|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.
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.
Example of sub-division to five (4 bit symbols)
Same data as used in Illustration 2b
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.
The following definition is an abstract of how a method built on a secondary compressibility model was conceived (as I currently view it) from the computable algorithmic perspective. It illustrates how one model of structure consolidation in order to achieve computable reducibility is analogous in structure to "Recursive data compression."
|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 modelPartition 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)
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