Pushdown Automata PDAs Fall 2005 Costas Busch RPI

Pushdown Automata PDAs Fall 2005 Costas Busch - RPI 1

Pushdown Automaton -- PDA Input String Stack States Fall 2005 Costas Busch - RPI 2

Initial Stack Symbol Stack stack head top bottom special symbol Appears at time 0 Fall 2005 Costas Busch - RPI 3

The States Input symbol Fall 2005 Pop symbol Costas Busch - RPI Push symbol 4

input stack top Fall 2005 Replace Costas Busch - RPI 5

input stack top Fall 2005 Push Costas Busch - RPI 6

input stack top Fall 2005 Pop Costas Busch - RPI 7

input stack top Fall 2005 No Change Costas Busch - RPI 8

Empty Stack input stack top Pop empty When the stack gets empty, the automaton HALTS (no more transitions after ) Fall 2005 Costas Busch - RPI 9

A Possible Transition input stack Fall 2005 top Pop Costas Busch - RPI 10

Non-Determinism PDAs are non-deterministic Allowed non-deterministic transitions Fall 2005 Costas Busch - RPI 11

Example PDA Fall 2005 : Costas Busch - RPI 12

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 2005 Costas Busch - RPI 13

Execution Example: Time 0 Input Stack current state Fall 2005 Costas Busch - RPI 14

Time 1 Input Stack Fall 2005 Costas Busch - RPI 15

Time 2 Input Stack Fall 2005 Costas Busch - RPI 16

Time 3 Input Stack Fall 2005 Costas Busch - RPI 17

Time 4 Input Stack Fall 2005 Costas Busch - RPI 18

Time 5 Input Stack Fall 2005 Costas Busch - RPI 19

Time 6 Input Stack Fall 2005 Costas Busch - RPI 20

Time 7 Input Stack Fall 2005 Costas Busch - RPI 21

Time 8 Input Stack accept Fall 2005 Costas Busch - RPI 22

A string is accepted if there is a computation such that: All the input is consumed AND The last state is an accepting state At the end of the computation, we do not care about the stack contents (the stack can be empty at the last state) Fall 2005 Costas Busch - RPI 23

The input string is accepted by the PDA: Fall 2005 Costas Busch - RPI 24

In general, is the language accepted by the PDA: Fall 2005 Costas Busch - RPI 25

Rejection Example: Time 0 Input Stack current state Fall 2005 Costas Busch - RPI 26

Rejection Example: Time 1 Input Stack current state Fall 2005 Costas Busch - RPI 27

Rejection Example: Time 2 Input Stack current state Fall 2005 Costas Busch - RPI 28

Rejection Example: Time 3 Input Stack current state Fall 2005 Costas Busch - RPI 29

Rejection Example: Time 4 Input Stack current state Fall 2005 Costas Busch - RPI 30

Rejection Example: Time 4 Input reject Stack current state Fall 2005 Costas Busch - RPI 31

The input string is rejected by the PDA: Fall 2005 Costas Busch - RPI 32

A string is rejected if there is no computation such that: All the input is consumed AND The last state is an accept state At the end of the computation, we do not care about the stack contents Fall 2005 Costas Busch - RPI 33

Another PDA example PDA Fall 2005 : Costas Busch - RPI 34

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 2005 Costas Busch - RPI 35

Execution Example: Time 0 Input Stack Fall 2005 Costas Busch - RPI 36

Time 1 Input Stack Fall 2005 Costas Busch - RPI 37

Time 2 Input Stack Fall 2005 Costas Busch - RPI 38

Time 3 Input Guess the middle of string Stack Fall 2005 Costas Busch - RPI 39

Time 4 Input Stack Fall 2005 Costas Busch - RPI 40

Time 5 Input Stack Fall 2005 Costas Busch - RPI 41

Time 6 Input Stack accept Fall 2005 Costas Busch - RPI 42

Rejection Example: Time 0 Input Stack Fall 2005 Costas Busch - RPI 43

Time 1 Input Stack Fall 2005 Costas Busch - RPI 44

Time 2 Input Stack Fall 2005 Costas Busch - RPI 45

Time 3 Input Guess the middle of string Stack Fall 2005 Costas Busch - RPI 46

Time 4 Input Stack Fall 2005 Costas Busch - RPI 47

Input Time 5 There is no possible transition. Input is not consumed Stack Fall 2005 Costas Busch - RPI 48

Another computation on same string: Input Time 0 Stack Fall 2005 Costas Busch - RPI 49

Time 1 Input Stack Fall 2005 Costas Busch - RPI 50

Time 2 Input Stack Fall 2005 Costas Busch - RPI 51

Time 3 Input Stack Fall 2005 Costas Busch - RPI 52

Time 4 Input Stack Fall 2005 Costas Busch - RPI 53

Time 5 Input No final state is reached Stack Fall 2005 Costas Busch - RPI 54

There is no computation that accepts string Fall 2005 Costas Busch - RPI 55

Another PDA example PDA Fall 2005 Costas Busch - RPI 56

Execution Example: Time 0 Input Stack Fall 2005 Costas Busch - RPI 57

Time 1 Input Stack Fall 2005 Costas Busch - RPI 58

Time 2 Input Stack Fall 2005 Costas Busch - RPI 59

Time 3 Input Stack accept Fall 2005 Costas Busch - RPI 60

Rejection example: Time 0 Input Stack Fall 2005 Costas Busch - RPI 61

Time 1 Input Stack Fall 2005 Costas Busch - RPI 62

Time 2 Input Stack Fall 2005 Costas Busch - RPI 63

Time 3 Input Stack Fall 2005 Costas Busch - RPI 64

Time 4 Input Stack Halt and Reject Fall 2005 Costas Busch - RPI 65

Pushing Strings Input symbol Fall 2005 Pop symbol Costas Busch - RPI Push string 66

Example: input stack Fall 2005 top pushed string Push Costas Busch - RPI 67

Another PDA example PDA Fall 2005 Costas Busch - RPI 68

Execution Example: Time 0 Input Stack current state Fall 2005 Costas Busch - RPI 69

Time 1 Input Stack Fall 2005 Costas Busch - RPI 70

Time 3 Input Stack Fall 2005 Costas Busch - RPI 71

Time 4 Input Stack Fall 2005 Costas Busch - RPI 72

Time 5 Input Stack Fall 2005 Costas Busch - RPI 73

Time 6 Input Stack Fall 2005 Costas Busch - RPI 74

Time 7 Input Stack Fall 2005 Costas Busch - RPI 75

Time 8 Input Stack accept Fall 2005 Costas Busch - RPI 76

Formalities for PDAs Fall 2005 Costas Busch - RPI 77

Transition function: Fall 2005 Costas Busch - RPI 78

Transition function: Fall 2005 Costas Busch - RPI 79

Formal Definition Pushdown Automaton (PDA) Accept states States Input alphabet Transition Initial Stack function state alphabet Fall 2005 Costas Busch - RPI Stack start symbol 80