Some Implications From The Theorem Of Pseudo-Periodicities
by
Huen Y.K.
CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A short communication - 1st released: 3/1/98. Revised 3/1)
Abstract
A pseudo-periodic number sequence is one which can only be described partially
by generating functions which in turn can only be completed by additional fudging terms
or generating functions [3]. The Theorem of Pseudo-Periodicities essentially proves that
in any binary
encoded string, sequence formed by one symbol can be curve-fitted by an exact generating
function but that belonging to the other symbol will have to be fudged. Since the
choice between the two symbols is arbitrary, at least 50% or greater of the number of
bits in
a string can be fitted by a pure generating function. Depending on
the nature of the original binary string, the percentage of perfect fit can range from 50% to
almost 100%. This imperfection can be explained in terms of the Second Law of
thermodynamics. This paper examines some implications arising from the
Theorem of Pseudo-Periodicities.
1. Introduction
Claude Elmwood Shannon pioneered modern information theory in 1948[1].
Shannon defined the rate of a language as:
..........................................H(M)
..................................r = ------- ........................................(1),
............................................N
in which N is the length of the message and H(M) is its entropy. All forms of
energy will eventually be downgraded to heat. An idea which is now not so popular
is that eventually the universe will reach perfect heat equilibrium .. a state of heat
death. However there do exist in this universe pockets of orders surrounded by
oceans of chaos or disorders. Life scientists propose that lifeforms are
capable of increasing the order of materials whilst at the same time generating
wastes[1]. This is in accordance with the Second Law of thermodynamics. The
author discovered via the Theorem of Pseudo-Periodcities that human languages
obey this law [2]. Any attempt to increase the order of part of a binary encoded string
will result in increased disorder in the remaining part so much so that
it is impossible to find a pure generating function to describe such binary strings.
However there are exceptions to this. There are number systems which are
so self-consistent that its entropy is practically zero. The author speculates that if such
number systems are used as human languages, there will be no uncertainties in meanings
and it will be a cryptographer's dream in view of its very high
compressibility. Imagine Shakespear's "Macbeth" being replaced
by a single generating function in sequence algebra which can then be
transmitted via the Internet and recovered by series expansions at the
receiver's end.
The assertion that the natural number system has zero entropy needs qualifications
as it seems to violate Godel's Theorem of Incompleteness [3]. Probably Godel's
Incompleteness theorem is also a consequence of the Second Law of thermodynamics.
In truth the entropy of the natural number system is almost zero but never reaches
absolute zero. This is because of the use of the symbol of infinity in the number
system. Sequence algebra deals with holistic number sequences. Integers never
exist in isolation but as members of a global set. All integers in the natural number
system are concommittent, i.e., properties result from the synergistic contributions of
all integers in the global set. The natural number system is very
much like our living body as both are composite systems. Life scientists assert
that unicellar protists eventually evolved
into multicellar protocists which ultimately became animals, fungi, and plants.
Life is the grand summations of
the little purposes of cooperating cells [2]. If integers are taken as cells, then the natural
number systems and its properties are the results of the total contributions from these
individual integers. In fact the equation of divisibles says as much (see equation (1)
below) [4].
..................................................ub
................................................-----
.................................................\.........1
................................Comp(z) := ) ----------- ..........................................(1).
................................................./.........i...i
................................................----- z (z - 1)
.................................................i = 2
The above equation states that the global composite number sequence is the
superposition of infinite number of pure periodic generating functions with monotonically
increasing intervals. Every composite number can only come about due to the
total contribution of integer divisors from 2 to the number itself. One can use the
world's most powerful computer to expand the above generating function but one
will never be able to complete the sequence even when the universe collapses
in heat death. The use of infinity in mathematics is unfortunate as it introduces
uncertainties. Suffice it to say that the entropy is very small and even if it is finite,
the use of Shannon's equation of the rate of language as given by eqatuion (2) will
return the value of zero [1]:
..........................................H(M)
..................................r = ------- ........................................(2),
............................................N
in which N is the length of the message and H(M) is its entropy.
2. Pseudo-Periodic Binary Sequences
For simplicity of discussions, we concentrate only on binary sequences since all
information can be encoded in binary ascii codes. The Theorem of Pseudo-Periodicities
might give the impression that these heavily fudged generating functions are quite
useless as fudging does not reveal fundamental principles. In fact entropy is a
double-edged sword. We discuss below the implications of the above theorem.
(i) Theory Of Everything (T.O.E.)
Big-Bang theorists think that they are on the way to discover T.O.E. the theory of
everything. From experimental and theoretical investigations, they are trying to
find "The Equation". This is an impossibility. Even localised information describable in binary
strings can never be curve-fitted exactly by an equation
or a group of equations, much less global information. In fact finding an exact description of one part will create
incompatibililty in the remaining parts which is a common experience amongst scientists and mathematicians.
It is the Second Law of thermodynamics at work. This exercise might be misguided.
