Michael L Cole

Systematical Background: Forming a recursive structure by reversibly setting, splitting and re-configuring composite binary input.

Illustration A - As in the Recursive data compression patent Series 2 (below) was formed by decomposing the compositional structure of the eight binary digits that make-up the eight-bit symbols of Series 1 (defined below). The new composition was formed by the distribution of the decomposed binary structure of Series 1 by being keyword configured positionally (bits 2^7 through 2^0) and sequentially (bits 1-8 each byte) in order of occurrence (high to Low). The newly formed symbology reduces the number of symbols by approximately 98% from 256 to five while retaining a completely lossless system of numerical notation.

Recursive compression and Intrinsic Order

Table 1a

1020 total A and B clustered bit states (columns 5 and 6 in Table 1a )

510 total A or B clustered bit states (column 5 or 6 in Table 1a )

2304 total AB clustered bit states (column 7 in Table 1a)

Total AB clustered bit states (column 7 total in Table 1a) divided by keyword variation (shown on top of this page and illustrated across in Table 1a) equals nine.

Block A and B divergence of clustered bits (columns 1 and 2 in Table 1a) total 72 bits per Recursive data keyword.

Illustration 2a

This binary variation is derived from the 52-byte string (top line of example). It demonstrates the lossless conversion from 27 symbols (top line) to 12 symbols (binary block and below).

This single recursive data compression variation (representative, see Table 1a) has many opportunities to succeed in microcosm; on strings of 256 bytes in length, containing 256 different symbols forming random sequences.

The probability of matching at least one bit per eight bits is .(point)996094

A 256 byte string comprised of 256 repeatable symbols each randomly selected at a random probability of 0.(point)996094 has a probability of not matching one bit (per eight bits) zero to three occurrences per string at probability of 0.(point)981252 or algebraically: Individual probability of the number of times that there is no match per 256 byte string comprised of the (as described) 256 repeatable symbols. The table below shows the random probabilities of the distribution of bytes from Column A and Column B. No match (as described) occurs each time a byte value from Column A (0-127) has an adjoining invertible match (i.e. 0, 255), this requires a byte value from Column B (128-255). The input for the next multi-part illustration is the first twelve characters from Illustration 2a which demonstrated that redundancy can be formed as a result of the recursive conversion of n symbols to n-1 (one or greater) symbols. Illustration 2b In the example the conversion steps through a succession of recursive compression stages from seven eight bit (sub-division to seven 4 bit) symbols to nine eight bit (sub-division to five 4 bit) symbols. Illustration 2b-2 Example of sub-division to five (4 bit symbols) Same data as used in Illustration 2b Illustration 2b-3 The patterns of succession can also be counted as they progress further through recursive sub-divisions and symbol stages. Example of recursive structure conversion step to six (12 bit symbols) and recursive sub-division to seven (6 bit symbols) Same data as used in Illustration 2b

An analysis of Recursive data compression (using the most primitive 8-bit keyword structure, no Sub-Structure (Illustration 1.g, 1.h) and utilizing only *-zip files) revealed that for every uncompressed file that compresses (at least three bytes using *-zip) there are 255 other file permutations which will compress equally ( 2 bytes).

Despite the fact that *-zipping a file ("a") which is zero bytes in length results in a *-zip file a minimum of twenty-two bytes in length, and despite the fact that I consider that to be a prima facie argument that all *-zip files are in theory compressible, I did not interpose any adaptation for that into this equational analysis.

Notwithstanding, there was an aggregate increase of 25,500 percent more *-zip files when using Recursive data compression. Additionally, for each *-zip file that re-compresses (by at least three bytes using *-zip) there are an additional 255 file permutations which will compress equally ( 2 bytes). i.e., If an arbitrary proportional assignment of one-percent of N length *-zip files can be re-compressed by at least three bytes using *-zip then the number of file permutations that can be compressed by at least 1 byte is (255 * 1%) or 2.55 times the aggregate number of *-zip files which are N bytes in length. That is another increase of 25,500 percent.

