# Transforming a Grammar to Chomsky Form

We show how to make a grammar:

- without unproductive symbols
- without unreachable symbols
- -free (no non-start nullable symbols)
- without single productions X::=Y (X,Y non-terminals)
- without productions of arity more than two
- with terminals occurring alone on right-hand side

## Elimination of Non-Productive Non-Terminals

Non-terminal is **non-productive** if it cannot generate any string of terminals

- if X::=w and w has only terminals, then X is productive
- if X:=p and p has only productive symbols, then X is productive

Start from empty set, apply these rules to compute productive symbols

- remaining symbols are non-productive
- delete them and all rules that contain them

## Elimination of Unreachable Non-Terminals

Non-terminal is **unreachable** if no derivation from start symbol contains it

- S is reachable
- if X is reachable and X::=p, then every symbol in p is reachable

Start from empty set, apply these rules to compute productive symbols

- remaining symbols are non-productive
- delete them and all rules that contain them

## Making Non-Terminals Non-Nullable

A grammar where is -free if

- no-non terminal is nullable, or
- only S is nullable and S does not appear on right-hand side

First, compute nullable nonterminals.

Then, for each production

create all productions of form

where

- is (as before), or,
- is nullable and is empty string (so it disappears)

Remove all right-hand sides that are “”

If S is nullable, introduce new initial symbol S' and add

## Eliminating Single Productions

Construct a directed graph of non-terminals between X,Y such that is production

Compute as the transitive closure of this graph

If X Y and Y ::= p is a rule, then add rule

X ::= p

Then remove single productions.

## Eliminating Non-Binary Productions

becomes

## Making Terminals Alone on Right-Hand Side

Just introduce a fresh non-terminal for terminal t

Use on RHS instead of t

Add rule

## Summary of Parsing General Context-Free Grammars

To check if :

- Convert to equivalent Chomsky normal form
- Use CYK Parsing Algorithm to check if