Designing Pushdown Automata for Even Palindromes

Jul 13, 2024

Lecture on Designing Pushdown Automata for Even Palindromes

Recap of Previous Lecture

  • Graphical notation of Pushdown Automata (PDA).
  • Transition diagrams for PDA.
  • Meanings of different symbols used.
  • Simple example of designing a PDA.

Today's Topic

  • Constructing a PDA for accepting even palindromes of the form L = W W^R for all W in the positive closure of {a, b}.

Understanding Palindromes

  • Words or sequences that read the same forward and backward.
  • Examples:
    • "noon" (N O O N)
    • "no lemon no melon"
    • "123321"
    • "abba"
    • "racecar"

Focus of Today's Example

  • Designing a PDA for even palindromes composed solely of symbols 'a' and 'b'.
  • Even palindrome: Palindromes with an even number of characters (e.g., "abba").
  • Positive closure (+): The sequence must contain at least one symbol.
  • Language representation: W W^R where W is the first half and W^R is the reverse.

Steps to Design the PDA

  1. Start State (q1):
    • Push Z_0 to the stack (initial symbol).
  2. State q2 (Reading First Half):
    • On input 'a', push 'a' to stack.
    • On input 'b', push 'b' to stack.
  3. Transition to State q3:
    • When midpoint is reached (details to be clarified later).
  4. State q3 (Reading Second Half):
    • On input 'a', pop 'a' from stack.
    • On input 'b', pop 'b' from stack.
  5. Final State (q4):
    • Check if Z_0 is at the top of the stack.
    • If yes, pop it and move to q4 (final state).

Example Walkthrough

Example 1: Accepting String "abba"

  • Start in q1, push Z_0 to stack.
  • Move to q2:
    • Input 'a', push 'a'.
    • Input 'b', push 'b'.
  • Reach midpoint, move to q3:
    • Input 'b', pop 'b'.
    • Input 'a', pop 'a'.
  • Reach end, check Z_0 and pop it.
  • String accepted.

Example 2: Rejecting String "abab"

  • Start in q1, push Z_0 to stack.
  • Move to q2:
    • Input 'a', push 'a'.
    • Input 'b', push 'b'.
  • Reach midpoint, move to q3:
    • Input 'a', check top of stack (not 'a', but 'b').
  • State cannot proceed, string rejected.

Addressing Doubts

  • Key Question: How do we determine the midpoint?
  • To be addressed in next lecture.

Next Lecture Preview

  • Detailed explanation on determining the midpoint in PDA transitions.
  • Viewer engagement: Drop ideas in the comment section.

Thank you for your attention! See you in the next lecture.