Grammar Transformations

Grammar Transformations

AutomataLab features an automated mathematical engine capable of transforming your Context-Free Grammars into standardized Normal Forms.

You can access these transformations by clicking the Tools menu above the Grammar Diagnostics panel. When you apply a transformation, AutomataLab overwrites the current grammar editor with the new syntactically equivalent grammar.

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1. Chomsky Normal Form (CNF)

A grammar is in Chomsky Normal Form if every production rule is of the exact form:

1. $A \rightarrow BC$ (Two non-terminals)
2. $A \rightarrow a$ (A single terminal)
3. $S \rightarrow \epsilon$ (Only if the start symbol $S$ derives epsilon and does not appear on the RHS of any rule).

The CNF Conversion Algorithm

When you click Convert to CNF, AutomataLab runs the classic 5-step conversion algorithm natively:

1. START: Eliminate the start symbol from right-hand sides by introducing a new start symbol $S_0 \rightarrow S$.
2. TERM: Eliminate rules with mixed terminals and non-terminals (e.g., $A \rightarrow B c$ becomes $A \rightarrow B T_c$ and $T_c \rightarrow c$).
3. BIN: Eliminate rules with more than 2 non-terminals (e.g., $A \rightarrow BCD$ becomes $A \rightarrow B X_1$ and $X_1 \rightarrow CD$).
4. DEL: Eliminate $\epsilon$-rules (null productions) by distributing them across the grammar.
5. UNIT: Eliminate unit rules (e.g., $A \rightarrow B$) by substituting the RHS of $B$ directly into $A$.

Why do we need CNF?

The primary reason to convert a grammar to CNF in AutomataLab is to use the CYK Parser. The CYK dynamic programming algorithm is strictly built around the binary branching structure of CNF. If you try to run the CYK parser on a non-CNF grammar, AutomataLab will throw an error and prompt you to convert it.

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2. Greibach Normal Form (GNF)

(Advanced) A grammar is in Greibach Normal Form if every production rule's right-hand side begins with exactly one terminal symbol, followed by zero or more non-terminals: $A \rightarrow a X_1 X_2 \dots X_n$

Why do we need GNF?

GNF is highly useful for theoretical proofs. Specifically, it proves that any Context-Free Language can be parsed by a Pushdown Automaton (PDA) without any $\epsilon$-transitions, because every step in GNF is guaranteed to consume exactly one input character!

While AutomataLab provides GNF conversion as an educational tool, none of the specific Parsers in the Parser Studio require GNF to function.

See Also

• Context-Free Grammars
• Mathematical Properties