hw5sol - CS 181 Winter 2005 Formal Languages and Automata...

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CS 181 — Winter 2005 Formal Languages and Automata Theory Problem Set #5 Solutions Problem 5.1. (10 points) Recall the shuffle operator from Homework 2. Prove or disprove: 1. Context free languages are closed under shuffle. Solution We will show that context free languages are not closed under shuffle! Consider the languages L 1 = { 0 n 2 n | n 0 } and L 2 = { 1 m 3 m | m 0 } . Then the shuffle of the two languages (denoted SHUFFLE ( L 1 , L 2 )) will contain the words that have an equal number of 0’s and 2’s, and equal number of 1’s and 3’s and morover, no 2 occurs before a 0 and no 3 occurs before a 1. Assume that SHUFFLE ( L 1 , L 2 ) is context-free. Then we know that context free languages are closed under intersection with regular languages. Consider intersecting SHUFFLE ( L 1 , L 2 ) with 0 * 1 * 2 * 3 * . Then we get precisely the language { 0 n 1 m 2 n 3 m | n, m 0 } . However, notice that this language is not context free. Intuitively, this is because we will need to recall n before m even though m is the last number we stored. This is not possible using a stack! To formally show this language is not context free, we can use the pumping lemma but we leave this as an exercise. 2. Let L be context free and R regular. Then the shuffle of L and R is context free. Solution We will show that in fact the shuffle of L and R is context-free. We do this by building a PDA for SHUFFLE ( L, R ) given a PDA for L and a DFA for R . Let P = ( Q L , Σ , Γ , δ L , q L , F L ) be a PDA accepting L and M = ( Q R , Σ , δ R , q R , F R ) be a DFA accepting R . Our construction of the PDA for shuffle will use ideas from the construction showing the context free languages are closed under intersection with regular languages and from our construction showing regular languages were closed under shuffle. In particular, we define the PDA N = ( Q L × Q R , Σ , Γ , δ, ( q L , q R ) , F L × F R ) . The transition function nondeterministically decides whether or not to feed the next input symbol to P or to M . In other words, (( q 0 , p 0 ) , b ) δ (( q, p ) , w, a ) if and only if one of the following hold ( q 0 , b ) δ L ( q, w, a ) and p 0 = p . In this case, we update the PDA and let the DFA idle.
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