[Fis] Calculating information content

[Karl Javorszky]

Calculating information content, example DNA



1)      Discussion about beer cans and living objects

Information is an abstract concept which can only be entertained in the
minds of entities that can deal with abstract concepts, usually humans. The
present discussion about whether inanimate objects can exchange information
is comparable to a discussion about whether plants and animals can master
trigonometry. We see that bees build and snails carry geometrical forms and
ferns form fractals, although no one could entertain the motion that they
know how to calculate, using the appropriate algorithms.

The existence of natural laws and the ability to speak about the existence
of natural laws are two different pairs of shoes. In the following I shall
show how information is a part of Nature, regardless of whether we
recognise it or not, like the laws of trigonometry are a part of Nature.


2)      Necessary half-steps

In order to understand the explication that follows, it is necessary that
the reader looks up and understands the following entries in the OEIS
(Online Encyclopaedia of Integer Sequences, www.oeis.org)

a)       OEIS/A000142: the number of permutations of n

b)      OEIS/A000254: the number of cycles in permutations of n

c)       OEIS/A235647: the number of cycles when reordering summands (a,b)
into summands (b,a)

d)      OEIS/A242615: the upper limits for the numbers of commutative and
sequenced assemblies of n objects.

It may be helpful to work through the treatise „Natural Orders – de
ordinibus naturalibus” (ISBN 9783990571378) to have a general overview of
the subject.


3)      Definition

>From “Natural Orders”:

8.3.3.3 Information is a description of what is not the case. [Let *x = a*
*k*. This is a statement, no information contained. Let *x = a**k* and
*k  **<symbol
for is_included_in>  ** {1,2,...,k,...,n}*. This statement contains the
information *k **<symbol for is_not_included_in>* *{1,2,...,k-1,k+1,...,n}*
.]

(Sorry for the included & not-included symbols not making it thru the
simplified  text editor in use here.)

4)      Discussion

4.1.) Comparison with the Shannon concept of information

If the alternatives are restricted to {0,1}, the above definition becomes
trivial, as non-0 = 1 and non-1 = 0. Therefore, the information concept can
only be demonstrated, if there are > 2 alternatives.

4.2.) Limited number of alternatives

If the number of alternatives is ∞, the information content can not be
calculated. Therefore, the number of alternatives must be limited. (n << ∞)

4.3.) Interpretation of a number describing a non-infinite extent

A number is a permutation of a limited number of digits. Let us use for
demonstration purposes a decimal system, where the available digits are
{0,1,2,3,4,5,6,7,8,9}.

The number is traditionally seen as a permutation of digits with the digits
representing a sample with replacement. We traditionally assume, that any
of the available digits can repeatedly appear on any of the positions of
the permutation that yields the number.

4.4.) Samples with a limited number of replacements

We introduce the concept of a hybrid between samples without replacement
and samples with replacement. There are many, but not endlessly many, of
the digits that can appear on any of the positions of the permutation that
constitutes the number.

4.5) Keeping track of the available digits

Assuming a limited number of each of the digits, the total number of
different permutations that can be built from the universe of digits is
limited. We may naively believe that each number can be constructed by a
permutation of samples of digits, because we are used to the concept that
the digits come from samples with replacements, but the number of
permutations of digits that can actually be constructed is severely
restricted by the limits on the number of available digits.

4.6.) Using cycles as limiting factors on the number of digits

Each of the cycles assigns to each of the elements in its corpus one
specific identifying symbol (a digit or a combination of digits). These
identifying symbols are restricted in their number by the number of
elements in the cycle.

4.7.) Information content and degrees of freedom

A limited number of elements can not have an unlimited number of different
sequences, nor can they have an unlimited number of different commutative
relations among each other. The upper limits for both restrict the number
of actually realisable permutations. Those permutations that can not be
actually realised are the information content of the assembly of elements
that already have been sampled. This is analogous to the concept of degrees
of freedom.


Karl
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://listas.unizar.es/pipermail/fis/attachments/20170327/159f2787/attachment.html>