Homework 3

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Homework 3

• due 1/4

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• 1 Minimize the following DFA. Indicate clearly
  which equivalence class corresponds to each state
  of the new automaton.

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• Let A x?a,bthe number of as in x 2 x
  the number of bs in x
• 2. Consider the context-free grammar G with start
  symbol S and productions
• S ? aAB aBA bAA e A ? aS bAAA
• B ? aABB aBAB aBBA bS
• Show that
• 2.1 L(G) ? A, and
• 2.2 A ? L(G) Hence L(G) A.
• Hints (for 2.1)
• For A we have A ? a ? a(a) A(a)
  12b(a)2B(a).
• For B and S, you should find similar
  assertions.
• 3. Find a nondeterministic pushdown automata
  (pda) which accepts the language A by empty
  stack.

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• 4. Consider the grammar
• S?ST a T? BS B? b
• the input ababa is a member of L(G).
• 2.1 Draw a parse tree for it.
• 2.2 Give a left-most derivation for the input
  ababa.
• 5. Run the CYK algorithm on the input string
  ababa for the above grammar. You should draw a
  chart and fill it completely.

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• 6. Refine the CKY algorithm so that it can count
  the number of parse trees of a given string.
• Hint In such modification, each cell T(i,j)
  would be a list of pairs (X, k) ,where X is a
  nonterminal and k is a positive number, meaning
  that the are k X-parse trees for the substring
  xij. Accordingly, the input would have exactly k
  parse trees if (S, k) belongs to the list T(0.n).
• 7. Use pumping lemma to show that the language A
  ww w ? a,b is not context free.
• 8. Design a PDA to accept the language
• B x?a,b x is even A, where A
  ww w ? a,b .
• Hint See page 10 of p2ch3.ppt