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Cap´ıtulo 5
- Elimine as produ¸c˜oes in´uteis na seguinte gram´atica:
S ▷ a | aA | B | C A ▷ aB | λ B ▷ Aa C ▷ cCD D ▷ dd | ddD
Resolu¸c˜ao;
I = {C, D}
S ▷ a | aA | B A ▷ aB | λ B ▷ Aa
G′^ = ⟨{S, A, B}, {a}, S, P ′⟩
- Elimine as λ-produ¸c˜oes de:
S ▷ AaB | aCaB A ▷ aBa | λ | B B ▷ bbA | λ C ▷ bbbC | AB
Resolu¸c˜ao;
Vλ = {A, B, C}
S ▷ AaB | aCaB A ▷ aBa | B B ▷ bbA C ▷ bbbC | AB
S ▷ aB | Aa | a | aaB | aCa | aa A ▷ aa B ▷ bb C ▷ bbb | B | A
S ▷ AaB | aCaB | aB | Aa | a | aaB | aCa | aa A ▷ aBa | B | aa B ▷ bbA | bb C ▷ bbbC | AB | bbb | B | A
G′^ = ⟨{S, A, B, C}, {a, b}, S, P ′⟩
- Elimine as produ¸c˜oes unit´arias nas duas gram´aticas anteriores.
Resolu¸c˜ao;
- S ▷ a | aA | B | C f echo − S = {B, C} A ▷ aB | λ f echo − A = ∅ B ▷ Aa f echo − B = ∅ C ▷ cCD f echo − C = ∅ D ▷ dd | ddD f echo − D = ∅
S ▷ a | aA | Aa | cCD A ▷ aB | λ B ▷ Aa C ▷ cCD D ▷ dd | ddD
G′^ = ⟨{S, A, B, C, D}, {a, c, d}, S, P ′⟩
- S ▷ AaB | aCaB f echo − S = ∅ A ▷ aBa | λ | B f echo − A = {B} B ▷ bbA | λ f echo − B = ∅ C ▷ bbbC | AB f echo − C = ∅
S ▷ AaB | aCaB A ▷ aBa | λ | bbA B ▷ bbA | λ C ▷ bbbC | AB
G′^ = ⟨{S, A, B, C}, {a, b}, S, P ′⟩
- Elimine produ¸c˜oes unit´arias, λ-produ¸c˜oes, produ¸c˜oes in´uteis e produ¸c˜oes esquerda-recursivas nas se- guintes gram´aticas livres do contexto: (a) S ▷ Sa | aAB A ▷ B | aa | λ B ▷ AbB | λ | C C ▷ aCCa | ABC
Resolu¸c˜ao;
Remo¸c˜ao de Produ¸c˜oes in´uteis
I = {C}
S ▷ Sa | aAB A ▷ B | aa | λ B ▷ AbB
Remo¸c˜ao de λ-produ¸c˜oes
Vλ{A, B}
S ▷ ASAS | B
A ▷ aAa B ▷ bB | C C ▷ aCb | ab
S ▷ SAS | ASS | SS A ▷ aa
S ▷ ASAS | B | SAS | ASS | SS A ▷ aAa | aa B ▷ bB | C C ▷ aCb | ab
Remo¸c˜ao de produ¸c˜oes unit´arias
S ▷ ASAS | B | SAS | ASS | SS f echo − S = {B} A ▷ aAa | aa f echo − A = ∅ B ▷ bB | C f echo − B = {C} C ▷ aCb | ab f echo − C = ∅
