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hw_04_sol - Problem Set 4 Fall 09 Due Thursday Nov 5 at...

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Unformatted text preview: Problem Set 4 Fall 09 Due: Thursday Nov 5 at 11:00 AM in class (i.e., Room 103 Talbot Lab) Please follow the homework format guidelines posted on the class web page: http://www.cs.uiuc.edu/class/fa09/cs373/ 1. [ Points : 10] Suppose G is a CFG with p rules and the body of each rule is not longer than n (Note: the rule S → aSbSA has a body length of 5). If we can derive ε from non-terminal S , then prove that we can derive it from S in no more than n p- 1 n- 1 steps. Solution: Consider all parse trees that derive ε from S and let T be one of them that has the minimum number of nodes. On every path from root to some leaf of T , no non-terminal appears twice (if it appears twice, then we can remove all nodes between the two occurrences of that non-terminal and still have a valid parse tree for ε ). Therefore the height of the tree is at most p- 1 . Since each node in the tree has at most n childs, the k-th level of T has at most n k nodes. Therefore the total number of nodes (which is equal to the total number of steps in the derivation) is no more than 1 + n + n 2 + ··· + n p- 1 = ( n p- 1) / ( n- 1) . 2. [ Points : 10] (a) Prove that this language is not context free: L = { a i b j c max( i,j ) | i,j ≥ } Solution: Assume to the contrary that it is context free. Therefore it follows the pumping lemma for CFLs and has a pumping length p . Consider the string s = a p b p c p ∈ L . Since | s | ≥ p , pumping lemma applies to s and we should be able to break s into s = xuyvz such that | uyv | < p , | uv | 6 = 0 and s i = xu i yv i z ∈ L for every i ≥ . We consider two cases (note that one of these cases always happens): • If | u | > and u contains some c 's, or | v | > and v contains some c 's, then s has exactly p a 's (note that | uyv | < p and since uyv has some c 's it cannot reach the area of a 's in the string) but less than p c 's therefore s / ∈ L . • Otherwise consider s 1 . Since u and v do not intersect c 's, s 1 has exactly p c 's. But the number of a 's or b 's in s 1 is more than p (since one of u and v has positive length and it intersects a 's or b 's). Therefore s 1 / ∈ L . So L does not follow the CFL-pumping lemma and therefore is not a CFL. (b) Prove that context free languages are not closed under the operator max , as de ned below: max( A ) = { w ∈ A | if wx ∈ A for some string x , then x = ε } . Hint: part(a) might be useful. 1 Solution: Consider this language A : A = { a i b j c k | k ≤ i or k ≤ j } We have seen a PDA for this language, so we know that A is context free. But note that if w = a i b j c k ∈ A and k < max( i,j ) , then wc ∈ A , therefore w / ∈ max( A ) . So the only possibility for w to be in max( A ) is when k = max( i,j ) . This means that max( A ) = L , where L is the language in part(a) that we know is not context free....
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• Fall '08
• Viswanathan,M
• Universal quantification, Formal language, Regular expression, Context-free grammar, context-free language

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hw_04_sol - Problem Set 4 Fall 09 Due Thursday Nov 5 at...

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