Pushdown Automata PDAs

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Presentation transcript:

Pushdown Automata PDAs Costas Busch - RPI

Pushdown Automaton -- PDA Input String Stack States Costas Busch - RPI

Formal Definition Non-Deterministic Pushdown Automaton NPDA Final states States Input alphabet Stack start symbol Transition function Initial state Stack alphabet Costas Busch - RPI

Initial Stack Symbol Stack Stack stack head top bottom special symbol Costas Busch - RPI

The States Pop symbol Input symbol Push symbol Costas Busch - RPI

input stack top Replace Costas Busch - RPI

input stack top Push Costas Busch - RPI

input stack top Pop Costas Busch - RPI

input stack top No Change Costas Busch - RPI

A Possible Transition input stack empty Pop top Costas Busch - RPI

The automaton Halts in state and Rejects the input string A Bad Transition input Empty stack HALT The automaton Halts in state and Rejects the input string Costas Busch - RPI

The automaton Halts in state and Rejects the input string A Bad Transition input Empty stack HALT The automaton Halts in state and Rejects the input string Costas Busch - RPI

No transition is allowed to be followed When the stack is empty Empty stack Costas Busch - RPI

A Good Transition input stack Pop top Costas Busch - RPI

Non-Determinism These are allowed transitions in a Non-deterministic PDA (NPDA) Costas Busch - RPI

NPDA: Non-Deterministic PDA Example: Costas Busch - RPI

Execution Example: Time 0 Input Stack current state Costas Busch - RPI

Time 1 Input Stack Costas Busch - RPI

Time 2 Input Stack Costas Busch - RPI

Time 3 Input Stack Costas Busch - RPI

Time 4 Input Stack Costas Busch - RPI

Time 5 Input Stack Costas Busch - RPI

Time 6 Input Stack Costas Busch - RPI

Time 7 Input Stack Costas Busch - RPI

Time 8 Input Stack accept Costas Busch - RPI

A string is accepted if there is a computation such that: All the input is consumed AND The last state is a final state At the end of the computation, we do not care about the stack contents Costas Busch - RPI

is accepted by the NPDA: The input string is accepted by the NPDA: Costas Busch - RPI

is the language accepted by the NPDA: In general, is the language accepted by the NPDA: Costas Busch - RPI

Another NPDA example NPDA Costas Busch - RPI

Execution Example: Time 0 Input Stack Costas Busch - RPI

Time 1 Input Stack Costas Busch - RPI

Time 2 Input Stack Costas Busch - RPI

Time 3 Input Guess the middle of string Stack Costas Busch - RPI

Time 4 Input Stack Costas Busch - RPI

Time 5 Input Stack Costas Busch - RPI

Time 6 Input Stack accept Costas Busch - RPI

Rejection Example: Time 0 Input Stack Costas Busch - RPI

Time 1 Input Stack Costas Busch - RPI

Time 2 Input Stack Costas Busch - RPI

Time 3 Input Guess the middle of string Stack Costas Busch - RPI

Time 4 Input Stack Costas Busch - RPI

There is no possible transition. Input Time 5 There is no possible transition. Input Input is not consumed Stack Costas Busch - RPI

Another computation on same string: Input Time 0 Stack Costas Busch - RPI

Time 1 Input Stack Costas Busch - RPI

Time 2 Input Stack Costas Busch - RPI

Time 3 Input Stack Costas Busch - RPI

Time 4 Input Stack Costas Busch - RPI

Time 5 Input No final state is reached Stack Costas Busch - RPI

There is no computation that accepts string Costas Busch - RPI

A string is rejected if there is no computation such that: All the input is consumed AND The last state is a final state At the end of the computation, we do not care about the stack contents Costas Busch - RPI

In other words, a string is rejected if in every computation with this string: The input cannot be consumed OR The input is consumed and the last state is not a final state The stack head moves below the bottom of the stack Costas Busch - RPI

Another NPDA example NPDA Costas Busch - RPI

Execution Example: Time 0 Input Stack Costas Busch - RPI

Time 1 Input Stack Costas Busch - RPI

Time 2 Input Stack Costas Busch - RPI

Time 3 Input Stack accept Costas Busch - RPI

Rejection example: Time 0 Input Stack Costas Busch - RPI

Time 1 Input Stack Costas Busch - RPI

Time 2 Input Stack Costas Busch - RPI

Time 3 Input Stack Costas Busch - RPI

Time 4 Input Stack Halt and Reject Costas Busch - RPI

Pushing Strings Pop symbol Input symbol Push string Costas Busch - RPI

Example: input pushed string stack top Push Costas Busch - RPI

Another NPDA example NPDA Costas Busch - RPI

Execution Example: Time 0 Input Stack current state Costas Busch - RPI

Time 1 Input Stack Costas Busch - RPI

Time 3 Input Stack Costas Busch - RPI

Time 4 Input Stack Costas Busch - RPI

Time 5 Input Stack Costas Busch - RPI

Time 6 Input Stack Costas Busch - RPI

Time 7 Input Stack Costas Busch - RPI

Time 8 Input Stack accept Costas Busch - RPI

Formalities for NPDAs Costas Busch - RPI

Transition function: Costas Busch - RPI

Transition function: Costas Busch - RPI

Formal Definition Non-Deterministic Pushdown Automaton NPDA Final states States Input alphabet Stack start symbol Transition function Initial state Stack alphabet Costas Busch - RPI