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It's worth noting that information, in Shannon and Weaver's definition,
does not equal 'meaning'. That's why it seems 'counterintuitive'.
"The fundamental problem of communication is that of reproducing
at one point either exactly or approximately a message selected at
another point. Frequently the messages have meaning; that is, they
refer to or are correlated according to some system with certain
physical or conceptual entities. These semantic messages are irrelevant
to the engineering problem. The significant aspect is that the actual
message is one selected from a set of possible messages."
The Mathematics of Communication - Weaver and Shannon, 1949
You could contrast this to the Boltzmann entropy where entropy is
a measure of disorder and is related to processes that are 'irreversible'
(not time-symmetric). This is related to 'addressability'.
As the article cited below notes, this is an essential aspect of
'distinctness' when all possible states are equally probable, a condition
Shannon asserts.
While a process appears to be reversible in theory, in practice, it can
be quite difficult or require so much precision as to be impossible.
Explosions are given as examples. The process is reversible as long
as particles don't interact. Entropy in this view is related to the
number of possible states of an isolated system. As systems interact,
the number of states becomes logarithmic. Are all states equally
probable given the initial states? If not, is that what 'meaning' means,
and is it a fundamental requirement for time?
So is the point that schemas increase the value of information
(make it meaningful) by reducing the number of potential states
regardless of the length (Cantorian madness in two steps)? Keep
in mind that apriori information is 'meaningful' and that in a
temporal system, the schema is apriori.
http://www.mathpages.com/home/kmath552/kmath552.htm
And you may also be a few steps from the 'why SGML is a better
design than XML' thread where SGML provides more entropy-reducing
constraints and XML is designed to cope with human laziness by
enabling more errors/interpretations/implementations of the
same information thus increasing the information by reducing
the meaningfulness. The puzzle you have to solve is that
this is done to enable local meanings instead of relying
on a global definition.
len
From: Roger L. Costello [mailto:costello@mitre.org]
Hi Folks,
I am trying to get an understanding of Claude Shannon's work on information
theory. Below I describe one small part of Shannon's work. I would like to
hear your thoughts on its ramifications to information exchange using XML.
INFORMATION
Shannon defines information as follows:
Information is proportional to uncertainty. High uncertainty equates
to a high amount of information. Low uncertainty equates to a low
amount of information.
More specifically, Shannon talks about a set of possible data.
A set comprised of 10 possible choices of data has less information than
a set comprised of a hundred possible choices.
This may seem rather counterintuitive, but bear with me as I give an
example.
In a book I am reading[1] the author gives an example which provides a nice
intuition of Shannon's statement that information is proportional to
uncertainty.
EXAMPLE
Imagine that a man is in prison and wants to send a message to his wife.
Suppose that the prison only allows one message to be sent, "I am fine".
Even if the person is deathly ill all he can
send is, "I am fine". Clearly there is no information in this message.
Here the set of possible messages is one. There is no uncertainty and there
is no information.
Suppose that the prison allows one of two messages to be sent, "I am fine"
or "I am ill". If the prisoner sends one of these messages then some
information will be passed to his wife.
Here the set of possible messages is two. There is uncertainty (of which
message will be sent). When one of the two messages is selected by the
prisoner and sent to his wife some information is
passed.
Suppose that the prison allows one of four messages to be sent:
1. I am healthy and happy
2. I am healthy but not happy
3. I am happy but not healthy
4. I am not happy and not healthy
If the person sends one of these messages then even more information will be
passed.
Thus, the bigger the set of potential messages the more uncertainty. The
more uncertainty there is the more information there is.
Interestingly, it doesn't matter what the messages are. All that matters is
the "number" of messages in the set. Thus, there is the same amount of
information in this set:
{"I am fine", "I am ill"}
as there is in this set:
{A, B}
SIDE NOTES
a. Part of Shannon's goal was to measure the "amount" of information.
In the example above where there are two possible messages the amount
of information is 1 bit. In the example where there are four
possible messages the amount of information is 2 bits.
b. Shannon refers to uncertainty as "entropy". Thus, the higher the
entropy (uncertainty) the higher the information. The lower the
entropy the lower the information.
QUESTIONS
1. How does this aspect (information ~ uncertainty) of Shannon's work relate
to data exchange using XML? (I realize that this is a very broad question.
Its intent is to stimulate discussion on the application of Shannon's
information/uncertainty ideas to XML data exchange)
2. A schema is used to restrict the allowable forms that an instance
document may take. So doesn't a schema reduce information?
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