Using the reverse engineering in this example allows the statistics to be traced. i.e., Assume a one megabyte file has a *-zip image which is a quarter-megabyte in length. Then 256 copies of the *-zip image are made, each with a unique 8-bit keyword appended to it, plus one optional ancillary byte. Each new quarter-megabyte file would then represent one (of 256) unique permutation of the original file. The result is a corresponding compression ratio of 4:1 for each of these 256 files.

Therefore, if *-zip can compress an arbitrary proportional assignment of one-percent (1%) of all uncompressed files N bytes in length (for every N) by at least three bytes then the least number of Recursive *-zip files that can be compressed at the same compression ratio using "Recursive data compression" is (256 * 1%) or 2.56 times (25,600 percent) the aggregate number of all *-zip files. The ratio is not changed when using a more realistic proportional assignment than one-percent.

In addition, if these *-zip files re-compress (referenced above) then there are at least 65534 extra compressible file permutations (not 255) in individual relation to these files, provided that they re-compressed by at least three bytes even if these files will not re-compress again.

illustration 1.a Illustration 1.g Sub-Structure IFSP - - Analysis (above) of quasi-optimum compression using IFSP Sub-Structure nets (3*25500) 76,500 percent more Recursively compressible *-zip files. Illustration 1.h Sub-Structure VUSP - Analysis (above) of quasi-optimum compression using VUSP Sub-Structure nets (6*25500) 153,000 percent more Recursively compressible *-zip files.

illustration 4.g - Equational analysis of quasi-optimum compression for the recursive computable reducibility model Partition delineated for 4 bits (below). illustration 4.a (above) Computable reducibility model of recursive structure consolidation delineated for 4 bits by illustration 4.a - 4.b in contrast with "Recursive data compression" secondary compressibility model (refer to illustration 1.a). The computable reducibility model of structure consolidation is implemented by recursively splitting binary segments as a transform mechanism and in its simplest form is analogous to the created redundancy strategy implemented by the recursive data compression secondary compressibility model when the created redundancy is (in part) the result of the lossless conversion of n symbols to n-1 (minus one or greater) symbols. Conversely, when the created redundancy of the secondary compressibility model is relied upon (in part) as the result of the lossless conversion of n symbols to n 1 (plus one or greater) symbols it becomes clear that the computable reducibility model of structure consolidation is analogous to (and must be implemented at the time of) keyword formulation. Therefore, I suggest that both sides of the analogy should be taken under advisement when devising variations to algorithms for formulating the keyword for Recursive data compression. Computable reducibility model of structure consolidation for 8 bits (delineated for 4 bits by illustration 4.a - 4.b) in contrast with "Recursive data compression" secondary compressibility model (illustration 1.a). As described in U.S. Patent 5,488,364, Recursive data compression, recursively splitting binary segments as a transform mechanism can create redundancy and as stated above this created redundancy is in part the result of the lossless conversion of n symbols to n-1 (minus one or greater) symbols. But as described in U.S. Patent 5,488,364 you can also recursively compress a file using *-zip where the file compresses (a program reported) one-hundred percent and the resulting *-zip file will also compress another (program reported) seventy to eighty-plus percent. This is accomplished by escalating the number symbols (n 1 or greater). This is more difficult to rationalize but the principles are the same. Therefore, the first subject will be, recursively compressing binary subdivisions that compress because they consist of less symbols. One symbol consists of a unique fixed-length string of binary digits. A block of symbols constitutes binary subdivision. In this scenario the resulting binary subdivisions compress because of they are less complex because they consist of less symbols. In the table below where column "A" holds numeral values greater than one. The relationship that binds the terms in column "B" to the corresponding value in the column "A" is formed by their equivalence (the sum of their parts). There are far fewer values in column "A" than there are (formed by sets) in column "B." This suggests that there are far fewer irreducible values (the sum of the terms) than there are reducible (non-random) values (the composition of the terms that equal the sum). The structure above is "a partition" and Peter Gustav Lejeune Dirichlet's (1805-1859) "Pigeonhole Principle." (Also known as "Dirichlet's Principle", "Dirichlet's Drawer Principle" and "Dirichlet's Letter Box Principle") communicates that the partition of a set of "n" objects into fewer than "n" (single object) mutually exclusive subsets is not possible. Arithmetic based partition structures and recursive compression may seem to have no place in modern reducibility theory, especially, since the use of arithmetic partitions based on the sum of their parts will propose an inconsistent presumption as to the number of irreducible values (random incompressible strings). A suitable homogeneous remedy for this would be to provide the same structure using terms that produce a product instead of a sum. The product table produces more irreducible values (random incompressible strings) 50/50 as anticipated. This mirrors the notions of contemporary algorithmic theories of reducibility. To see exactly how all this works with illustrated examples, please read about reducing the number of symbols in a string using lossless recursive data compression.

