CS172 hw1 - CS 172 Computability and Complexity Spring 2008...

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CS 172: Computability and Complexity, Spring 2008 S. A. Seshia & A. Kulkarni HW 1: Introduction to DFAs and NFAs Assigned: January 24, 2008 Due in class: January 31, 2008 Note: Take time to write clear and concise solutions. Confused and long-winded answers may be penalized. Consult the course webpage for course policies on collaboration. 1. (4 points) A palindrome is a binary string which is equivalent to its reversal. For example, 1, 1001 and 00100 are palindromes, but 1011 is not. Prove that any even-length palindrome must have the same number of ones in its first half as in its second half. (be sure to include your proof idea/sketch as well as the complete proof) 2. (6 points) (a) Design and draw a DFA which reads in a non-negative integer written in binary, least significant bit first, and accepts exactly when the input is divisible by four. (There is no sign bit in the representation.) Specify what Q , Σ , δ , q 0 , and F are for your DFA. (b) Now suppose that before the least significant bit, there is a bit indicating the sign of the integer: 0 for a negative number, and 1 for a positive number. How would you modify
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