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hw2(1) - University of Central Florida School of Computer...

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University of Central Florida School of Computer Science COT 4210 Spring 2004 Prof. Rene Peralta Solutions to Homework 2 Consider integers written in base 3 with no leading 0s. Let L be the set of such strings which represent even numbers. 1. Construct a DFA that accepts L . The red state means “I am odd”. The yellow state means “I am even”. 1
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2. Construct a left-linear grammar for L . Associate with each state U the set of strings whose computation can end in U. Then we have (let B,R,Y denote be blue,red,yellow states, resp.) B λ R ( Y + B )1 + R (2 + 0) Y B 2 + R 1 + Y (2 + 0) which simplifies to R ( Y + λ )1 + R (2 + 0) Y 2 + R 1 + Y (2 + 0) The starting symbol of the grammar is Y . 3. Write a regular expression for L . Using Adler’s rule we have R = ( Y + λ )1(2 + 0) * then Y = 2 + ( Y + λ )1(2 + 0) * 1 + Y (2 + 0) factoring Y Y = 2 + 1(2 + 0) * 1 + Y (1(2 + 0) * 1 + 2 + 0) and finally Y = (2 + 1(2 + 0) * 1)(1(2 + 0) * 1 + 2 + 0) * 2
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4. Write a regular expression for L r . Above is the “reverse” of the automata for L . Just for fun, we can
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