Symbol processing

[gerry at sees.bangor.ac.uk]

> I have some thoughts about symbol processing I'd like to discuss.  These
> thoughts arose from the desire to develop a system which would accept
> a stream of symbols and display a compressed but meaningful and
> maniputable display of the stream.
>
> It's very tempting to me to assign each symbol a unique integer and start
> examining the stream for structure using math techniques such as Fourier
> Analysis or Wavelets.  But as soon as you make the assignments, with the
> intent of using these math techniques, you are assigning meaning you
> probably had not intended to do.   For example, by assigning two symbols
> consecutive integer values, you are saying these symbols are similar two
> each other.  When you apply math operations to these symbols like
> averaging the values of two symbols, you wind up with a number which
> is meaningless because the initial values were arbitrary.  Consequently,
> I believe one should focus on the probabilities that a symbol or group
> of symbols will appear in a stream.
>
> I'm sure Fourier Analysis and Wavelets will be valuable tools in this
> endeavor, but they have to be used appropriatly of course.  Fourier
> techniques work best on data which is highly repetitive, while wavelets
> are better able to deal with discontinuities.
>
> If you are interested in learning about wavelets here are some resources:
>
> 1.  http://www.wired.com/wired/3.05/departments/geek.page.html.  This
>     is a short introduction to wavelets.
>
> 2.  A longer introduction can be found in a postscript paper written by
>     Colm Mulcahy called "Plotting and Scheming with Wavelets" at
>     http://www.spelman.edu/~colm/papers.html.
>
> 3.  A well written introductory book that explains the mathematics of
>     wavelets is the second edition of "The World According to Wavelets" by
>     Barbara Burke Hubbard.

Dear All,

Re Chuck's last posting, I like the general idea he has sketched about
a system which can compress a stream of symbols and display a graphical
and manipulable representation of the resulting knowledge structure.

Re Chuck's remarks about compression techniques, I'll put down a few thoughts
in that area along the lines I discussed with him before:

If one's primary interest is in practical techniques for compression, then
there is no doubt that concepts like 'wavelets', 'fractals' and others
can be useful and relevant.

But for my own work, I have (so far) steered clear of concepts like
these because they seem to exploit properties of pre-established
mathematical concepts. By contrast, I am trying to build a theory from
'primitive' ideas like matching symbols and unifying symbols and I
don't want to muddy the water by using relatively complicated,
ready-made mathematical concepts.

MATHS AS A MEANS OF COMPRESSING INFORMATION

I think it is interesting that mathematics can often provide a very
effective means of compressing information. For example, a formula
like s = 0.5 g.t^2 expresses in a very compact form the relationship
between distance and time for a free-falling object. Without the formula,
you would need a very large table of distances and times.

So it seems that it is no accident that "mathematics is the language of
science", given that science is, generally speaking, trying to find
succinct (compressed) descriptions of the world.

Why or how should mathematics have this ability to compress information?
For me, a working hypothesis is that many, or possibly all,
of the devices used in mathematics may be understood in terms of the
kinds of pattern matching ideas that I am exploring.

If that turns out to be true, then it may provide an explanation of
why or how the various 'mathematical' techniques for information
compression work as well as they do. It should be possible to trace a
link from the kinds of primitive operations that I am using
to the workings of wavelets, fractals, linear predictive coding and
other mathematical techniques for compression.

MATHEMATICS AS "NOTHING BUT" MATCHING AND UNFICATION OF PATTERNS?

I have to admit that I have not got very far in demonstrating that
at least some parts of maths may be understood in terms of matching patterns.
I'll say something briefly about four things which seem to be relevant:

1 Understanding Turing machines in terms of information compression by
multiople alignment, unification and search (ICMAUS).

At http://www.sees.bangor.ac.uk/~gerry/sp_summary.html, there is
a downloadable script called "Information compression by
multiople alignment, unification and search as a general theory of
computing".

In this article I have tried to show that a 'Post Canonical System'
can be understood in terms of ICMAUS. Since it is known that the PCS
model is equivalent to the Turing model, the article is arguing indirectly
that the Turing model of 'computing' may be understood as ICMAUS.

2 Another article at that site is "Logic as information compression by
multiple alignment, unification and search" which argues that logic
can be understood as ICMAUS.

Since logic is often regarded as the foundation of mathematics, this would
argue that ICMAUS is a foundation for mathematics.

3 The simplest kind of number system is the 'unary' system where zero is '0',
1 is '1', 2 is '11', 3 is '111' and so on.

This number system can be defined with two 'rules' like this:

     $ -> 0
     $ -> $1

The second one is recursive which allows the rules to 'generate' the 
infinite set of unary numbers. In the article mentioned above about Turing
machines, I have shown how the two rules can be modelled
as ICMAUS.

4 It is possible to model the workings of a 1-bit adder by means of a simple
table. Since any table can be modelled with ICMAUS, it is possible to
model the 1-bit adder with ICMAUS. The same goes for 2-bit adders etc but
the tables get rather large.

That's all for now. I hope there is something of interest there.

Gerry Wolff