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自動機理論與正規語言
期中考試
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一、 是非題(30%)
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1. Let F denote a function, S1 be the domain of F, and S2 be its range. If the
domain of F is all of S1, then F is a total function on S1.
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2. A walk is a path in which no edge is repeated, and a path is simple if no
vertex is repeated.
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3. A deterministic automation is one in which each move is uniquely
determined by the current configuration.
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4. Compared to an accepter, a transducer is a more general automation,
capable of producing strings of symbols as output.
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5. A string in a language L will be called a sentence of L.
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6. If functions f and g have the same order of magnitude, then
f (n)  ( g (n)) .
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7. O(n)  O(n)  2O(n) .
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8. A language L is called regular if and only if there exists some
non-deterministic finite accepter M such the L = L(M).
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9. The transition function of a non-deterministic finite accepter (nfa)
should be a total function.
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10. The transition function of a non-deterministic finite accepter (nfa) is
defined as  : Q    2 Q .
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11. Digital computers are completely nondeterministic.
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12. Language L  {a n : n  0}  {b n a : n  1} can be accepted by an four state
nfa.
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13. Two finite accepters are said to be equivalent if they both accept the
same language.
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14. The language accepted by a deterministic finite accepter (dfa)
M  (Q, ,  , q0 , F ) is L(M )  {w  * :  * (q0 , w)  F} .
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15.  ,  , and a   are all regular expressions. These are called
primitive regular expressions.
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16. Let r be a regular expression, then L(r) is a regular language.
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17. Let L be a regular language, then there exists a regular expression r
such that L=L(r).
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18. A grammar G  (V , T , S , P) is said to be right-linear if all producations
are of the form A  xB , A  x , where A, B  V , and x  T * .
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19. Regular grammars should be linear.
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20. Let G be a right-linear grammar, then L(G) is a regular language.
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21. A generalized transition graph is a transition graph whose edges are
labeled with regular expressions.
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22. The family of regular languages is closed under union, intersection,
concatenation, complementation, and star-closure.
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23. The family of regular languages is closed under reversal, difference,
right quotient, and homomorphism.
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24. If L is a language on  , then its homomorphic image is defined as
h( L)  {h( w) : w  L} .
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25. Let L1 and L2 be a language on the same alphabet, then right quotient
of L1 with L2 is defined as L1 L2  {x : xy  L2 for some y  L1} .
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26. A regular language is given in a standard representation if and only if it
is described by a finite automaton, a regular expression, or a regular
grammar.
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27. A context-free grammar G  (V , T , S , P) is said to be a simple grammar
if all its productions are of the form A  ax , where AV , a  T , and
x T * .
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28. If every grammar that generates a context-free language L is
ambiguous, then the language is called inherently ambiguous.
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29. Language L  {a n b m c k : n  2m  k , n, m, k  0} is not context-free.
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30. A grammar G  (V , T , S , P) is said to be contest-free if all productions
in P have the form A  x , where AV and x  (V  T ) * .
二、 請寫出下列 grammars 所產生出的 languages。(9%)
1. (3%)The grammar G  ({S},{a, b}, S , P) , with productions S  aSa , S  bSb ,
S .
Ans:
2. (3%)The grammar G  ({S , A, B},{a, b}, S , P) , with productions S  aAB , A  bBb ,
B  A| .
Ans:
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(3%) The grammar G  ({S , A, B},{a, b}, S , P) , with productions S  Aab ,
A  Aab | B , B  a .
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三、 (10%)針對下列各 regular languages，請寫出適當的 left-linear grammars。
1.(5%) L  {w : na (w) and nb (w) are both even}
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2.(5%) L(aa * (ab  a)*)
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四、 (4%)請寫出下列 finite automata 所能接受的 language。
a, b
b
q1
q0
b
q2
a
a
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五、 (8%)請寫出下列 regular languages 的 regular expressions。
1. (4%) L  a n b m : (n  m) is even 
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2. (4%) L(r )  w {0,1}* : w contains an even number of 0' s
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六、 (8%)請畫出可接受下列語言的 transition graph。(不可以畫 generalized
transition graph)
1. (4%) L(( a  b) * b(a  bb)*)
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2. (4%) L  w : na (w)  nb (w)
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七、 (4%) Let L1  L(a * baa*) and L2  L(aba*) . Find L1 L2 .
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八、 (5%) 給定一 grammar G  ({S , A, B},{a, b}, S , P) , with productions S  aAB ,
A  bBb , B  A |  。請畫出 string abbbb 的 derivation tree.
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九、 (6%)針對下列各 languages，請寫出適當的 context-free grammars。
1. (3%) L  {a n b m | 2n  m  3n, n, m  0}
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2. (3%) L  {w  {a, b}* : na (w)  nb (w) and na (v)  nb (v), where v is any prefix of w}
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十、 (16%)下列証明請任選二題做做看，答題時請註明題號。(請一定只選二題，
每多做一題扣 5%)。
1. Please show that if L1 and L2 are regular languages, then so is L1  L2 .
2. Let G  (V , , S , P) be a right-linear grammar, the L(G) is a regular language.
3. Procedure: nfa-to-dfa
4. A pumping lemma: Let L be an infinite regular language. Then these exists some
positive integer m such that any w L with | w | m can be decomposed as
w  xyz , with | xy | m , and | y | 1 , such theat wi  xy i z is also in L for all
i  0,1,2,3,... .
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