Spontaneous Generation Of Number Sequences In A Primeval Number Soup

by

Huen Y.K.

CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~activweb/
Related URL-sites: http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg

(A short communication - 1st released: 23/11/97.)


Abstract

If you have a 1000-faced dice, what is the probability of generating the natural number sequence from 0 to 999 in a thousand consecutive throws? The answer is approximately one in 10^3000. However, if number sequences can be described in closed formulations, the probability of generating such formulas by random throws of dice is much higher. Imagine a random number generator issuing an endless string of ASCII decimal codes from 0 to 127. Theoretically, if a monkey is asked to type randomly on this 128 keyed typewriter and given enough time, any published scientific or mathematic papers could eventually be reproduced. The probability of generating closed forms sequence algebraic expressions for Nat(z), Even(z), Odd(z), Comp(z), Oddcomp(z), Prime(z) and other higher mathematical sequences is child's play compared to the spontaneous generation of biochemical components as the building blocks for life. Here is a web-interactive computer program that will detect spontaneous generations of sequences in closed forms. The present software is just a number game. However if instead of detecting number sequences we detect DNA sequences, the implication might be far more reaching. This paper simply shows that spontaneous generation of specific information by a perfectly random process is quite feasible especially if such information can be described in closed forms.


1. Introduction

1. The Primeval Number Soup

The primeval number soup to be studied is modelled by an infinite 1-dimensional random number string. Viable building blocks are primitive number sequences which have closed form expressions. Primitve number sequences such as Nat(z), Even(z) and Odd(z) have closed form representations which can be written in 1-dimensional ASCII decimal number codes as follows:

Nat(z) = z/(z-1). ASCII decimal code of the RHS is given by "122 47 28 122 45 49 29". Since each character has 1/128 chance of occurring, therefore the probability of generating the RHS expression for Nat(z) is one in 562949953421312.

Even(z) = z^2/(z^2-1). ASCII decimal code of the RHS is given by "122 94 50 47 28 122 94 50 45 49 29". Since each character has 1/128 chance of occurring, therefore the probability of generating the RHS expression for Even(z) is one in 151115727451828646838272.

Odd(z) = z/(z^2-1). ASCII decimal code of the RHS is given by "122 47 28 122 94 50 45 49 29". Since each character has 1/128 chance of occurring, therefore the probability of generating the RHS expression for Odd(z) is one in 9223372036854775808.

One could use this method to predict the probabilities of generating Comp(z), Oddcomp(z), Prime(z) and other higher mathematical sequences. It shows that the probabilities of generating these sequence formulas spontaneously is quite high.

If there are no closed forms, the probability of generating a 100 digit natural number sequence such as {0,1,2,3,.....................98,99} would be one in 2400889076051242533081 4857516899576105018151691160832802708344378095698991146369156964318347 7429899954431074921514308546363396623271386703410973337887610163634765 5507370061701735589111476563621882077520149696001 which is a 211 digit number. The probabilities of generating Even(z) and Odd(z) will be identical since zeroes are also counted as digits.

The 1-dimensional primeval soup will be modelled by a random number generator issuing endlessly a random alphnumeric string where each character in the ASCII decimal code falls within the range from 0 to 127 with equal probability. As this string grows, a search algorithm will systematically pattern match for z/(z-1) and report the hit when encountered. Give below in table 1 are samples of primitive and higher ordered sequences formed by sequence products which could be included in pattern matching for an extended version.

Table 1 - Some Primitive And Compound Sequence Expressions

---------------------------------------------------------------------------------------------
Nat(z), Even(z), and Odd(z)
Comp(z), Oddcomp(z) and Prime(z)
Nat(z)^n, Even(z)^n, Odd(z)^n
Nat(z)^p*Even(z)^q, Nat(z)^p*Odd(z)^q, Even(z)^p*Odd(z)^q
Nat(z)^p*Even(z)^q*Odd(z)^r
Nat(z)*Comp(z), Nat(z)*Oddcomp(z), Nat(z)*Prime(z)
Comp(z)^n, Oddcomp(z)^n, Prime(z)^n
........................
---------------------------------------------------------------------------------------------

Table 1 gives an inkling of an infinite pool of higher ordered number sequences which can be built from primitive sequences. The present software only detect spontaneous generations of Nat(z) by ASCII decimal code pattern matching. Each time the left commandbutton is clicked, a random 60000 alphnumeric character string will be generated. Then the software will pattern search for the Nat(z) formula given by z/(z-1) starting from the beginning of the string. Whether there is a match or not will be reported on the screen. The same software can be expanded to detect higher order sequences but this will be left to interested readers. The software can be tested online or after downloading. The source codes written in VBScript and ActiveX are contained in the two files bios.htm and bios.alx file.

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(2) - Spontaneous Generation Of NumberSequences In A Primeval Number Soup -- by Huen Y.K. (date released : 23.11.97) (bios.htm 1K, bios.alx 5K).

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3. Summary

Bios.htm is linked with bios.alx using Miscrosoft's ActiveX Control Pad. The software will run online but if you need to download it, it will not do automatic registering in your Win95 registry. This is deliberately done so that you have control over whether you wish to relink it or not after studying the source codes. To relink, please change the file path in bios.htm to suit the directory path chosen to store the two files. Then relink using ActiveX Control Pad.

4. Reference:

Remarks: The two papers given below might give you the needed backgroun in sequence algebra.

1. Huen Y.K.: A matrix map for primes and nonprimes, Int. J. Math.Educ.Sci.Technol., 1994, Vol.25, No.6, pp 913 - 920.

2. Huen Y.K..: Visual algebra and its applications, Int. J. Math. Educ. Sci. Technol., 1997, Vol.28, No.3, 333-344.

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