Home9ClosurePumping

# Home9ClosurePumping - CS 341 Automata Theory Elaine Rich...

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CS 341 Automata Theory Elaine Rich Homework 9 Due Thursday, Nov. 2 at11:00 1) Build a deterministic PDA that accepts L \$, where L = { w { a , b }* : # a ( w ) = # b ( w )}. 2) Determine, for each of the following languages, whether it is (I) Regular, (II) Context free but not regular, or (III) not context free. Prove your answer. a) L = {( ab ) n a n b n : n > 0} b) L = { a i b n : i , n > 0 and i = n or i = 2 n } (a) L = { xy : x , y { a , b }* and | x | = | y |} c) L = { xwx R : x , w {0, 1} + } d) L = {0 i 1 j : i , j 0 and j = i 2 } e) L = ¬ L 0 , where L 0 = { ww : w { a , b }*} 3) Consider the language L = { ba m 1 ba m 2 b ba mn : n 2, m 1, …, mn 0, and mi mj for some i , j } a) Give a PDA that accepts L . b) Write a context-free grammar that generates L . c) Prove that L is not regular. 4) Give an example of a context-free language L ( ≠Σ *) that contains a subset L 1 that is not context-free. Prove that L is context free. Describe L 1 and prove that it is not context-free.

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CS 341 Automata Theory Elaine Rich Homework 9 Answers 1) Build a deterministic PDA that accepts L \$, where L = { w { a , b }* : # a ( w ) = # b ( w )}. The idea is to use a bottom of stack marker so that M can tell what it should be counting. If the top of the stack is an a , it is counting a ’s. If the top of the stack is a b , it is counting b ’s. If the top of the stack is #, then it isn’t counting anything yet. So if it is reading an a , it should start counting a ’s. If it is reading a b , it should start counting b ’s. M = ({0, 1, 2}, { a , b }, { a , b }, 0, {2}, ), where = {((0, ε , ε ), (1, #)), ((1, \$, #),(2, ε )), /* starting and ending. ((1, a , a ), (1, aa )), ((1, b , a ), (1, ε )), /* already counting a ’s. ((1, a , b ), (1, ε )), ((1, b , b ), (1, bb )), /* already counting b ’s. ((1,
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Home9ClosurePumping - CS 341 Automata Theory Elaine Rich...

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