Simplification of CFGs

Simplification of Context-Free Grammars (CFGs) aims to remove redundant or unnecessary components from a grammar without changing the language it generates. This process makes the grammar more concise and efficient for parsing and analysis. Common simplifications include removing useless symbols, eliminating ε-productions, and eliminating unit productions.

Removing Useless Symbols

Useless symbols are non-terminals or terminals that do not contribute to generating strings in the language. These can be removed through the following steps:

  1. Identify Non-Terminals That Do Not Generate Terminal Strings:
    • A non-terminal A is useful if it derives a string of terminals (directly or indirectly).
    • Eliminate all non-terminals that cannot generate terminal strings.
  2. Identify Unreachable Symbols:
    • A non-terminal A is reachable if there exists a derivation from the start symbol S that includes A.
    • Remove all non-terminals that cannot be reached from the start symbol.

Example: Consider the grammar:

</>
Copy
S → AB | C
A → aA | ε
B → b
C → cC
D → d

Here:

  • D is not reachable and can be removed.
  • C cannot derive a terminal string and can also be removed.

The simplified grammar is:

</>
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S → AB
A → aA | ε
B → b

Eliminating ε-Productions

An ε-production is a production of the form A → ε, where A is a non-terminal, and ε is the empty string. Eliminating ε-productions ensures that the grammar generates the same language without using the empty string in derivations.

Steps to Eliminate ε-Productions:

  1. Identify Nullable Non-Terminals: A non-terminal A is nullable if it derives ε directly or indirectly.
  2. Modify Productions: For every production involving a nullable non-terminal, create new productions by omitting the nullable non-terminal.
  3. Remove ε-Productions: Eliminate all productions of the form A → ε except when the start symbol can derive ε.

Example: Consider the grammar:

</>
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S → AB | ε
A → aA | ε
B → b

Here, A is nullable. Modify the productions:

</>
Copy
S → AB | B
A → aA | ε
B → b

After removing A → ε and adjusting other rules, the grammar becomes:

</>
Copy
S → AB | B
A → aA
B → b

Eliminating Unit Productions

A unit production is a production of the form A → B, where A and B are non-terminals. Unit productions can be eliminated by replacing them with the productions of B.

Steps to Eliminate Unit Productions:

  1. Identify Unit Productions: Find all productions of the form A → B.
  2. Replace with B’s Productions: Replace A → B with the productions of B.
  3. Remove Unit Productions: Remove the original unit productions from the grammar.

Example: Consider the grammar:

</>
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S → A
A → B
B → b

Replace A → B and S → A with the productions of B:

</>
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S → b
A → b
B → b

Key Takeaways

  • Simplification of CFGs involves removing useless symbols, ε-productions, and unit productions.
  • Removing useless symbols improves grammar efficiency and clarity.
  • Eliminating ε-productions and unit productions makes the grammar more concise and easier to parse.
  • These simplifications preserve the language generated by the grammar.