Question

Convert the context free grammar to an equivalent grammar in Chomsky normal form: (show your work)

S → AxAyA | BC

A → BB | a
B → bd | ε
C → SC | c

Answer 1

Chomsky Normal Form format
S0 -> epsilon
NT -> NT NT
NT -> T

where NT = Non terminal and T = terminal

• Given Grammar, put S0 as starting state
  

S0 -> S
S -> AxAyA | BC
A -> BB | a
B -> bd | ε
C -> SC | c

• Remove epsilon productions

Removing B -> ε

S0 -> S
S -> AxAyA | BC | C
A -> BB | a | ε | B
B -> bd
C -> SC | c

Removing A -> ε

S0 -> S
S -> AxAyA | BC | AxAy | AxyA | xAyA | xy | Axy | xAy | xyA | C
A -> BB | a | B
B -> bd
C -> SC | c

Remove Unit productions

S0 -> AxAyA | BC | AxAy | AxyA | xAyA | xy | Axy | xAy | xyA | SC | c
S -> AxAyA | BC | AxAy | AxyA | xAyA | xy | Axy | xAy | xyA | SC | c
A -> BB | a | bd
B -> bd
C -> SC | c

• Change Ts on the right side production to NTs if it is not unit production

S0 -> AXAYA | BC | AXAY | AXYA | XAYA | XY | AXY | XAY | XYA | SC | c
S -> AXAYA | BC | AXAY | AXYA | XAYA | XY | AXY | XAY | XYA | SC | c
A -> BB | a | ED
B -> ED
C -> SC | c
X -> x
Y -> y
E -> b
D -> d

• Change into CNF

S0 -> FI | BC | FG | FH | XI | XY | FY | XG | XH | SC | c
S -> FI | BC | FG | FH | XI | XY | FY | XG | XH | SC | c
A -> BB | a | ED
B -> ED
C -> SC | c
X -> x
Y -> y
E -> b
D -> d
F -> AX
G -> AY
H -> YA
I -> GA

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