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hw_04 - Problem Set 4 Fall 09 Due Thursday Nov 5 at 11:00...

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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. 2. [ Points : 10] (a) Prove that this language is not context free: L = { a i b j c max( i,j ) | i,j ≥ } (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. 3. [ Points : 10] Describe an algorithm that given a grammar G determines whether L ( G ) has at least 373 strings or not. 4. [ Points : 10] Given a grammar G in Chomsky normal form: (a) Design an algorithm (via modifying the CYK algorithm) that for any given number...
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