hw_05 - Problem Set 5 Fall 09 Due: Thursday Dec 3 at 11:00...

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Unformatted text preview: Problem Set 5 Fall 09 Due: Thursday Dec 3 at 11:00 AM in class (i.e., Room 103 Talbot Lab) Please follow the homework format guidelines posted on the class web page: http://www.cs.uiuc.edu/class/fa09/cs373/ 1. [ Points : 15] (a) For each of the following PCP problems, either nd a solution or prove that a solution does not exist. i. 11 101 11 11011 110 1 ii. 10 1 10 01 01 10 10 iii. 1110 1 1 0111 (b) Prove that PCP is undecidable even if we restrict its alphabet to two symbols, for example = { , 1 } . (c) Prove that PCP is decidable if we restrict its alphabet to one symbol, for example = { } . 2. [ Points : 15] Consider the alphabet = { a,b,c,d } . Let R = { ( ab, ) , ( ba, ) , ( cd, ) , ( dc, ) } . Starting with a string w * , we can convert w to some other string w by applying the following rule: If ( x,y ) R or ( y,x ) R , and x is a substring of w , i.e. w = w 1 xw 2 , then w = w 1 yw 2 . The single player Group Game is started by some string w in * . At each round the player changes the string by applying the above rule. The game ends after a nite number of rounds. We say thethe string by applying the above rule....
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hw_05 - Problem Set 5 Fall 09 Due: Thursday Dec 3 at 11:00...

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