(ii) DNA-sequences
Molecular biologists think that DNA-sequences contain all the information
for metabolic chemistry of life. Since DNA-sequences can also be reduced to binary
strings, again we will be stone-walled by the Theorem of Pseudo-Periodicity.
One will never be able to develop an equation or an explanation system which
will predict all the information from DNA-sequences. If we assume that the 50% or
more of the contents can be described exactly by an equation, then all it says
is that we will succeed to unravel at least half or more of its mysteries. The
rest will wallow in uncertainties. Fortunately life scientists do not harbour the
same pretension as the Big-Bang theorists.
(iii)The Natural Number System
Is it true that the natural number system is the only system with zero
entropy? That will depend on whether one is a conventional number theorist
or a sequence algebraist. Sequence algebra is the algebra of holistic number
sequences. To state that z/(z-1) is the closed formulation for Nat(z), there is
no uncertainty here. There can only be one meaning to this formulation. That
is why entropy is zero. On the other hand how comes there are so many
unsolved problems in number theory? The answer is that sequence algebra
is a first order linear theory of holistic number sequences. Within this domain
all sequences have closed deterministic forms. What cannot be described by
sequence algebra turns out to be nonlinear problems. Even here sequence
algebra is making progress such as the prediction of amicable pairs [3]. In posing
a number theoretic problem, one must declare whether it is linear or nonlinear.
After all we have plenty of nonlinear mathematical problems outside number
theory itself which are still unsolved.
(iv)Cryptographic Systems
Cryptography is probably the only discipline which could benefit from entropy.
Messages with zero entropy are easy to crack. If binary encoded message are
encoded as pseudo-periodical generating functions, the amount of entropy can
be quantified. This is a useful measure which enables crytographers to design
encryption systems with controllable uncertainties. Just have a look at the
generating function given by equation (3) and you will appreciate what can be done
to befuddle code crackers:
.............................................1...........1
......................Crypta(z) := ------ + ------ + Fudging(z)..........................(3),
............................................i.............j
..........................................A - 1.....B - 1
where A and B represents the two binary symbols, i and j the intervals and Fudging(z)
contains terms which take care of uncertainties or entropy. If the three terms are transmitted
along different encrypted channels, these will be meaningless when intercepted. Only
when these are combined and series expanded does one get a decrypted
message. To make things more difficult, the original binary message could be
encrypted by a conventional encryption algorithm before one applies the pseudo-periodic
algorithm. Obviously if entropy is high, one might get a very long expression for
Fudging(z). Since the first two terms are very compact, a longer string for Fudging(z) is
not a big issue. In fact a longer string in the third term means that one could
apply pseudo-periodic curve-fitting a second time making the encryption even more
confusing. Get the picture?
(v) Data-Compression For Bitmaps
Instead of compressing bitmaps by conventional methods, how about doing it by
pseudo-periodic functions? Remember that since all generating functions have uniform
algebraic structures, one needs not encode the expressions in full. All you
need to transmit are the intervals such as the value 5 in the generating function 1/(z^5-1)
since series
expansion software at the user's end can be programmed to understand such
information. The software will repaint on the screen three times with the first two
time using periodic bit sequences and the picture is touched up using the bits from
the Fudging(z). This method is suitable for black and white bitmaps but with
some imagination one could develop an algorithm for multicoloured bitmaps.
(vi) Intergalactic Radio Transmissions
This sounds far-fetched but there were programs to sent messages to distant civilisations and it was hoped that
such messages could be understood by those who intercepted the messages. There is
no guarantee that the aliens will turn on the radio at the right time. If one is turned on
during the middle of a message, it is not useful. Messages written in a periodic
language is more useful here. If you want the alien to know that you know sequence
algebra, you should send only the equation of divisibles repeatedly as 1/(z^i*(z^i-1))...
1/(z^i*(z^i-1))...1/(z^i*(z^i-1))...1/(z^i*(z^i-1))... . An intelligent alien will be able to
decode this message far better than one in spatial language such as: "Hello, the composite
number sequence is 4, 6, 8, 9, ...........".
6. Conclusions
The Theorem of Pseudo-Periodicities has some disturbing implications for scientists
in search of the Theory of Everything. Godel's theorem of incompleteness already
warned against this but scientists paid little attention to it for political reasons. Now
the present theorem echoes Godel's thesis. It seems that God introduce entropy
into this world to make sure that Man will not outsmart Him. Kepler reminds us that
science is asymptotic: it never arrives at but only approaches the tantalizing goal of
final knowledge [2]. Even if Man has only acquired 1/10th of the total knowledge
known at that time, he would have landed in the moon. The assumption that once we
have T.O.E. all learning stops is a farcical one. From the vast ocean of knowledge
you could do wonder with a miniscule part of it if used wisely. Do you know that
the world's eminent violinist Fritz Kriesler had a very small repetoire and he practised
only 4 hours a day compared to
some modern violinists? In this world we need not know everything but what we
know we have better know it well and use it wisely.
7. References
Comments: Not all references in this list are directly referred in the main paper.
These are provided to readers as background papers in sequence algebra. These papers
can be easily hyperlinked whilst you are in the web.
1. Schneier Bruce: Applied Cryptography, Protocols, Algorithms, and Source Code in C, chapter 11,
pp 233 to 234, Wiley (2nd ed.), 1996
2. Margulis Lynn and Dorion Sagan (1995): What Is Life? The Orion
Publishing Group, Printed in Italy, pp12-33.
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3.
Is A Periodic Language A Pipe Dream?
- Huen Y.K. (Date Released 2/1/98, 24 Kbytes)
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4. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci.
Technol., 1994, VOL. 25, NO.6, pp 913-920.
5. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ.
Sci. Technol., 1996, VOL.27, NO. 5, 685-691.
6. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1996,
VOL.??, NO.?, ???-??? (In the press as proof paper mes 100421).
7. The twin prime problem revisited, INT.J.MATH.EDUC.SCI.TECHNOL.,199?,VOL.??, NO.?,???-???, proof paper
mes-0488 (10 pages).
8. Is Pie Periodic?, INT.J.MATH.EDUC.SCI.TECHNOL.,199?,VOL.??,NO.?,???-???, (in the press).
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9. Evaluations Of Normc( ) Function
In Macsyma 2.2
- Huen Y.K. (Date Released 17/12/97, 14 Kbytes)
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10.
List Processing In Sequence Algebra
- Huen Y.K. (Date Released 23/12/97, 20 Kbytes)
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11. A Simple Introduction To Sequence
Algebra - by Huen Y.K.
(date release: 15.3.97) (38 KBytes, 11*A4 pages).
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12. The Canonical Generating Function
or CGF(z) ... - by Huen Y.K.
(date released : 27.5..97) (24 KBytes, 7*A4s).
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13. Visual Solutions Of Number Theoretic
Problems ..... - by Huen Y.K. (date released : 3.6.97) (38.3 KBytes, 10*A4s).
========================================================
14. Final Value Theorem Applied To Number
Sequences... - by Huen Y.K. (date released : 5.6.97) (29.4 KBytes, 9*A4s).
========================================================
15. Unsolved Problems In Sequence
Algebra - by Huen Y.K. (date released : 6.6.97) (29.4 KBytes, 9*A4s).
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16. Methods Of Developing Sequence
Algebraic Formulations For Comp(z) and Prime(z) - by Huen Y.K. (date released : 20.6.97) (36.8 KBytes, 10*A4s).
========================================================
17. Composite Number Sequence
Challenge 1/97 - by Huen Y.K. (date released : 28.6.97) (24.8 KBytes, 7*A4s).
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18. Lemmata, Corollaries, And
Theorems In Sequence Order Analysis. - by Huen Y.K. (date released : 6.7.97) (38.3 KBytes, 12*A4s).
========================================================
19. Improved Formulations For Comp(z)
and Prime(z)
- by Huen Y.K. (date released : 16.9.97) (17 KBytes ).
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20. Detecting False Reports
in Primality Tests By The Oddcomp(z) Method.
- by Huen Y.K. (date released : 18.9.97, Revised 20/9) (26 KBytes ).
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21. The Throwing Power Of
Oddcomp(z).
- by Huen Y.K. (date released : 24.9.97 ) (15 Kbytes).
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22. Sequence Algebraic
Approach To Prime Number Theorem
- by Huen Y.K. (date released : 28.9.97 ) (21 Kbytes).
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23. Generating Functions -
Closed Forms vs Open Forms
- by Huen Y.K. (date released : 1.10.97 ) (21 Kbytes).
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24. Generating Large
Odd Composite With Two Prime Factors
- by Huen Y.K. (date released : 3.10.97 ) (13.5 Kbytes).
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25. In Search Of Counter-
Examples In Maple's Isprime Function.
- by Huen Y.K. (date released : 4.10.97 ) (18 Kbytes).
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26. A Sequence Algebraist's
View Of Lehmann's Primality Test
- by Huen Y.K. (date released : 6.10.97 ) (26 Kbytes).
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27. On Odd(z), Oddcomp(z),
Seq1(z) and Seq2(z)
- by Huen Y.K. (date released : 10.10.97 ) (17 Kbytes).
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28. How To Generate A Short
And Contiguous Oddcomp(z) Sequence?
- by Huen Y.K. (date released : 15.10.97 ) (13 Kbytes).
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(29) A Sketch Of Test-Tube
Evolution In A Primeval Number Soup - by Huen Y.K.
(date released : 25.11.97) (paper35.htm 1 K).
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(30) In Search Of Primes From
The XGS Sequence - by Huen Y.K.
(date released : 8.12..97) (paper38.htm 23 K).
=====================END OF PAPER ======================