hw5sol - EE 376B/Stat 376B Information Theory Prof T Cover...

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EE 376B/Stat 376B Handout #14 Information Theory Thursday, May 26, 2011 Prof. T. Cover Solutions to Homework Set #5 1. Random program. Will the sun rise tomorrow? Suppose that a random program (symbols i.i.d. uniform over the symbol set) is fed into the nearest available computer. To our surprise the first n bits of the binary expansion of 1 / 2 are printed out. Roughly what would you say the probability is that the next output bit will agree with the corresponding bit in the expansion of 1 / 2 ? Solution: Random program The arguments parallel the argument in Section 7.10, and we will not repeat them. Thus the probability that the next bit printed out will be the next bit of the binary expansion of 2 is cn cn +1 . 2. Images. Consider an n × n array x of 0’s and 1’s . Thus x has n 2 bits. ( a ) ( b ) ( c ) Find the Kolmogorov complexity K ( x | n ) (to first order) if (a) x is a horizontal line. (b) x is a square. (c) x is the union of two lines, each line being vertical or horizontal. (d) x is a random array. (e) x is a rectangle lined up with the axes. (f) x is the union of two such rectangles meeting in a corner. (g) Highly optional: X is a quantized circle. 1
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Solution: Images. (a) The program to print out an image of one horizontal line is of the form For 1 i n { Set pixels on row i to 0; } Set pixels on row r to 1; Print out image. Since the computer already knows n , the length of this program is K ( r | n ) + c , which is log n + c . Hence, the Kolmogorov complexity of a line image is K (line | n ) log n + c. (1) (b) For a square, we have to tell the program the coordinates of the top left corner, and the length of the side of the square. This requires no more than 3 log n bits, and hence K (square | n ) 3 log n + c.
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