Theory of Computation Lecture 10

Pushdown Automata Overview

• Pushdown automata (PDA) are an extension of nondeterministic finite automata featuring a stack for additional memory.
• They can recognize non-regular languages due to the stack allowing unbounded storage.
• PDAs are equivalent in computational power to context-free grammars, facilitating two methods for proving language context-freeness.

Components of Pushdown Automata

• Control: Represents states and transition functions within the automaton.
• Input Tape: Contains the input string read by the PDA.
• Stack: Enables the storage and retrieval of symbols, thus allowing the PDA to manipulate data beyond finite memory limits.

Stack Operations

• Pushing: Writing a symbol onto the stack, which moves existing symbols down.
• Popping: Removing the top symbol from the stack, allowing remaining symbols to shift up.
• A stack can theoretically hold an unlimited amount of information, enhancing the PDA's recognition abilities.

Example Language Recognition

• The language {0^n 1^n | n ≥ 0} cannot be recognized by finite automata due to their limited memory.
• A PDA can recognize this language by pushing 0s onto the stack and popping them for each corresponding 1 read.
• Acceptance criteria: Input is accepted if the stack is empty when input ends, rejecting it if the stack is empty while 1s remain or if 0s appear after 1s.

Formal Definition of Pushdown Automata

• A PDA is defined similarly to a finite automaton but incorporates a stack.
• Input and stack alphabets are specified, allowing separate sets of symbols.
• Transition functions are defined based on the current state, next input symbol, and top stack symbol.

Transition Function Characteristics

• Transition functions may operate without reading an input or stack symbol (denoted by ε).
• They specify state changes and stack operations using the notation "a,b → c", indicating that upon reading 'a' from the input, the top stack symbol 'b' is replaced with 'c'.
• State diagrams for PDAs reflect how the automaton transitions while utilizing the stack, analogous to those of finite automata but adapted for stack operations.

In this quiz, we explore the concept of pushdown automata, a vital computational model that incorporates a stack for enhanced memory capabilities. Understand the distinctions between pushdown automata and nondeterministic finite automata as introduced in Lecture 10. Test your knowledge on the key principles and applications of this computational theory.