematics to set our own apparatus in motion:  a “meta-meta-idea,” about how meta-ideas themselves come about.  I

“Metaphors are learned when two experiences are learned at once.  If a metaphoric link would result in a contradiction in the target domain, it will not be learned.  Neurally, contradictions are mutual inhibitions.  Any would-be link that would lead to a contradiction with the inherent structure of the target domain will be inhibited; thus it will never be learned.”  (Lakoff and Johnson, Metaphors We Live By, 258)

One External Idea (The "XOR Trick")

Translation into mathematics:  XOR relations - the only purely symmetric ones between integers in (bit-string-based) arithmetic - guide all pattern-making.  In particular, they guide the Universal Law of Imaginary Multiplication (ULIM) that must be assumed if we’re to build structures of indefinitely large dimension (hence, unbounded numbers of CPUs - meaning, here, “Cayley-Dickson Process” units!)

This is a relation suggestive of (and quite as fundamental as) that of “Pythagorean Triples” for the Greeks:  “Associative Triples” (henceforth, “trips”) are hypercomplex units whose indices form an XOR triple - i.e.,

(Subscript[i, A]) x  (Subscript[i, B])= (Subscript[i, [A xor B]])

When vectors are “multiplied” (treated as instances of Quaternion units, basically), the familiar “I, J, K” rule is just another way of saying the 3 axes are imaginaries with indices 1, 2, 3:  that is,

(Subscript[i, 1]) x (Subscript[i, 2]) = ( + (Subscript[i, 3])  ) … but  (Subscript[i, 2]) x  (Subscript[i, 1]) = ( -  (Subscript[i, 3]) ) .

The convention is to write such triplets in parentheses as (a,b,c), so that any pair multiplied in cyclical left-to-right order has the positive value of the 3rd unit as resultant, but any pair multiplied in reverse (right-to-left) order results in the negative value of the 3rd.  Further convention:  assume a, b, and c are indices of imaginary units (so that we can suppress all the italics and subscripting usual in working with imaginaries).

Cayley-Dickson Process and Fano-Plane Graphics

With the 8-D Octonions, we have multiple trips in evidence - in fact, 7:

(1,2,3); (1,4,5); (1,6,7); (2,4,6); (2,5,7); (3,4,7); (3,6,5).

All such triples, for all 2N-ions (with N=3  Octonions, of course!) can be built by two simple rules which are the gist of the Cayley-Dickson Process:

Rule 1:  Appending a high-bit (new power of 2) and multiplying prior indices on its left, yields positive-signed units whose index is their sum:  starting with the Real Unit (index = 0) and adding 20 = 1, we get the 2-D Imaginaries; adding 21 = 2, we get the Quaternions’ (1,2,3).  With the Octonions, adding a unit indexed 22 = 4 to the Quaternions’ index- set of (1,2,3), we get 3 of the 6 new trips - with (1,2,3), no longer new, also included (an “obvious” fact, once you think about it, which we call Rule 0):

(1, 4, 5); (2, 4, 6); (3, 4, 7)

Rule 2:  Taking any Rule 0 trip (i.e., any trip inherited from the (2^(N-1))-ions), if we add the new high-bit to both indices in a product-pair, the result will have opposite sign from the product-pair without the new high-bit added in.  This is how we get the other 3 trips in the Octonions:

(1, 2, 3) -> (2+4, 1+4, 3); adding and rearranging yields (3, 6, 5)
(1, 2, 3) -> (1+4, 2, 3+4); adding and rearranging yields (2, 5, 7)
(1, 2, 3) -> (1, 2+4, 3+4); adding and rearranging yields (1, 7, 6)

For the Octonions, with 7 imaginary units, the simplest possible finite projective plane (called the “Fano plane” after the Italian geometer who first explored it) serves to represent the workings of Rules 0, 1, and 2 pictorially... provided we adopt the right conventions!  (Like our "XOR multiplication"!!)  Here's a perfectly valid rendering of the Fano plane as an Octonion multiplication table, for instance, which we can't use -- but which we can take as a starting point for implementing code.

We start things rolling by going to the Wolfram Demonstration Projects, and look at a contribution of Oleksandr Pavlyk.  We see that our "XOR Trick" isn't played along any of the 7 lines!  In particular, the Rule 0  trip is anything but:  (1, 2, 4) instead of (1, 2, 3)!  But Pavlyk's demo is perfectly correct: there are 480 ways to represent Octonion multiplication with the Fano Plane representation:  with the Quaternions, there are only 2 -- (1, 2, 3) or (1, 3, 2); with the 16D Sedenions, there are over 200 billion!

This is why we need our XOR convention (which can be used for any (2^N)-ions!)  One of the first coding tasks to implement our workstation, then:  mutate Pavlyk's code to give us ways to generate Fano representations like those below:

Let' s look at these 4 Fano diagrams in detail, going clockwise from the upper left.  The first shows the Quaternions' Rule 0 trip in the center circle, with the Rule 1 trips running from midpoints, through the center, and bisecting the angles.  The sides being bisected, meanwhile, are Rule 2 trips.  But the label says the center is not G, but S -- not the Generator, but the Strut Constant.  And we' re further told this is in the Sedenions, not the Octonions.  That' s because it' s a 2 - D rendering of a Box - Kite in 16 - D, with two terms at each node, not one.  One term < G (= 8 for Sedenions) and is not shown; it is always automatically derivable, though:  for its index is the low term L xor (G + S).  We will see such Box-Kite computations in a minute.

The second graphic abstracts this situation:  there are 7 different Octonions to pick from that we can put in the center, as the Strut Constant S:  put another way, the high-index term U at any vertext is G plus the strut-opposite of L.  And, in the Box-Kite representation, the 3 nodes making up the side trips and the central "Rule 0" trip are taken to be triangles on an octahedral grid -- and as such, they touch like same-colored checkerboard squares.  The Box-Kite, then, suppresses the visualization of the central node's contents, to focus on these four triangles, whose opposite numbers are empty -- "Vents" through which Sedenion winds blow.

The third graphic puts the G of the Sedenions in the center -- which puts the U terms of the unique Sedenion box-kite with the given "Rule 0" trip in the center in the angles.  Add these terms to the L terms for the same nodes and you get the pair of terms which make the zero-divisors there in the box-kite.  (If we put, not G, but X (= G + S) in the center, we get the U terms of the "Rule 0" units themselves.  So we can generate all 6 U terms for a Box-Kite's nodes by simple graphical means.)

The fourth graphic takes a Rule 2 trip from the side of the third, and puts it in the central "Rule 0" circle of a 32-D (Pathion) Box-Kite.  This is the simplest instance of a new phenomenon:  a "Type II" Box-Kite, with 2 of its 3 struts reversed.  This small difference is the basis of all the fractals in higher-dimensional ZD theory.

How is this possible?  Because "Type I" Box-Kites have a powerful degree of symmetry, manifest in "twisting":  if the units at one node, (U + L), exchange one of their terms with a node they're joined to by an edge on the Box-Kite (U* + L*), the result is a switch to a different Box-Kite, but sharing the same system of low-index terms and the same generator, but with a different Octonion unit serving as Strut Constant.  The seven Sedenion Box-Kites, then, form one brocade ... but with "Type II" Box-Kites, such "keeping things within the family" no longer obtains:  "broken" Box-Kites lacking ZDs entirely can be reached by such "twisting" -- and these "holes" provide the mulch from which fractal structures emerge.

Graphical Narrative:  From Box-Kites to Skies