FL2006-MidTermExamSol

							                                  Midterm Examination                                2006/12/1
                            Formal Language and Automata Theory

姓名:________________ 學號:_____________________                              分數:___________


1. (6 pts) Let  = {a, b}. Then 3 = {aaa,aab,aba,abb,baa,bab,bba,bbb};
                               + - 3* = {a,b,aa,cb,bb,ba}.

2. (10pts) Let A and B be two languages. Then which of the followings always hold?
    (a) (A+)* = A+      (b) (A*)+ = A+       (c) (A U A2 )⊆B => A*⊆ B.
   (d) A(BC) = AB  AC            (e) A(BC) = AB AC
 ANS:_________d___________________________


3. (45 pts) Given the following NFA machine M allowing  transitions :

                      M                     a           b           c
                     p           {}          p           q           r
                       q          p           q           r           {}
                       rF         q            r          {}          p


    (a) Draw a state transition diagram for the NFA M. [8 pts]




    (b) Let M = (Q, , , S, F). Then what are the contents of Q, , , S and F, respectively.
        [7 pts]
        Q = _____________________________            = _____________________
        S = {p}                                     F =__________________________
        =




    (c) List all the strings of length ≤ 3 accepted by M. [5 pts]
        ANS: _c,ac,ca,bc,bb,cb,...______________________________


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     (d) Find a right linear grammar equivalent to M. Remember to designate the start symbol;
        otherwise it is assumed to be the LHS of the first rule. [10 pts]
       Ans:
         p  ap | bq | cr
         q br | aq | p
         r  ar | cp | q | e


     (e) Let pe denote -closure({p}). It is known that for the machine M, pe = {p} is the
         -closure of p. What are the -closures of the other two states? [4pts]

         ANS:        qe = ____{p,q}__________ re = _____{p,q,r}________


     (f) Convert the automata M to a DFA M' by filling in the following table: [11pts]

         M'                       a                   b                      c
         Se = pe = {p}            p                  pq                     pqr
         pq                       pq                  pqr                    pqr
         pqrF                     pqr                 pqr                    pqr




Notes: 1. Useless states of new machine must not be listed in the table. 2. The table rows are
provided to you for your convenience; you may probably need additional rows or may not use
up all table rows. 3. S is the set of initial states of M and you must have known that its
-closure Se is the initial state of M'. 4. Remember to mark final states.

4.   [14 pts] Consider the following NFA.


              M:             a           b
             p              q           r
                q            r          p,q
                rF           p           r


     (a) [7pts] construct a strongly left linear grammar equivalent to M.
        Ans:
         Sr
         p  ra | qb | e
         q  pa | qb
         r  pb | qa | rb

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     (b) Give a regular expression equivalent to M. [7 pts]


     Ans:   [ab*b + (ab*a + b)b* a]* [ab*a + b] b*



5. [20pts] For any n ≥ 1, let An = { x 1 an-1 | x ∈ {0,1}* , a ∈ {0,1} } be the set of all
   strings of which the n-th bit counted from the right is 1.
    (a) Find a NFA with n+1 states accepting An. [5pts]




     Let M be any DFA accepting An. We want to show that M must have at least 2n states.
Thus the exponential explosion of states while converting a NFA to an equivalent DFA in
worst case is inevitable. The proof proceeds as follows.
   Let x = xn xn-1… x1 and y= yn yn-1… y1 be any two distinct strings of length n, and let p
and q be the states of M that are reached when x and y are consumed by M, respectively. In
other words, (s,x) = p and (s,y)=q, where s in the initial state of M.

  (b) Show why it must be the case that p ≠ q. [10pts]


  Pf: Since x ≠ y, there exist an index k such that xk ≠ yk. Without loss of generality, let
xk = 0 and yk = 1. Now the string x 0n-k ∉ An since the last n-th bit of it = xk = 0 and y 0n-k ∈
An since the last n-th bit of it = yk = 1. But if p = q, we would have (s,x0n-k) = (p,0n-k)
=(q,0n-k) =(s,y0n-k) and hence either both accept or both reject. This is impossible. Hence p
could not be equal to q.


  (c) According to (b), why must M have at least 2n states? [5pts]


  Since there are 2n distinct strings z of length n, each reaching a distinct state (s,z) on any
machine M recognizing An. As a result, M must contain at least 2n states.



6. [20pts] Let A = {x | x  {a,b}* and contains as many a’s as b’s.}, and the grammar G
   given by: S   | aSb | bSa | SS.
    (a) Give a parse tree of the input 'ababba'. [5pts]




     

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        Show that L(G) = A in two steps:
        (b) L(G)⊆A. hint: by induction on n that if S n  then  contains as many a's as b's.
        (c) A ⊆ L(G). hint: show that if x∈A, then there exists a derivation of it. [7pts]

       Ans:
         (b) We will show that if S *  then #a( ) = #b(). As a result, if ∈ L(G), then
              S *  and must be a terminal string, and hence belongs to A.


        The proof is by ind. on n such that Sn 
        For the case that n = 0: S0  iff  = S. Then #a( ) = #b() = 0.
        For the case n = k+1. By ind.hyp, #a(') = #b(') if Sk '
        Now if Sn  , It must be one of the following cases:
        S k S SS = .
        S k S aSb = .
        S k S bSa = .
        S k S  = .
 In all cases, we can find that #a() = #b() given the ind.hyp. that #a(S) = #b(S)


       (c) The proof is by induction on the length of x ∈ A.
         If |x| = 0, then x =  and S  . (ok)
         If x = x1x2…,xn is a string of length n > 0.
         Then there are 3 cases to consider depending on the values of x1 and xn.
       Case 1: x1xn = ab. Then x2…xn-1 is a string < n and also belongs to A. Hence, by ind.hyp.,
            S * x2 … xn-1. As a result S aSb * x1…xn = x.
       Case 2: x1xn = ba. Similar to Case 1.
       Case3: x1xn = aa or bb. Then there must exist a split of x into two nonempty substrings yz
          such that x = yz and both y and z are members of A. Since y and z are shorter than x,
by ind.hyp., there exist derivations for either string: S * y and S * z. As a result, we can
form a derivation of x as follows: SSS*yS * yz = x.




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7. [20 pts] Let A2 = {x | x  {a,b}* and contains twice as many a’s as b’s.}.
    (a) Using [the game-theoretical argument of ] the pumping lemma to show that A2 is not
        regular.


       Ans: The game proceeds as follows:
       1. Suppose D pick a number k > 0.
       2. Y choose a string xyz ∈ A2 with x = a2k, y = bk and z = .
       3. Suppose D decompose y into u v w with |v| = t ≠ 0.
       4. Y let i = 0.
       Now the resulting string xuviwz = a 2k bk - t ∉ A2, and Y wins.
       Since Y has a winning strategy in this game, by pumping lemma, A2 is not regular.




    (b) Design a context free grammar G without more than 5 rules to generate the language
        A2, i.e., L(G) = A2.


     Ans:    S  e | aaSb | abSa | baSa | SS.




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