S ▷ ASAS | bB | aCb | ab | SAS | ASS | SS A ▷ aAa | aa B ▷ bB | aCb | ab C ▷ aCb | ab
Remo¸c˜ao de produ¸c˜oes esquerda-recursivas
S ▷ ASAS | bB | aCb | ab | ASS | ASASZS | bBZS | aCbZS | abZS | ASSZS ZS ▷ AS | S | ASZS | SZS A ▷ aAa | aa B ▷ bB | aCb | ab C ▷ aCb | ab
G′^ = ⟨{S, ZS , A, B, C}, {a, b}, S, P ′⟩
(c) S ▷ aA | aB | SaA A ▷ B | aa | λ B ▷ bB | λ | C C ▷ aCa | ABC
Resolu¸c˜ao;
Remo¸c˜ao de Produ¸c˜oes in´uteis
I = {C}
S ▷ aA | aB | SaA A ▷ B | aaλ B ▷ bB | λ
Remo¸c˜ao de λ-produ¸c˜oes
Vλ{A, B}
S ▷ aA | aB | SaA A ▷ B | aa B ▷ bB
S ▷ a | Sa B ▷ b
S ▷ aA | aB | SaA | a | Sa A ▷ B | aa B ▷ bB | b
Remo¸c˜ao de produ¸c˜oes unit´arias
S ▷ aA | aB | SaA | a | Sa f echo − S = ∅ A ▷ B | aa f echo − A = {B} B ▷ bB | b f echo − B = ∅
S ▷ aA | aB | SaA | a | Sa A ▷ bB | b | aa B ▷ bB | b
Remo¸c˜ao de produ¸c˜oes esquerda-recursivas
S ▷ aA | aB | a | aAZS | aBZS | aZS Zs ▷ aA | a | aAZS | aZS A ▷ bB | b | aa B ▷ bB | b
G′^ = ⟨{S, ZS , A, B}, {a, b}, S, P ′⟩
(d) S ▷ AbAaBa | ABSa | aAAbC A ▷ aAa | λ B ▷ bbB | BCa | aD | λ C ▷ cAE | aCCa D ▷ E | F E ▷ aCb F ▷ F aF | F SaA G ▷ SaH H ▷ aA | λ
Resolu¸c˜ao;
Remo¸c˜ao de Produ¸c˜oes in´uteis
I = {C, D, E, F, G, H}
S ▷ AbAaBa | ABSa A ▷ aAa | λ B ▷ bbB | λ
G′^ = ⟨{M 0 , M 1 , M 2 , M 3 , S, A}, {a, b}, {S}, P ′⟩
c) S ▷ abAB A ▷ bAB | λ B ▷ BAa | A | λ
Resolu¸c˜ao;
fecho-B = {A}
S ▷ abAB A ▷ bAB | λ B ▷ BAa | bAB | λ
M 0 ▷ a M 1 ▷ b M 2 ▷ M 0 M 1 M 3 ▷ AB S ▷ M 2 M 3 A ▷ M 1 M 3 | λ M 4 ▷ BA B ▷ M 4 M 0 | M 1 M 3 | λ
G′^ = ⟨{M 0 , M 1 , M 2 , M 3 , M 4 , S, A, B}, {a, b}, {S}, P ′⟩
d) S ▷ AaA | AbA A ▷ AaC | bAAa | λ B ▷ bC | A C ▷ D | a | ba D ▷ bAb | ba
Resolu¸c˜ao;
fecho-B = {A} fecho-C = {D}
S ▷ AaA | AbA A ▷ AaC | bAAa | λ B ▷ bC | AaC | bAAa | λ C ▷ a | ba | bAb D ▷ bAb | ba
I = {B, D}
