cs373 fa10 hw9sol - Solutions for Problem Set 9 CS 373:...

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Unformatted text preview: Solutions for Problem Set 9 CS 373: Theory of Computation Assigned: November 9, 2010 Due on: November 16, 2010 at 10am Homework Problems Problem 1 . [Category: Comprehension] Consider the following grammar over the terminals { , # } and start symbol S . S → TT | U T → T | T | # U → U 00 | # 1. For each of the following strings, answer whether or not they belong to the language defined by the grammar: 00#0#00, 0#0, 000#000000. If they do, give a derivation and parse tree for the string. If not, provide a short, informal justification for why the string cannot be generated. [6 points] 2. What is the language of the grammar? You need not prove your answer, but you should provide a short informal justification for your answer. [2 points] 3. Is the language of the grammar regular? Prove your answer. [2 points] Solution: 1. 00#0#00 is in the language as can be seen by the following derivation. S ⇒ TT ⇒ TT ⇒ 00 TT ⇒ 00# T ⇒ 00#0 T ⇒ 00#0 T ⇒ 00#0 T 00 ⇒ 00#0#00 0#0 is not derivable in the grammar because the only strings that can be generated with a single #-symbol are those from U . But then the strings obtained from U have at least two 0s after the # (if there are any 0s after the #). String 000#000000 is derivable as follows. S ⇒ U ⇒ U 00 ⇒ 00 U 0000 ⇒ 000 U 000000 ⇒ 000#000000 2. The strings derivable from T are L 1 = { i #0 j | i,j ≥ } , and from U are L 2 = { i #0 2 i | i ≥ } . Thus, L ( G ) = L 1 L 1 ∪ L 2 3. L ( G ) is not regular. We will prove this using closure properties. Consider the following sequence of languages....
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This note was uploaded on 10/04/2011 for the course CS 373 taught by Professor Viswanathan,m during the Fall '08 term at University of Illinois, Urbana Champaign.

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cs373 fa10 hw9sol - Solutions for Problem Set 9 CS 373:...

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