Pushdown Automata PDAs Costas Busch LSU 1 Pushdown

Pushdown Automata PDAs Costas Busch - LSU 1

Pushdown Automaton -- PDA Input String Stack States Costas Busch - LSU 2

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

The States Input symbol Pop symbol Costas Busch - LSU Push symbol 4

input stack top Replace Costas Busch - LSU 5

input stack top Push Costas Busch - LSU 6

input stack top Pop Costas Busch - LSU 7

input stack top No Change Costas Busch - LSU 8

Non-Determinism PDAs are non-deterministic Allowed non-deterministic transitions Costas Busch - LSU 9

Example PDA : Costas Busch - LSU 10

Basic Idea: • Push the a’s on the stack 2. Match the b’s on input with a’s on stack 3. Match found Costas Busch - LSU 11

Execution Example: Time 0 Input Stack current state Costas Busch - LSU 12

Time 1 Input Stack Costas Busch - LSU 13

Time 2 Input Stack Costas Busch - LSU 14

Time 3 Input Stack Costas Busch - LSU 15

Time 4 Input Stack Costas Busch - LSU 16

Time 5 Input Stack Costas Busch - LSU 17

Time 6 Input Stack Costas Busch - LSU 18

Time 7 Input Stack Costas Busch - LSU 19

Time 8 Input Stack accept Costas Busch - LSU 20

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 Costas Busch - LSU 21

Rejection Example: Time 0 Input Stack current state Costas Busch - LSU 22

Rejection Example: Time 1 Input Stack current state Costas Busch - LSU 23

Rejection Example: Time 2 Input Stack current state Costas Busch - LSU 24

Rejection Example: Time 3 Input Stack current state Costas Busch - LSU 25

Rejection Example: Time 4 Input Stack current state Costas Busch - LSU 26

Rejection Example: Time 4 Input reject Stack current state Costas Busch - LSU 27

There is no accepting computation for The string is rejected by the PDA Costas Busch - LSU 28

Another PDA example PDA : Costas Busch - LSU 29

Basic Idea: • Push v on stack 2. Guess middle of input 3. Match on input with v on stack 4. Match found Costas Busch - LSU 30

Execution Example: Time 0 Input Stack Costas Busch - LSU 31

Time 1 Input Stack Costas Busch - LSU 32

Time 2 Input Stack Costas Busch - LSU 33

Time 3 Input Guess the middle of string Stack Costas Busch - LSU 34

Time 4 Input Stack Costas Busch - LSU 35

Time 5 Input Stack Costas Busch - LSU 36

Time 6 Input Stack accept Costas Busch - LSU 37

Rejection Example: Time 0 Input Stack Costas Busch - LSU 38

Time 1 Input Stack Costas Busch - LSU 39

Time 2 Input Stack Costas Busch - LSU 40

Time 3 Input Guess the middle of string Stack Costas Busch - LSU 41

Time 4 Input Stack Costas Busch - LSU 42

Input Time 5 There is no possible transition. Input is not consumed Stack Costas Busch - LSU 43

Another computation on same string: Input Time 0 Stack Costas Busch - LSU 44

Time 1 Input Stack Costas Busch - LSU 45

Time 2 Input Stack Costas Busch - LSU 46

Time 3 Input Stack Costas Busch - LSU 47

Time 4 Input Stack Costas Busch - LSU 48

Time 5 Input No accept state is reached Stack Costas Busch - LSU 49

There is no computation that accepts string Costas Busch - LSU 50

Pushing & Popping Strings Input symbol Pop string Costas Busch - LSU Push string 51

Example: input stack pop string top push string Replace Costas Busch - LSU 52

Equivalent transitions pop push Costas Busch - LSU 53

Another PDA example PDA Costas Busch - LSU 54

Execution Example: Time 0 Input Stack current state Costas Busch - LSU 55

Time 1 Input Stack Costas Busch - LSU 56

Time 3 Input Stack Costas Busch - LSU 57

Time 4 Input Stack Costas Busch - LSU 58

Time 5 Input Stack Costas Busch - LSU 59

Time 6 Input Stack Costas Busch - LSU 60

Time 7 Input Stack Costas Busch - LSU 61

Time 8 Input Stack accept Costas Busch - LSU 62

Formalities for PDAs Costas Busch - LSU 63

Transition function: Costas Busch - LSU 64

Transition function: Costas Busch - LSU 65

Formal Definition Pushdown Automaton (PDA) Accept states States Input alphabet Transition Initial Stack function state alphabet Costas Busch - LSU Stack start symbol 66

Instantaneous Description Current state Remaining input Costas Busch - LSU Current stack contents 67

Example: Instantaneous Description Time 4: Input Stack Costas Busch - LSU 68

Example: Instantaneous Description Time 5: Input Stack Costas Busch - LSU 69

We write: Time 4 Time 5 Costas Busch - LSU 70

A computation: Costas Busch - LSU 71

For convenience we write: Costas Busch - LSU 72

Language of PDA Language accepted by PDA Initial state Costas Busch - LSU : Accept state 73

Example: PDA : Costas Busch - LSU 74

PDA : Costas Busch - LSU 75

Therefore: PDA : Costas Busch - LSU 76