S ▷ AaA | AbA A ▷ AaC | bAAa | λ C ▷ a | ba | bAb
M 0 ▷ a
M 1 ▷ b M 2 ▷ AM 0 M 3 ▷ M 1 A S ▷ M 2 A | AM 3 A ▷ M 2 C | M 3 M 2 | λ C ▷ a | M 1 M 0 | M 3 M 1
G′^ = ⟨{M 0 , M 1 , M 2 , M 3 , S, A, C}, {a, b}, {S}, P ′⟩
e) S ▷ aA | aB | SaAB A ▷ B | aa | λ B ▷ bB | Ca | λ C ▷ aCa | ABC
Resolu¸c˜ao;
fecho-A = {B}
S ▷ aA | aB | SaAB A ▷ aa | λ | bB | Ca B ▷ bB | Ca | λ C ▷ aCa | ABC
M 0 ▷ a M 1 ▷ b M 3 ▷ AB M 4 ▷ M 0 C M 5 ▷ SM 0 S ▷ M 0 A | M 0 B | M 5 M 3 A ▷ M 0 M 0 | λ | M 1 B | CM 0 B ▷ M 1 B | CM 0 | λ C ▷ M 4 M 1 | M 3 C
G′^ = ⟨{M 0 , M 1 , M 3 , M 4 , M 5 , S, A, B, C}, {a, b}, {S}, P ′⟩
f) S ▷ SaS | ABAa A ▷ bAAa | λ B ▷ bCb | C C ▷ aC | ba
Resolu¸c˜ao;
fecho-B = {C}
S ▷ SaS | ABAa A ▷ bAAa | λ B ▷ bCb | aC | ba C ▷ aC | ba
M 0 ▷ a
D ▷ M 1 M 3 | AM 3 | M 0 C | M 1 B | b
G′^ = ⟨{M 0 , M 1 , M 2 , M 3 , M 4 , M 5 , S, A, B, C, D}, {a, b}, {S}, P ′⟩
h) S ▷ aC A ▷ ACb | bA | λ B ▷ bAa C ▷ BA | ba
Resolu¸c˜ao;
M 0 ▷ a M 1 ▷ b M 2 ▷ AC M 3 ▷ M 1 A S ▷ M 0 C A ▷ M 2 M 1 | M 1 A | λ B ▷ M 3 M 0 B C ▷ BA | M 1 M 0
G′^ = ⟨{M 0 , M 1 , M 2 , M 3 , S, A, B, C}, {a, b}, {S}, P ′⟩
- Transformar as gram´aticas resultantes do exemplo anterior `a forma normal de Greibach
a) M 0 ▷ a M 1 ▷ b M 2 ▷ SM 1 S ▷ M 0 M 2 | M 0 M 1
Resolu¸c˜ao;
A 4 ▷ a A 3 ▷ b A 2 ▷ A 1 A 3 A 1 ▷ A 4 A 2 | A 4 A 3
A 4 ▷ a A 3 ▷ b A 2 ▷ A 4 A 2 A 3 | A 4 A 3 A 3 A 1 ▷ A 4 A 2 | A 4 A 3
A 4 ▷ a A 3 ▷ b A 2 ▷ aA 2 A 3 | aA 3 A 3 A 1 ▷ aA 2 | aA 3 G′^ = ⟨{A 1 , A 2 , A 3 , A 4 }, {a, b}, {A 1 }, P ′⟩
b) M 0 ▷ a M 1 ▷ b M 2 ▷ M 1 A M 3 ▷ M 0 S S ▷ M 3 M 2 | M 0 A A ▷ M 0 M 2 | b
Resolu¸c˜ao;
A 6 ▷ a A 3 ▷ b A 2 ▷ A 3 A 5 A 4 ▷ A 6 A 1 A 1 ▷ A 4 A 2 | A 6 A 5 A 5 ▷ A 6 A 2 | b
A 6 ▷ a A 3 ▷ b A 2 ▷ bA 5 A 4 ▷ aA 1 A 1 ▷ aA 1 A 2 | aA 5 A 5 ▷ aA 2 | b G′^ = ⟨{A 1 , A 2 , A 3 , A 4 , A 5 , A 6 }, {a, b}, {A 1 }, P ′⟩
