hw5sol

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern