hw1sol - Problem Set 1 Solutions CS373 - Spring 2011 Due:...

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Unformatted text preview: Problem Set 1 Solutions CS373 - Spring 2011 Due: Thursday Feb 10 at 2:00 PM in class (151 Everitt Lab) Please follow the homework format guidelines posted on the class web page: http://www.cs.uiuc.edu/class/sp11/cs373/ 1. Encoding input and building a DFA [ Category : Design, Points : 15] Let = { , 1 } . i) Create an injective function g that maps the natural numbers to members of * such that | g ( x ) | > | g ( y ) | only if x > y . (1 Point) Solution: There are many functions that meet the requirements. The key is to create a function that makes the next two parts easy. One such function g ( n ) encodes the individual digits of the number n 's representation in base 5 , then discards any leading zeros. The character map would be 000 , 1 001 , 2 010 , 3 011 , 4 100 . The rst few outputs of the function g would be g (1) = 1 , g (2) = 10 , g (3) = 11 , g (4) = 100 , g (5) = 1000 , and g (6) = 1001 . ii) Design a DFA that recognizes the following language L = { w * | w = g ( n ) , where n = 5 k for a natural number k } . You must use the injective function that you created in part i. To recieve full credit, the DFA can have at most six states. (5 Points) Solution: Using the mapping function above, the DFA is the one recognizing 1000(000) * , shown in Figure 1. Note that any string that is not in the image of g must be rejected by the DFA. Also note that g (1) must be rejected, as 1 = 5 , and is not a natural number. There exists a di erent mapping g that encodes the desired powers of 5 as members of 111(111) * , and the DFA for this takes fewer than six states. iii) Design a DFA that recognizes the following language L = { w * | w = g ( n ) , where n = 25 k for a natural number k } . You must use the injective function that you created in part i. To recieve full credit, the DFA can have at most nine states. (3 Points) Solution: Using the mapping from the solution to part i, the DFA that recognizes members of g (25 k ) is the one that recognizes 1000000(000000) * , shown in Figure 2. Again, using a di erent encoding that still respects the length requirements, it is possible to use fewer than the allowed amount of states. iv) Create a di erent injective function h that maps the natural numbers to members of * . There are no restrictions on h except that it must be injective. (1 Point) Solution: Since there are no more length requirements, h can map 5 k to 1 k , and all other numbers n to n . There are numerous functions that make part v easy. 1 1 1 1 1 1 , 1 Figure 1: A DFA that recognizes 1000(000) * . v) Design a DFA that recognizes the following language L = { w * | w = h ( n ) , where n = 5 k for a natural number k } . You must use the injective function that you created in part iv. To recieve full credit, the DFA can have at most three states. (5 Points) Solution: Using the encoding from part iv, the DFA that recognizes h (5 k ) for natural k is just the DFA that recognizes 11 * . A sample DFA that uses three states is shown....
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This note was uploaded on 04/28/2011 for the course CS 373 taught by Professor Viswanathan,m during the Spring '08 term at University of Illinois, Urbana Champaign.

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hw1sol - Problem Set 1 Solutions CS373 - Spring 2011 Due:...

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