- Converter as seguintes gram´aticas `a forma normal de Greibach
a) S ▷ aaS | SS | aa
Resolu¸c˜ao;
M 0 ▷ a S ▷ M 0 M 0 S | SS | M 0 M 0
A 2 ▷ a A 1 ▷ A 2 A 2 A 1 | A 1 A 1 | A 2 A 2
A 2 ▷ a A 1 ▷ A 2 A 2 A 1 | A 2 A 2 | A 2 A 2 A 1 A 3 | A 2 A 2 A 3 A 3 ▷ A 1 | A 1 A 3
A 2 ▷ a A 1 ▷ A 2 A 2 A 1 | A 2 A 2 | A 2 A 2 A 1 A 3 | A 2 A 2 A 3 A 3 ▷ A 2 A 2 A 1 | A 2 A 2 | A 2 A 2 A 1 A 3 | A 2 A 2 A 3 | A 2 A 2 A 1 A 3 A 3 | A 2 A 2 A 3 A 3
A 2 ▷ a A 1 ▷ A 2 A 2 A 1 | A 2 A 2 | A 2 A 2 A 1 A 3 | A 2 A 2 A 3 A 3 ▷ aA 2 A 1 | aA 2 | aA 2 A 1 A 3 | aA 2 A 3 | aA 2 A 1 A 3 A 3 | aA 2 A 3 A 3
A 2 ▷ a A 1 ▷ aA 2 A 1 | aA 2 | aA 2 A 1 A 3 | aA 2 A 3 A 3 ▷ aA 2 A 1 | aA 2 | aA 2 A 1 A 3 | aA 2 A 3 | aA 2 A 1 A 3 A 3 | aA 2 A 3 A 3
G′^ = ⟨{A 1 , A 2 , A 3 }, {a}, {A 1 }, P ′⟩
c) S ▷ aA | bBb A ▷ aBa | aa B ▷ Abb | Cab C ▷ Aba | Bab | aba
Resolu¸c˜ao;
M 0 ▷ a M 1 ▷ b S ▷ M 0 A | M 1 BM 1 A ▷ M 0 BM 0 | M 0 M 0 B ▷ AM 1 M 1 | CM 0 M 1 C ▷ AM 1 M 0 | BM 0 M 1 | M 0 M 1 M 0
A 6 ▷ a A 2 ▷ b A 1 ▷ A 6 A 5 | A 2 A 3 A 2 A 5 ▷ A 6 A 3 A 6 | A 6 A 6 A 3 ▷ A 5 A 2 A 2 | A 4 A 6 A 2 A 4 ▷ A 5 A 2 A 6 | A 3 A 6 A 2 | A 6 A 2 A 6
A 6 ▷ a A 2 ▷ b A 1 ▷ A 6 A 5 | A 2 A 3 A 2 A 5 ▷ A 6 A 3 A 6 | A 6 A 6 A 3 ▷ A 5 A 2 A 2 | A 4 A 6 A 2 A 4 ▷ A 5 A 2 A 6 | A 5 A 2 A 2 A 6 A 2 | A 4 A 6 A 2 A 6 A 2 | A 6 A 2 A 6
A 6 ▷ a A 2 ▷ b A 1 ▷ A 6 A 5 | A 2 A 3 A 2 A 5 ▷ A 6 A 3 A 6 | A 6 A 6 A 3 ▷ A 5 A 2 A 2 | A 4 A 6 A 2 A 4 ▷ A 5 A 2 A 6 | A 5 A 2 A 2 A 6 A 2 | A 6 A 2 A 6 | A 5 A 2 A 6 A 7 | A 5 A 2 A 2 A 6 A 2 A 7 | A 6 A 2 A 6 A 7 A 7 ▷ A 6 A 2 A 6 A 2 | A 6 A 2 A 6 A 2 A 7
A 6 ▷ a A 2 ▷ b A 1 ▷ A 6 A 5 | A 2 A 3 A 2 A 5 ▷ A 6 A 3 A 6 | A 6 A 6 A 3 ▷ A 5 A 2 A 2 | A 4 A 6 A 2 A 4 ▷ A 5 A 2 A 6 | A 5 A 2 A 2 A 6 A 2 | A 6 A 2 A 6 | A 5 A 2 A 6 A 7 | A 5 A 2 A 2 A 6 A 2 A 7 | A 6 A 2 A 6 A 7 A 7 ▷ aA 2 A 6 A 2 | aA 2 A 6 A 2 A 7
