Enter Grammar
Use one production per line. Use spaces for symbols with more than one character.
Formula Used
Chomsky normal form allows only these production patterns:
- A → B C, where A, B, and C are variables.
- A → a, where a is one terminal.
- S₀ → ε, only when the language contains the empty string.
The conversion applies these transformations: add S₀, remove ε rules, remove unit rules, remove useless symbols, replace terminals in mixed rules, and split long productions into binary pairs.
How to Use This Calculator
- Enter one grammar rule on each line.
- Use an arrow like S -> A B or S → A B.
- Separate alternatives with the vertical bar symbol.
- Set the start symbol and helper prefix.
- Choose the conversion steps you want to apply.
- Press the convert button and review the result above the form.
- Use the CSV or PDF button to save the final grammar.
Example Data Table
| Input rule | Possible CNF action | Reason |
|---|---|---|
| S -> A S A | Split into binary helper rules | The right side has three variables. |
| A -> B | Remove unit production | A single variable on the right side is not final CNF. |
| B -> ε | Expand nullable combinations | Empty productions are removed except allowed start empty rule. |
| S -> a B | Create terminal helper | A terminal appears inside a longer rule. |
Why Chomsky Normal Form Helps
Chomsky normal form gives a context free grammar a strict shape. Every production becomes a terminal rule, a pair rule, or the allowed empty rule for the start symbol. This shape removes noise from grammar design. It also makes many parsing algorithms easier to follow. The CYK algorithm is the best known example. It expects binary branches, so long right sides must be split.
A converter helps because manual work can be slow. One missing nullable variable can change the language. One forgotten unit rule can leave the grammar outside the required form. A calculator keeps each step visible. You can compare the source grammar with the final grammar. You can also see how many rules were added or removed.
Cleaner Grammar Steps
The process usually starts with a new start symbol. This protects the original start symbol when it appears on a right side. The next step removes epsilon productions. Nullable variables are found first. Then new alternatives are made by removing nullable symbols in safe combinations. The empty string is kept only when the language needs it.
After that, unit productions are removed. A unit production sends one variable directly to another variable. These rules do not fit the final shape. The converter follows their closure. It copies useful non unit rules back to the source variable. Useless symbols are also removed. A symbol is useless when it cannot make a terminal string, or when it cannot be reached.
Reliable Parser Preparation
Terminals inside long productions need special handling. A rule like A to a B cannot stay as it is. The terminal a receives a helper variable. Then the long rule uses that helper instead. This keeps terminal rules separate from variable pair rules. It also makes the final grammar easier to scan.
Long right sides are then broken into binary chains. A rule with three or more symbols becomes several pair rules. Fresh helper variables hold the remaining suffixes. The language should stay the same, except for normal changes caused by the optional empty string rule. Good output shows these helper rules clearly.
Practical Checks for Students
Use a consistent notation before converting. Put one nonterminal on the left side of each line. Separate alternatives with a vertical bar. Use spaces when symbols have more than one character. This removes ambiguity during token reading. Review the nullable set and unit closure after conversion. These details often explain surprising output.
Chomsky normal form is not always the shortest grammar. It is a disciplined form for proof, parsing, and analysis. The final grammar may contain more variables than the original grammar. That is normal. What matters is that each rule follows the required shape. This calculator is designed to show those choices without hiding the important transformations.
It helps students test examples before formal proofs, projects, or parser code in class lab work.
FAQs
What is Chomsky normal form?
It is a restricted grammar form. Each production must be a variable pair, a single terminal, or one allowed start empty rule.
Can this calculator handle epsilon rules?
Yes. It finds nullable variables, builds safe combinations, and removes empty rules. The start empty rule can be kept when required.
What is a unit production?
A unit production has one variable on the right side. A -> B is a unit production. It must be removed for final CNF.
Why are helper variables created?
Helper variables replace terminals in longer rules. They also split long right sides into binary rules that match the required structure.
Does the conversion keep the same language?
The intended language is preserved by standard transformations. The only special case is the empty string, which depends on the selected start option.
How should I write multi-character symbols?
Use spaces between symbols. For example, write Expr -> Term Plus Expr. This avoids confusion with compact single-letter notation.
Can I download the result?
Yes. Use the CSV button for table data. Use the PDF button for a simple printable copy of the final grammar.
Why does the output have more rules?
CNF often needs helper variables. Long productions and mixed terminal rules are expanded, so the final grammar can become larger.
What is a useless symbol?
A useless symbol cannot produce terminal strings or cannot be reached from the start symbol. Such symbols do not affect parsing.
Is CNF used by CYK parsing?
Yes. The CYK algorithm works naturally with binary grammar rules. CNF gives the grammar the structure that CYK expects.
Can I skip some conversion steps?
Yes. The checkboxes let you disable steps. The review message will warn you when skipped steps leave rules outside final CNF.