1. Chapter 8. PROPERTIES OF CONTEXT-FREE LANGUAGES Exercises
DS020 오토마타형식언어
Chapter 8. PROPERTIES OF CONTEXT-FREE LANGUAGES
Exercises
September 25, 2003

2. 7. Show that the following languages on are not context-free.
8.1 TWO PUMPING LEMMAS
7. Show that the following languages on are not context-free.
w = a(m-1)^3bm
Solution
(m-1) m
a … … a b … b
1) |vxy|
2) |v x y|
3) |vxy|
i = 0
If |v| = 0  i = 2, else if |y| = 0  i = 0, otherwise (*)
i = 2
i = 2  a는 (m-1)3 + k1개, b는 (m+k2)개
(m+k2-1)3 ≥ m3 = (m-1)3 + 3m2 -3m + 1 > (m-1)3 + k1
따라서 pumping된 문자열은 L에 속하지 않게 됨  Not CFL
September 25, 2003

3. w = ambmcm+1 a … a b … b c … c |vxy| i = 2 2) |v x y| 3) |vxy|
8.1 TWO PUMPING LEMMAS
w = ambmcm+1
Solution
m m m+1
a … a b … b c … c
|vxy|
2) |v x y|
3) |vxy|
4) |v x y|
5) |vxy|
i = 2
If |y| = 0  i = 2, otherwise i = 0
i = 0
September 25, 2003

4. w = ambm+1cm+1 a … a b … b c … c |vxy| i = 2 2) |v x y|
8.1 TWO PUMPING LEMMAS
w = ambm+1cm+1
Solution
m m m+1
a … a b … b c … c
|vxy|
2) |v x y|
3) |vxy|
4) |v x y|
5) |vxy|
i = 2
If |y| = 0  i = 2, otherwise i = 0
i = 0
If |v| = 0  i = 2, otherwise i = 0
September 25, 2003

5. 11. Show that the language is context-free but not linear.
8.1 TWO PUMPING LEMMAS
11. Show that the language is context-free but not linear.
Solution
1) context free grammar
S  S1 S2
S1  a S1 b | λ
S2  a S2 b | λ
2) not linear
w = ambm ambm
m m m m
a … a b … b a … a b … b
|uv| |yz| i = 2
September 25, 2003

6. 8.2 CLOSURE PROPERTIES AND DECISION ALGORITHMS
FOR CONTEXT-FREE LANGUAGES
6. Which of the language families we have discussed are not closed under reversal?
September 25, 2003

7. 8.2 CLOSURE PROPERTIES AND DECISION ALGORITHMS
FOR CONTEXT-FREE LANGUAGES
10. Show that the family of linear language is not closed under intersection.
September 25, 2003

8. 8.2 CLOSURE PROPERTIES AND DECISION ALGORITHMS
FOR CONTEXT-FREE LANGUAGES
19. Is the family of deterministic context-free languages closed under homomorphism?
September 25, 2003