A 6 ▷ a A 2 ▷ b A 1 ▷ aA 5 | bA 3 A 2 A 5 ▷ aA 3 A 6 | aA 6 A 3 ▷ aA 3 A 6 A 2 A 2 | aA 6 A 2 A 2 | aA 3 A 6 A 2 A 6 A 6 A 2 | aA 6 A 2 A 6 A 6 A 2 | aA 3 A 6 A 2 A 2 A 6 A 2 A 6 A 2 | aA 6 A 2 A 2 A 6 A 2 A 6 A 2 | aA 2 a 6 A 6 A 2 | aA 3 A 6 a 2 A 6 A 7 A 6 A 2 | aA 6 A 2 A 6 A 7 A 6 A 2 | aA 3 A 6 A 2 A 2 A 6 A 2 A 7 A 6 A 2 | aA 6 A 2 A 2 A 6 A 2 A 7 A 6 A 2 | aA 2 A 6 A 2 A 7 A 6 A 2
A 4 ▷ aA 3 A 6 A 2 A 6 | aA 6 A 2 A 6 | aA 3 A 6 A 2 A 2 A 6 A 2 | aA 6 A 2 A 2 A 6 A 2 | aA 2 A 6 | aA 3 A 6 A 2 A 6 A 7 | aA 6 A 2 A 6 A 7 | aA 3 A 6 A 2 A 2 A 6 A 2 A 7 | aA 6 A 2 A 2 A 6 A 2 A 7 | aA 2 A 6 A 2 A 7 A 7 ▷ A 6 A 2 A 6 A 2 | A 6 A 2 A 6 A 2 A 7
G′^ = ⟨{A 1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7 }, {a, b}, {A 1 }, P ′⟩
d) S ▷ SaS | AB A ▷ bAa | λ B ▷ ab | C C ▷ Ba
Resolu¸c˜ao;
Vλ = {A}
S ▷ SaS | AB A ▷ bAa B ▷ ab | C C ▷ Ba
S ▷ B A ▷ ba
S ▷ SaS | AB | B A ▷ bAa | ba B ▷ ab | C C ▷ Ba
M 0 ▷ a M 1 ▷ b S ▷ SM 0 S | AB | M 1 M 0 | BM 0 A ▷ M 1 AM 0 | M 1 M 0 B ▷ M 0 M 1 | BM 0
A 4 ▷ a A 5 ▷ b A 1 ▷ A 1 A 4 A 1 | A 2 A 3 | A 4 A 5 | A 3 A 4 A 2 ▷ A 5 A 2 A 4 | A 5 A 4 A 3 ▷ A 4 A 5 | A 3 A 4
A 4 ▷ a A 5 ▷ b A 1 ▷ A 2 A 3 | A 4 A 5 | A 3 A 4 | A 2 A 3 A 6 | A 4 A 5 A 6 | A 3 A 4 A 6 A 2 ▷ A 5 A 2 A 4 | A 5 A 4 A 3 ▷ A 4 A 5 | A 4 A 5 A 7 A 6 ▷ A 4 A 1 | A 4 A 1 A 6 A 7 ▷ A 4 | A 4 A 7
A 4 ▷ a A 5 ▷ b A 1 ▷ A 2 A 3 | A 4 A 5 | A 3 A 4 | A 2 A 3 A 6 | A 4 A 5 A 6 | A 3 A 4 A 6
Cap´ıtulo 6
c) L = {ambn/m ≤ n ≤ 3 m}
Resolu¸c˜ao;
M = ⟨{q 0 , q 1 , q 2 , q 3 , q 4 , q 5 , q 6 , q 7 , q 8 }, {a, b}, {A, B}, δ, {q 0 }, z, {q 1 , q 7 }⟩
d) L = {ambna^2 m+1/n, m ≥ 1 }
Resolu¸c˜ao;
M = ⟨{q 0 , q 1 , q 2 , q 3 , q 4 }, {a, b}, {A}, δ, {q 0 }, z, {q 4 }⟩
e) L = {ambn/n = m + 1 ou m = n + 1}
Resolu¸c˜ao;
M = ⟨{q 0 , q 1 , q 2 , q 3 , q 4 , q 5 , q 6 , q 7 }, {a, b}, {A}, δ, {q 0 }, z, {q 3 }⟩
