pitfalls of complexity

[Rob Freeman]

Hi Mike,

You give a number of definitions yourself. Which is good. It shows you
are thinking about it:

"complex" = "nonuniform with multitudes of parts"

"specified complexity" = "with a code and system behind it."

"Randomness by definition is the lack of pattern."

I wonder where you get these definitions from? You structure your
argument around them. Do they mean anything? For instance, what does
"lack of pattern" mean?

You are absolutely right we need to think about what we mean by some
of these things. The senses in which I am using "randomness" and
"complexity" are closest to those of Kolmogorov and Chaitin. Wikipedia
will get us started:

"Kolmogorov randomness

Kolmogorov randomness (also called algorithmic randomness) defines a
string (usually of bits) as being random if and only if it is shorter
than any computer program that can produce that string. This
definition of randomness is critically dependent on the definition of
Kolmogorov complexity. To make this definition complete, a computer
has to be specified, usually a Turing machine. According to the above
definition of randomness, a random string is also an "incompressible"
string, in the sense that it is impossible to give a representation of
the string using a program whose length is shorter than the length of
the string itself. However, according to this definition, most strings
shorter than a certain length end up to be (Chaitin-Kolmogorovically)
random because the best one can do with very small strings is to write
a program that simply prints these strings."
http://en.wikipedia.org/wiki/Kolmogorov_complexity#Kolmogorov_randomness

You are astute in that there is an implicit assumption in my use of
these definitions. I am assuming language can be thought of as a
computable process. Some may object. Of course by contrast the
advantage of accepting these definitions is exactly that it does at
least hold out the hope, the first for years, of identifying a
computable process we can use to model language.

-Rob

On Thu, Jul 2, 2009 at 9:59 PM, <Mike_Cahill at sil.org> wrote:
> Dear Rob,
>
> I mostly just listen to people on this list, but can't resist a comment on
> your concluding paragraphs:
>
>      "Another nice thing about randomness as an explanation for homonymy,
>      idiosyncrasy, and most other problems which vex us in language, is
>      that it short circuits the debate on complexity. The idea is that if
>      a
>      system exhibits random patterns it is already maximally complex.
>      Stephen Wolfram calls this "computational irreducibility." He has
>      gone
>      into it quite extensively. Though not for language. His shock claim
>      is
>      that the vast majority of systems are already maximally, and thus
>      equally, complex (in the sense of being universal computers.)
>
>      If that is true and systems exhibiting random patterns, in particular
>      language, are computationally irreducible, then it may not be a
>      question of comparing the complexity of languages and deciding if
>      they
>      become more or less complex over time. The important question may be
>      do they exhibit random patterns. Because if they do they may already
>      be maximally complex."
>
> This connects randomness, computational irreducibility, and complexity.
> Randomness = maximally complex. This of courses hinges on what you mean by
> complex, and there can be at least two ways of looking at that. If an
> entity is nonuniform with multitudes of parts, then it appears complex. But
> that apparent complexity could be the result of randomness (think of sand
> dunes shaped by wind) or it could be what you might call "specified
> complexity," with a code and system behind it (think letters written on a
> beach). The DNA of any organism is quite complex, but is far from random.
> If you randomized the peptide chains that compose it, that molecule might
> still have the appearance of complexity, but scrambled (random!) DNA is
> unsystematic and useless.
>
> A specified complexity would be amenable to analysis and computational
> treatment, while random complexity would not. So I'd agree with the
> statement that "systems exhibiting random patterns ... are computationally
> irreducible". But the part I left out, in the ellipsis ("in particular
> language") is what the point is. I'm not so sure that randomness short
> circuits the debate on complexity as much as avoids it.
>
> Finally, when you state that
>      And why is randomness so important? Because paradoxically it allows
>      us
>      to find more patterns (if patterns are regular then rules limit how
>      many we can find.)
>
> I'm wondering what you mean by randomness. Randomness by definition is the
> lack of pattern.
>
> A nice exercise to start the morning with.
>
> Mike Cahill