Pushdown Automata Overview

How Pushdown Automata (PDA) Works

• PDAs are more powerful than Finite Automata (FA), capable of accepting languages that FA cannot.
• All languages accepted by FA are also accepted by PDA.

Components of PDA

• Input Tape: Composed of cells or symbols with a read-only head that moves left to right.
• Finite Control: Contains a pointer to the current symbol being read.
• Stack: A structure for temporary storage with infinite capacity, allowing push and pop operations only from one end.

Formal Definition of PDA

• Q: Finite set of states.
• ∑: Input set.
• Γ: Stack symbols that can be pushed or popped.
• q0: Initial state.
• Z: Start symbol in Γ.
• F: Set of final states.
• δ: Mapping function for state transitions.

Instantaneous Description (ID)

• Described as a triple (q, w, α):
  • q: Current state.
  • w: Remaining input.
  • α: Stack contents, with the top on the left.
• Moves are represented by the symbol ⊢ (one move) or ⊢* (sequence of moves).

Example 1: PDA for Language {a^n b^2n | n ≥ 1}

• Push two 'a's onto the stack for each 'a' read.
• For each 'b' read, pop one 'a' from the stack.
• Transition functions:
  • δ(q0, a, Z) = (q0, aaZ)
  • δ(q0, a, a) = (q0, aaa)
  • δ(q0, b, a) = (q1, ε)
  • δ(q1, b, a) = (q1, ε)
  • δ(q1, ε, Z) = (q2, ε)
• Final configuration indicates acceptance when the stack is empty.

Example 2: PDA for Language {0^n 1^m 0^n | m, n ≥ 1}

• Push all '0's onto the stack when reading '0's.
• Do nothing when reading '1's.
• Pop '0's from the stack when encountering '0's after reading '1's.
• Transition functions:
  • δ(q0, 0, Z) = δ(q0, 0Z)
  • δ(q0, 0, 0) = δ(q0, 00)
  • δ(q0, 1, 0) = δ(q1, 0)
  • δ(q1, 0, 0) = δ(q1, ε)
  • δ(q0, ε, Z) = δ(q2, Z) (indicates acceptance)

PDA Acceptance Criteria

• Acceptance by Final State: PDA enters a final state after reading the entire input.
• Acceptance by Empty Stack: The stack becomes empty after processing the input string.

Example of PDA Acceptance by Empty Stack

• Accepts strings where the number of '0's is twice that of '1's.
• Two scenarios for handling '1's and '0's:
  • If '1' precedes '0's, push two '1's onto the stack for each '1' and pop for every two '0's.
  • If '0' precedes '1's, push the first '0' onto the stack, read the second '0', and pop when a '1' is read.
• Transition functions cover both scenarios to ensure acceptance.

Explore the fundamental concepts of Pushdown Automata (PDA) and their components. This quiz covers the structure, definition, and functionality of PDAs, demonstrating their superiority over Finite Automata. Test your understanding of states, input sets, stack operations, and instantaneous descriptions.