Excerpt

Kolmogorov Complexity defines a random binary sequential string as being less patterned than a non-random binary sequential string. Accordingly, the non-random binary sequential string will retain the information about it's original length when compressed, where as the random binary sequential string will not retain such information. In introducing a radix 5 based system to a sequential string of both random and non-random series of strings using a radix 5, or quinary, based system. When a program is introduced to both random and non-random radix 5 based sequential strings that notes each similar subgroup of the sequential string as being a multiple of that specific character and affords a memory to that unit of information during compression, a sub-maximal measure of Kolmogorov Complexity results in the random radix 5 based sequential string. This differs from conventional knowledge of the random binary sequential string compression values.

PACS numbers: 89.70 Eg, 89.70 Hj, 89.75 Fb, 89.75 Kd Traditional literature regarding compression values of a random binary sequential string have an equal measure to length that is not reducible from the original state^{[1]}. Kolmogorov complexity states that a random sequential string is less patterned than a non-random sequential string and that information about the original length of the non-random string will be retained after compression ^{[2]}. Kolmogorov complexity is the result of the development of Algorithmic Information Theory that was discovered in the mid-1960's ^{[3]}. Algorithmic Information Theory is a subgroup of Information Theory that was developed by Shannon in 1948^{[4]}.

Recent work by the author has introduced a radix 5 based system, or a quinary system, to both random and non-random sequential strings ^{[5]}. A patterned system of segments in a binary sequential string as represented by a series of l's and 0's is rather a question of perception of subgroups within the string, rather than an innate quality of the string itself. While Algorithmic Information Theory has given a definition of patterned verses patternless in sequential strings as a measure of random verses non-random traits, the existing standard for this measure for Kolmogorov Complexity has some limits that can be redefined to form a new sub-maximal measure of Kolomogorov Complexity in sequential binary strings ^{[6]}. Traditional literature has a non-random binary sequential string as being such: [111000111000111] resulting in total character length of 15 with groups of l's and 0's that are sub-grouped in units of threes. A random binary sequence of strings will look similar to this example: [110100111000010] resulting in a mixture of sub groups that seem 'less patterned' than the non-random sample previously given.

Compression is the quality of a string to reduce from it's original length to a compressed value that still has the property of 'decompressing' to it's original size without the loss of the information inherent in the original state before compression. This original information is the quantity of the strings original length before compression, bit length, as measured by the exact duplication of the l's and 0's found in that original sequential string. The measure of the string's randomness is just a measure of the patterned quality found in the string.

The quality of 'memory' of the original pre-compressed state of the binary sequential string has to do with the quantity of the number of l's and 0's in that string and the exact order of those digits in the original string are the measure of the ability to compress in the first place. Traditional literature has a nonrandom binary sequential string as being able to compress, while a random binary sequential string will not be able to compress. But if the measure of the number and order of digits in a binary sequence of strings is the sole factor for defining a random or non-random trait to a binary sequential string, then it is possible to 'reduce' a random binary sequential string by some measure of itself in the form of sub-groups. These sub-groups, while not being as uniform as a non-random sub-group of a binary sequential string, will nonetheless compress from the original state to one that has reduced the redundancy in the string by implementing a compression in each subgroup of the random binary

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^{[1]} S. Kotz and N.I. Johnson, Encyclopedia of Statistical Sciences (John Wiley & Sons, New York, 1982).

^{[2]} abide.

^{[3]} R.J. Solomonoff, Inf. & Cont. 7, 1-22 & 224-254 (1964), A.N. Kolmogorov, Pro. Inf. & Trans. 1, 1-7 (1965) and G.J. Chaitin, Jour. ACM 16, 145-159 (1969).

^{[4]} C.E. Shannon, Bell Labs. Tech. Jour. 27, 379-423 and 623-656 (1948).

^{[5]} B.S. Tice "The use of a radix 5 base for transmission and storage of information", Poster for the Photonics West Conference, San Jose, California Wednesday January 23, 2008.

^{[6]} S. Kotz and N.I. Johnson, Encyclopedia of Statistical Sciences (John Wiley & Sons, New York, 1982).

- Quote paper
- Professor Bradley Tice (Author), 2012, Random and Non-random Sequential Strings Using a Radix 5 Based System, Munich, GRIN Verlag, https://www.grin.com/document/199140

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