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Homework 6

Homework 6. due 12/21. 1 Pumping lemma for CFL. 1.1 Show that the language L 1 = { xx  {0,1}* | x  {0,1}* } is not context free. Instances of L 1 include 0101, 110110 etc, but do not include 00110 and 0111.

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Homework 6

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  1. Homework 6 due 12/21

  2. 1 Pumping lemma for CFL 1.1 Show that the language L1 = { xx  {0,1}* | x  {0,1}* } is not context free. Instances of L1 include 0101, 110110 etc, but do not include 00110 and 0111. 1.2 Show that the language L2 = { xy  {0,1}* | x  {0,1}* and y is the 1's complement of x.} is not context free. Instances of L3 include 0110, 110001 etc, but do not include 0101, 010 and 0111.

  3. 2. Context free grammar composition • Given two grammar : • G1: S  bS | Ab A  a | BA | SB B  bA | bB • G2: T  Tb | aTC C aC | bT 2.1 Find a CFG G3 such that L(G3) = L(G1)  L(G2). 2.2 Find a CFG G4 such that L(G4) = { xy | x L(G1)*, y L(G2) }. • Notes • you can reuse production rules and symbols in G1 and G2. • You need not study the specific detail of G1 and G2, as studied in the lecture, there is a systematic method to compose both grammars directly from the general definition of two input CFGs.

  4. 3. CYK algorithm 3. Given the grammar G in Chomsky normal form • S  AB ABB | a B  AB | b | AA 3.1 Apply the CYK algorithm on the input string aabba to determine if it is in L(G) by completing the following table. 3.2 Is aabba a member of L(G) ? why ? 3.3 Is this grammar ambiguous ? why ?

  5. 4 Parse trees and derivations • Given the following grammar : • S  AB | e A  aB B  Sb 4.1 Find a left-most derivation for the string aabbbb 4.2 Find a right-most derivation for the string aabbbb. 4.3 Find a parse tree for aabbbb.

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