When the example uses n=4 the correspondence results in 12 extra members. illustration 4.b (above)

Illustration of recursive compression utilizing a designed recursive structure reduces the number of symbols in a string by ninety-eight percent (98%) Illustration shows how recursive compression utilizing a designed recursive structure reduces the number of symbols in a string by ninety-eight percent (98%)

According to the US Patent Office, expired* U.S. Patent 5,488,364, is referenced by the following patents:

1 - 8,364,836 Withdrawn, System and methods for accelerated data storage and retrieval

2 - 8,326,984, Selective compression for network connections

3 - 8,321,445, Generating content snippets using a tokenspace repository

4 - 8,275,909, Adaptive compression

5 - 8,275,897, System and methods for accelerated data storage and retrieval

6 - 8,117,173, Efficient chunking algorithm

7 - 8,112,619, Systems and methods for accelerated loading of operating systems and application programs

8 - 8,112,496, Efficient algorithm for finding candidate objects for remote differential compression

9 - 8,090,936, Systems and methods for accelerated loading of operating systems and application programs

10 - 8,073,926, Virtual machine image server

11 - 8,073,047, Bandwidth sensitive data compression and decompression

12 - 8,054,879, Bandwidth sensitive data compression and decompression

13 - 8,024,483, Selective compression for network connections

14 - 8,010,668, Selective compression for network connections

15 - 7,882,084, Compression of data transmitted over a network

16 - 7,873,065, Selectively enabling network packet concatenation based on metrics

17 - 7,849,462, Image server

18 - 7,783,781, Adaptive compression

19 - 7,777,651, System and method for data feed acceleration and encryption

20 - 7,714,747, Data compression systems and methods

21 - 7,705,753, Methods, systems and computer-readable media for compressing data

22 - 7,613,787, Efficient algorithm for finding candidate objects for remote differential compression

23 - 7,555,531, Efficient algorithm and protocol for remote differential compression

24 - 7,370,120, Method and system for reducing network latency in data communication

25 - 7,321,952, System and method for data phase of memory and power efficient mechanism for fast table lookup

26 - 7,296,114, Control of memory and power efficient mechanism for fast table lookup

27 - 7,296,113, Memory and power efficient mechanism for fast table lookup

28 - 7,292,162, Data coding system and method

29 - 6,301,394, Method and apparatus for compressing data

30 - 6,008,657, Method for inspecting the elements of piping systems by electromagnetic waves

31 - 5,977,889, Optimization of data representations for transmission of storage using differences from reference data

32 - 5,666,560, Storage method and hierarchical padding structure for direct access storage device (DASD) data compression

33 - 5,627,534, Dual stage compression of bit mapped image data using refined run length and LZ compression

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Random data compression is like a magician pulling two rabbits out of one hat.

FACT: THE PHRASE RANDOM DATA COMPRESSION NEVER APPEARS IN THE RECURSIVE DATA COMPRESSION PATENT. IT DOES NOT EVEN CONTAIN THE PHRASE RANDOM DATA.

If the pigeonhole principle counting argument equation as applied in many textbooks for recursive compression is not disproved by my following equation, then I guess that I am a dummy. Because the counting argument equation below clearly shows that (as applied in many textbooks for recursive compression) the pigeonhole principle counting argument equation does not work in at least one instance.

Reference: Intrinsic order of dispersion in recursive compression, dual combinational elements for KEYWORD: "00000000" with "11111111" in any value dispersion table, illustrating two empty (null) bit states (containing no elements and no members) equivalent to one bit.