Normal Form of CFG

CFGs with standardized production rules.

Chomsky Normal Form

Aka. CNF. Named after Noam Chomsky.

A CFG $G$ is in CNF iff all productions of $G$ has either 1 terminal or 2 non-terminals in RHS.

• $A \rightarrow BC$ or
• $A \rightarrow a$

CFG to CNF Converstion

• If start symbol $S$ occurs in the RHS of some production rule,
  • add $S'$ as the start symbol
  • add a production rule $S'\rightarrow S$
• Remove Λ-productions
• Remove unit productions
• Remove long productions by using new non-terminals
  For a production rule $A\rightarrow B_1B_2B_3\dots B_n$, replace it with $A\rightarrow B_1C$ and $C\rightarrow B_2B_3\dots B_n$. Do this repeatedly until all long productions are removed.
• For production rules of the form $A\rightarrow aB$
  Replace it with $A\rightarrow XB$ and a new production rule $X\rightarrow a$

Greibach’s Normal Form

Aka. GNF. Named after Sheila Greibach.

A CFG $G$ is in GNF iff all productions are of the form: $A\to aA_{1}A_{2}\cdots A_{n}$