Automata and Formal Languages. Homework 4 (1.10.2009) 1. Let

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					         Automata and Formal Languages. Homework 4 (1.10.2009)

1. Let L be the language recognized by the following DFA:
                                            a                  b

                                                          a


                                                      b        a




                                                      b


  Construct a DFA that recognizes the language

   (a) h−1 (L) where h is the homomorphism h(0) = bab, h(1) = aba.
   (b) L/baa∗ .

2. Prove that if L ⊆ Σ∗ is regular so is the language

   (a) Prefix(L) = {u ∈ Σ∗ | ∃v ∈ Σ∗ : uv ∈ L },
   (b) Suffix(L) = {v ∈ Σ∗ | ∃u ∈ Σ∗ : uv ∈ L },
    (c) Subword(L) = {w ∈ Σ∗ | ∃u, v ∈ Σ∗ : uwv ∈ L }.

3. Show that the family of regular languages is closed under the following operations:

   (a) Half(L) = {w | ww ∈ L},
   (b) Shuffle(L1 , L2 ) = {u1 v1 u2 v2 . . . uk vk | u1 u2 . . . uk ∈ L1 , v1 v2 . . . vk ∈ L2 }.

  (Language Half(L) contains all words such that ww is in L, and language Shuffle(L1 , L2 )
  contains all words that can be obtained by shuffling together the letters of some words
  u ∈ L1 and v ∈ L2 . The shuffle needs not to be perfect, i.e., the words ui and vi in the
  definition are elements of Σ∗ and not necessarily single letters.)

4. Prove that there are algorithms to determine

   (a) if a given regular language L is prefix-closed, i.e., uv ∈ L =⇒ u ∈ L,
   (b) if a given regular language L is prefix-free, i.e., uv ∈ L, v = ε =⇒ u ∈ L,
    (c) if all words of a given regular language L are square-free, i.e, ww ∈ L for all
        w ∈ Σ∗ .
5. Show that there are algorithms to determine

   (a) if a given regular language L is a code (see Problem 4 of the homework set 1 for
       the definition of codes),
   (b) if a given NFA is unambiguous (see Problem 4 of the homework set 2 for the
       definition of unambiguity).

6. Find the minimum state DFA for the language recognized by the following NFA:

                                          a                a

                                              b

                                          b       a
                                      a



7. Prove that for every n ≥ 1 there exists an NFA with n states such that the equivalent
   minimum state DFA has 2n states. (Hint: recall Problem 6 in the second homework
   set.)