Pushdown Automata PDAs Fall 2006 Costas Busch RPI

Pushdown Automata PDAs Fall 2006 Costas Busch - RPI 1

Pushdown Automaton -- PDA Input String Stack States Fall 2006 Costas Busch - RPI 2

Initial Stack Symbol Stack stack head top bottom special symbol Appears at time 0 Fall 2006 Costas Busch - RPI 3

The States Input symbol Fall 2006 Pop symbol Costas Busch - RPI Push symbol 4

input stack top Fall 2006 Replace Costas Busch - RPI 5

input stack top Fall 2006 Push Costas Busch - RPI 6

input stack top Fall 2006 Pop Costas Busch - RPI 7

input stack top Fall 2006 No Change Costas Busch - RPI 8

Pop from Empty Stack input Pop stack Automaton halts! top If the automaton attempts to pop from empty stack then it halts and rejects input Fall 2006 Costas Busch - RPI 9

Non-Determinism PDAs are non-deterministic Allowed non-deterministic transitions Fall 2006 Costas Busch - RPI 10

Example PDA Fall 2006 : Costas Busch - RPI 11

Basic Idea: 1. Push the a’s 2. on the stack 2. Match the b’s on input with a’s on stack 3. Match found Fall 2006 Costas Busch - RPI 12

Execution Example: Time 0 Input Stack current state Fall 2006 Costas Busch - RPI 13

Time 1 Input Stack Fall 2006 Costas Busch - RPI 14

Time 2 Input Stack Fall 2006 Costas Busch - RPI 15

Time 3 Input Stack Fall 2006 Costas Busch - RPI 16

Time 4 Input Stack Fall 2006 Costas Busch - RPI 17

Time 5 Input Stack Fall 2006 Costas Busch - RPI 18

Time 6 Input Stack Fall 2006 Costas Busch - RPI 19

Time 7 Input Stack Fall 2006 Costas Busch - RPI 20

Time 8 Input Stack accept Fall 2006 Costas Busch - RPI 21

A string is accepted if there is a computation such that: All the input is consumed AND The last state is an accepting state we do not care about the stack contents at the end of the accepting computation Fall 2006 Costas Busch - RPI 22

Rejection Example: Time 0 Input Stack current state Fall 2006 Costas Busch - RPI 23

Rejection Example: Time 1 Input Stack current state Fall 2006 Costas Busch - RPI 24

Rejection Example: Time 2 Input Stack current state Fall 2006 Costas Busch - RPI 25

Rejection Example: Time 3 Input Stack current state Fall 2006 Costas Busch - RPI 26

Rejection Example: Time 4 Input Stack current state Fall 2006 Costas Busch - RPI 27

Rejection Example: Time 4 Input reject Stack current state Fall 2006 Costas Busch - RPI 28

There is no accepting computation for The string Fall 2006 is rejected by the PDA Costas Busch - RPI 29

Another PDA example PDA Fall 2006 : Costas Busch - RPI 30

Basic Idea: 1. Push v 2. Guess 2. on stack middle of input 3. Match on input with v on stack 4. Match found Fall 2006 Costas Busch - RPI 31

Execution Example: Time 0 Input Stack Fall 2006 Costas Busch - RPI 32

Time 1 Input Stack Fall 2006 Costas Busch - RPI 33

Time 2 Input Stack Fall 2006 Costas Busch - RPI 34

Time 3 Input Guess the middle of string Stack Fall 2006 Costas Busch - RPI 35

Time 4 Input Stack Fall 2006 Costas Busch - RPI 36

Time 5 Input Stack Fall 2006 Costas Busch - RPI 37

Time 6 Input Stack accept Fall 2006 Costas Busch - RPI 38

Rejection Example: Time 0 Input Stack Fall 2006 Costas Busch - RPI 39

Time 1 Input Stack Fall 2006 Costas Busch - RPI 40

Time 2 Input Stack Fall 2006 Costas Busch - RPI 41

Time 3 Input Guess the middle of string Stack Fall 2006 Costas Busch - RPI 42

Time 4 Input Stack Fall 2006 Costas Busch - RPI 43

Input Time 5 There is no possible transition. Input is not consumed Stack Fall 2006 Costas Busch - RPI 44

Another computation on same string: Input Time 0 Stack Fall 2006 Costas Busch - RPI 45

Time 1 Input Stack Fall 2006 Costas Busch - RPI 46

Time 2 Input Stack Fall 2006 Costas Busch - RPI 47

Time 3 Input Stack Fall 2006 Costas Busch - RPI 48

Time 4 Input Stack Fall 2006 Costas Busch - RPI 49

Time 5 Input No accept state is reached Stack Fall 2006 Costas Busch - RPI 50

There is no computation that accepts string Fall 2006 Costas Busch - RPI 51

Pushing & Popping Strings Input symbol Fall 2006 Pop string Costas Busch - RPI Push string 52

Example: input stack pop string Fall 2006 top push string Replace Costas Busch - RPI 53

Equivalent transitions pop push Fall 2006 Costas Busch - RPI 54

Another PDA example PDA Fall 2006 Costas Busch - RPI 55

Execution Example: Time 0 Input Stack current state Fall 2006 Costas Busch - RPI 56

Time 1 Input Stack Fall 2006 Costas Busch - RPI 57

Time 3 Input Stack Fall 2006 Costas Busch - RPI 58

Time 4 Input Stack Fall 2006 Costas Busch - RPI 59

Time 5 Input Stack Fall 2006 Costas Busch - RPI 60

Time 6 Input Stack Fall 2006 Costas Busch - RPI 61

Time 7 Input Stack Fall 2006 Costas Busch - RPI 62

Time 8 Input Stack accept Fall 2006 Costas Busch - RPI 63

Formalities for PDAs Fall 2006 Costas Busch - RPI 64

Transition function: Fall 2006 Costas Busch - RPI 65

Transition function: Fall 2006 Costas Busch - RPI 66

Formal Definition Pushdown Automaton (PDA) Accept states States Input alphabet Transition Initial Stack function state alphabet Fall 2006 Costas Busch - RPI Stack start symbol 67

Instantaneous Description Current state Fall 2006 Remaining input Costas Busch - RPI Current stack contents 68

Example: Instantaneous Description Time 4: Input Stack Fall 2006 Costas Busch - RPI 69

Example: Instantaneous Description Time 5: Input Stack Fall 2006 Costas Busch - RPI 70

We write: Time 4 Fall 2006 Time 5 Costas Busch - RPI 71

A computation: Fall 2006 Costas Busch - RPI 72

For convenience we write: Fall 2006 Costas Busch - RPI 73

Language of PDA Language accepted by PDA Initial state Fall 2006 Costas Busch - RPI : Accept state 74

Example: PDA Fall 2006 : Costas Busch - RPI 75

PDA Fall 2006 : Costas Busch - RPI 76

Therefore: PDA Fall 2006 : Costas Busch - RPI 77