Jim:
...maybe we should take a look at the reason why these methods are so powerful.
And they are powerful.
Boris:
Compressive power of digitization is inversely proportional to the magnitude of
a base, that is, directly proportional to the number of incremental-power
digits used. It's a form of comparison to a domain-invariant comparands:
digits, where the difference is carry. The power comes from substitution of
multiple "unary" inputs with a single higher-power digit. Digital addition
*conserves* that compression, but doesn't increase it.
Same for multiplication: it only conserves compression generated by prior
comparison of the addends, if they're found to be identical. Again, the power
comes from substitution of multiple identical inputs with a single higher-power
multiplicand. It's different from digitization only in that the comparand here
input-specific, not domain-invariant.
This is my core point again, - compressive power always comes from inverse
operations: digitization, individual comparisons, then division (iterative
comparison) & so on. Direct operations: digital addition, multiplication,
elevation to power... only conserve that power by avoiding decompression.
From: Jim Bromer
Sent: Sunday, August 12, 2012 5:17 PM
To: AGI
Subject: [agi] Addition and Multiplication Are the Engines of Computation
The Positional Notation System of numbers, like Binary and Decimal numbers, is
a representational system which is a compression of the most natural way to
represent a count, the unary system where a single mark is used to denote each
individual item that is being counted. This Positional Notation system has a
complexity value of a lossless exponential increase in efficiency for each
digit of the binary representation. The method of addition, when converted
into a true Boolean Form almost certainly has an exponential increase in
complexity for both the number of bits of the addends and the number of
addends. Multiplication represents an exponential increase in efficiency over
the method of addition for that special class of addends which represents one
multiplicand being added over and over again by the number of times represented
by the value of the other multiplicand. These standard algorithms of addition
and multiplication are both lossless. Algebra, which might represent one of the
earliest programmable systems imagined, is so effective just because you can
use addition, subtraction, multiplication and division of the coefficients of
the literal variables of an algebraic statement.
So the binary system, addition and multiplication are really the engines of
computation. I believe that the reason these methods are so powerful is
because they can use extremely efficient compressed representations of numbers
without needing to decompress them everytime they are used. I have tried to
come up with some kind of terminology to represent this and I have suggested
that addition and multiplication are procedural compression methods and
transformational compression methods because they are able to use compressed
data in its compressed form. Perhaps I should say that they are procedural
methods that can act on a kind of compressed data.
The reason why I mention this is because it may help to better define what is
needed to make AGI feasible. I believe that many schemes which have used
efficient numerical methods on objects of AGI have failed because they really
did not adequately model the kinds of things that need to be modeled in an AGI
program. So my thesis is that if we cannot just throw numerical methods at AGI
programs and make them work, then maybe we should take a look at the reason why
these methods are so powerful. And they are powerful.
Jim Bromer
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