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De ning algorithmic entropy in .NET Printing datamatrix 2d barcode in .NET De ning algorithmic entropy

De ning algorithmic entropy generate, create 2d data matrix barcode none for .net projects ASP.NET Web Application Framework The concept of information, Data Matrix ECC200 for .NET which has been described extensively in 3, also evolved beyond Shannon s view, and independently of his classical theory. An alternative de nition, which is referred to as algorithmic information, is attributed to G.

Chaitin, R. Solomonoff, and A. Kolmogorov.

1 Algorithmic information opened the way to the eld of algorithmic information theory (AIT). From AIT s perspective, any source event x is treated as an object variable. The event may consist in any symbol sequence, whether random or deterministic.

The focus of AIT is not on the source X , the statistical ensemble of all possible events or symbolic sequences, but on this particular sequence x.. See, for instance: http://home.mira.net/ reynella/debate/informat.htm and useful links therein. 7.2 The Turing machine Entering the AIT domain simp ly requires one to acknowledge the following three basic de nitions: (a) The complexity K (x) is de ned as the smallest size of a program q(x) necessary to generate the sequence x. (b) Such a program is a nite set of binary instructions with a length of . q(x). bits. (c) The program can be implemented by a Turing machine (TM). The smallest program size, min q(x). = K (x), which is called th e complexity of x, is equivalently referred to as algorithmic information content, algorithmic complexity, algorithmic entropy, or Kolmogorov complexity (also called Kolmogorov Chaitin complexity and sometimes noted KC(x) instead).2 The Turing machine, which is named after its inventor, A. Turing,3 can be viewed as the most elementary and ideal implementation of a computer, and is, as we shall see, of in nite computation power (due to speed, not memory size).

To clarify and develop all of the above, we ought rst to understand what a Turing machine looks like and how it works; this is addressed in the next section. Then we will investigate Kolmogorov complexity and its properties. Interestingly, I will show that Kolmogorov complexity is in fact incomputable! Finally, I will show that Kolmogorov complexity and Shannon s entropy are symptotically bounded, a most remarkable feature considering the previous property.

. The Turing machine The Turing machine (TM) is a n abstract, idealized, or paper version of the simplest and most elementary computing device. As Fig. 7.

1 illustrates, it consists of a tape and a read/write head. The tape is of inde nite length and it contains a succession of memory cells, into which are written the bit symbols 0 or 1.4 By convention, cells that were never written or were left blank are read as containing the 0 bit.

The tape can be made to move left or right by one cell at a time. The operations of the head and tape are de ned by a table of instructions {I1 , I2 , . .

. , I N } of nite size N , also called an action table. The action table is not a program to be read sequentially.

Rather, it is a set of instructions corresponding to different possibilities to be considered by the machine, as I shall clarify. At each instruction step, the machine s head is initially positioned at a single tape cell..

See, for instance: http://en .net framework ECC200 .wikipedia.

org/wiki/Kolmogorov_complexity, http://szabo.best.vwh.

net/ kolmogorov.html. See, for instance: http://plato.

stanford.edu/archives/spr2002/entries/turing-machine/, http://en.wikipedia.

org/wiki/Universal_Turing_machine, www.turing.org.

uk/turing/, www.alanturing.net/, www.

cs.usfca.edu/ www.

AlanTuring.net/turing_archive/pages/Reference%20Articles/What%20is%20a%20Turing20Machine. html.

More generally, the symbols that can be put into the cells, including a conventional blank symbol, could be selected from any nite alphabet.. Algorithmic entropy and Kolmogorov complexity Instructions State 3 Head Tape Tape Cells Figure 7.1 Schematic representation of Turing machine. The machine is said to be in an input state si , which corresponds to a speci c instruction in the action table. This instruction tells the machine to perform three basic operations altogether: (a) Given the cell s content, what new content (namely, 1 or 0) is to be written into the cell; (b) In what direction the tape should be moved, namely, left or right; (c) Into what new state s j the machine should be moved. For instance, given the input state s1 the corresponding instruction could be: (a) If reading 0 then write 1, (b) Move tape to the left, (c) Go into state s3 .

If the cell reading is 0, the machine changes the contents, then moves according to the two other actions, (b) and (c). If the cell reading is 1, then it halts. To cover this other possibility, a second instruction can be introduced into the action table, for instance; (a) If reading 1 then write 0, (b) Move tape right, (c) Go into state s2 .

These two sets of instructions in the action table can be summarized as follows: s1 ; 0 1; L; s3 s1 ; 1 0; R; s2 . (7.1).

Since the instructions move .net framework Data Matrix 2d barcode the TM into new states s2 , s3 , the instructions corresponding to input states s2 , s3 should also be found in the table. There are no restrictions concerning.

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