jsll-chap3

S S ( S )
∴ CFG G = ({(, ), S, T }, {(, )}, R, S ) where R = {S → T | , T → () | (T ) | T T } 6. (Problem 3.2.4) (b) id + (id + id) ∗ id E
E T F id
+
T
T F
11. (Problem 3.5.2) (c) {www : w ∈ {a, b}∗ } L = {www : w ∈ {a, b}∗ }
p p
CFL p
p
s = a ba ba b s (1) b (2) v v, y y y b uxz a
2
s = uvxyz
|vxy | 2
2 2
p v s∈ /L a 3 CFL a b uxz 2 b
2
|vy | > 0 uv xy z uv xy z ∈ /L {www : w ∈ {a, b}∗ }
5
*
F id
( E T F id (id ∗ id + id ∗ id)
E + T F id
)
2
E T F
(
E
)
E T T F id 7. (Problem 3.3.1) (a) (s, aba, ) (s, aba, ) (s, aba, ) ∴ aba (b) aba aa (s, aa, ) (s, aa, ) (s, aa, ) abb (s, aab, ) (s, aab, ) baa (s, baa, ) (s, aa, a) (f, a, a) 3 (f, bb, ) (s, bb, a) (a) (f, a, ) (s, a, a) (s, a, a) (f, , a) (s, , aa) aba (s, ba, a) (f, ba, ) (s, ba, a) (s, a, aa) 3 (s, a, aa) * F id
+ T F id来自T * F id(s, , aaa) (f, , aa)
Rejected. rejected
(s, b, aa) (s, , aaa) rejected
(f, , ) accepted
bab (s, bab, ) baaaa (s, baaaa, ) (s, aaaa, a) (s, aaa, aa) (f, aa, aa) (f, a, a) (f, , ) accepted (c) L = {xay | x, y ∈ {a, b}∗ ∧ |x| = |y |} 8. (Problem 3.3.2) (b) One of the PDA that accepted L = {am bn : m M = {K, Σ, Γ, ∆, q, F } n 2m} is: F = {r } and (s, ab, a) (f, b, a) (f, , ) accepted
4
(( )( ))
PDA (p, (( )( )), ) (q, ( )( )), S ) (q, )( )), S )S )) (q, ( )), S )) (q, )), ))) (q, (( )( )), S ) (q, ( )( )), SS ) (q, ) ( )), )S )) (q, )), S ))) (q, , ) (q, ( )), (S ))) (q, ), )) (q, (( )( )), (S )) (q, ( )( )), (S )S ))
December 10, 2002 by Wang Weiwei
Chap.3, Context-free Languages
1. (Problem 3.1.1) (a) aa, baa, aba, aab (b) S ⇒ AA ⇒ bAA ⇒ bAAb ⇒ bAbAb ⇒ bAbbAb ⇒ babbAb ⇒ babbab S ⇒ AA ⇒ bAA ⇒ bAAb ⇒ bAbAb ⇒ bAbbAb ⇒ bAbbab ⇒ babbab S ⇒ AA ⇒ bAA ⇒ bAbA ⇒ babA ⇒ babbA ⇒ babbAb ⇒ babbab S ⇒ AA ⇒ bAA ⇒ bAAb ⇒ baAb ⇒ babAb ⇒ babbAb ⇒ babbab (c) S ⇒ AA ⇒ · · · ⇒ bm AA · · · ⇒ bm Abn A ⇒ · · · ⇒ bm Abn Abp ⇒ bm Abn abp ⇒ bm abn abp 2. (Problem 3.1.3) (b) Context-free Grammar for {wwR | w ∈ {a, b}∗ } G = {V, Σ, R, S } 3. (Problem 3.1.5) (a) S ⇒ aB ⇒ abS ⇒ abaB ⇒ ababS ⇒ ababbA ⇒ ababba 4. (Problem 3.1.9) (f) Context-free gramma for {am bn : m G = {V, Σ, R, S } 5. (Problem 3.2.2) 2n} where V = {a, b, S } Σ = {a, b} R = {S → aSa | bSb | }
where R = {S → , S → aSb, S → aaSb}
G = {V, Σ, R, S } ∵ S ⇒ (S ) ⇒ ()
V = {S, (, )}
Σ = {(, )}
R = {S → | SS | (S )}
S ⇒ SS ⇒ S (S ) ⇒ S () ⇒ ()
1
S ( S ) S
10. (Problem 3.5.1) (a) {am bn : m = n} La = {am bn : m = n} L2 = {am bn : m > n} La = L1 ∪ L2 L1 Σ = {a, b} (V, Σ, R1 , S ) R1 = {S → T b, T → | T b | aT b} L2 = L(G2 ) L1 L2 G2 = (V, Σ, R2 , S ) R2 = {S → aT, T → | aT | aT b} CFL 3.5.1 La CFL L2 CFL N = {s, r} L1 = L(G1 ) G1 = V = Σ∪N L1 = {am bn : m < n}
M = {{p, q }, {(, )}, {(, ), S }, ∆, p, {q }}
∆
=
{((p, , ), (q, S )), ((p, , S ), (q, SS )), ((q, , S ), (q, (S ))), ((q, , S ), (q, )), ((q, (, (), (q, )), ((q, ), )), (q, ))}
where K = {q, r} Γ = {a, b, Z } ∆ = {((q, a, ), (q, aa)), ((q, a, ), (q, a)), ((q, , ), (r, e)), ((r, b, a), (r, ))}
PDA
a, Z/aaZ , Z/Z b, aZ/Z
a, Z/aZ
9. (Problem 3.4.1) 3.4.1 3.1.4 PDA