Excerpt

## ABSTRACT

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 2 based system to a sequential string of both random and non-random series of strings using a radix 2, or binary, based system. When a program is introduced to both random and non-random radix 2 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 2 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 2 based system, or a binary 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 lose of the information inherent in the original state before compression.

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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, Aspects of Kolmogorov Complexity: The Physics of Information. (River Publishers, Denmark, 2009).

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

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- Professor Bradley Tice (Author), 2012, Compression and Geometric Data, Munich, GRIN Verlag, https://www.grin.com/document/199139

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