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How to convert CFG to Greibach Normal Form?
A Context Free Grammar (CFG) is said to be in Greibach Normal Form(GNF), if production rules satisfy one of the following criteria −
Only a start symbol can generate ε. For example, if S is the start symbol then S → ε is in GNF.
A non-terminal can generate a terminal. For example, if A is Non terminal and a is terminal then, A → a is in GNF.
A non-terminal can generate a terminal followed by any number of non-terminals. For Example, S → aAS is in GNF.
Case 1
G1 = {S → aAB | aB, A → aA| a, B → bB | b}
The production rules of G1 satisfy the rules specified for GNF, then the grammar G1 is in GNF.
Case 2
G2 = {S → aAB | aB, A → aA | ε, B → bB | ε}
The production rule of G2 is not satisfying the rules specified for GNF as
A → ε and B → ε contain ε(only the start symbol can generate ε).
So, the grammar G2 is not in GNF.
Steps for converting CFG into GNF
Step 1 − Convert the grammar into CNF. If the given grammar is not in CNF, convert it into CNF.
Step 2 − If the grammar consists of left recursion, eliminate it. If the context free grammar contains any left recursion, eliminate it.
Step 3 − In the grammar, convert the given production rule into GNF form. If any production rule in the grammar is not in GNF form, convert it.
Example
Consider the context free grammar
S→SS|(S)|a
Convert this grammar to Greibach Normal Form.
Solution
Given below is an explanation for conversion of CFG to GNF −
Step 1: Converting to CNF: S->SS|XSY|a X->( Y->) Step 2: Remove left recursion from S->SS S->XSYP/aP P->SP/ε X->( Y->) Step 3: Remove null production P->ε S->XSYP/aP P->SP/S X->( Y->) Step 4: Convert to GNF as S->XSYP is not in GNF, Replace X with ( S->(SYP/aP P->SP/S X->( Y->) Step 5: Convert to GNF as P->SP is not in GNF, Replace S with (SYP/aP S->(SYP/aP P->(SYPP/aPP/(SYP/aP X->( Y->)
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