Convert the PDA for the language of palindromes over {0, 1} into an equivalent CFG
To convert a Pushdown Automaton (PDA) for the language of palindromes over {0, 1} into an equivalent Context-Free Grammar (CFG), follow these steps:
1. Create non-terminal symbols for the different parts of the PDA:
- S: Start symbol and the overall goal is to reach this symbol at the end
- A: A non-terminal symbol to represent a palindrome
2. Define the production rules for the CFG:
- S -> ε: This production rule allows the CFG to generate an empty string, which is always a palindrome.
- S -> 0S0: This rule represents the production of a palindrome by adding a '0' to both sides of an existing palindrome.
- S -> 1S1: Similar to the previous rule, this rule adds a '1' to both sides of an existing palindrome.
- S -> 0 | 1: This rule allows the CFG to generate single characters '0' and '1', as they are also palindromes.
- A -> ε: This rule allows the CFG to generate an empty string, which is a palindrome.
- A -> 0A0: This rule represents the production of a palindrome by adding a '0' to both sides of an existing palindrome.
- A -> 1A1: Similar to the previous rule, this rule adds a '1' to both sides of an existing palindrome.
- A -> 0 | 1: This rule allows the CFG to generate single characters '0' and '1', as they are also palindromes.
3. The CFG is represented by the tuple (V, Σ, R, S), where:
- V: Set of non-terminal symbols {S, A}
- Σ: Set of terminal symbols {0, 1}
- R: Set of production rules defined in step 2
- S: Start symbol 'S'
And that's it! You have successfully converted the PDA for the language of palindromes over {0, 1} into an equivalent CFG.