Pushdown Automata PDAs 1 Pushdown Automaton PDA Input

Pushdown Automata PDAs 1

Pushdown Automaton -- PDA Input String Stack States 2

Initial Stack Symbol Stack bottom special symbol 3

The States Input symbol Pop symbol Push symbol 4

input stack top 5

input stack top 6

input stack top 7

input stack top 8

Non-Determinism 9

NPDA: Non-Deterministic PDA Example: 10

Execution Example: Time 0 Input Stack current state 11

Time 1 Input Stack 12

Time 2 Input Stack 13

Time 3 Input Stack 14

Time 4 Input Stack 15

Time 5 Input Stack 16

Time 6 Input Stack 17

Time 7 Input Stack 18

Time 8 Input Stack accept 19

A string is accepted if: • All the input is consumed • The last state is a final state We do not care about the stack contents 20

The input string is accepted by the NPDA: 21

In general, is the language accepted by the NPDA: 22

Another NPDA example NPDA 23

24

Execution Example: Time 0 Input Stack 25

Time 1 Input Stack 26

Time 2 Input Stack 27

Time 3 Input Stack 28

Time 4 Input Stack 29

Time 5 Input Stack 30

Time 6 Input Stack accept 31

Pushing Strings Input symbol Pop symbol Push string 32

Example: input stack top pushed string 33

Another NPDA example NPDA 34

35

Execution: Time 0 Input Stack current state 36

Time 1 Input Stack 37

Time 3 Input Stack 38

Time 4 Input Stack 39

Time 5 Input Stack 40

Time 6 Input Stack 41

Time 7 Input Stack 42

Time 8 Input Stack accept 43

Formalities for NPDAs 44

Transition function: 45

Transition function: 46

Formal Definition Non-Deterministic Pushdown Automaton NPDA Final states States Input alphabet Stack alphabet Transition function Stack start symbol 47

Instantaneous Description Current state Remaining input Current stack contents 48

Example: Instantaneous Description Time 4: Input Stack 49

Example: Instantaneous Description Time 5: Input Stack 50

We write: (Time 4) (Time 5) 51

A computation: 52

For convenience we write: 53

Formal Definition Language of NPDA Start state : Final state 54

Example: thus NPDA : 55

thus NPDA : 56

Therefore: NPDA : 57

Converting Context-Free Grammars to NPDAs 58

An example grammar: What is the equivalent NPDA? 59

Grammar: NPDA: 60

The NPDA simulates leftmost derivations of the grammar L(grammar) = L(NPDA) 61

Grammar: A leftmost derivation: 62

NPDA execution: Time 0 Input Stack current state 63

Time 1 Input Stack 64

Time 2 Input Stack 65

Time 3 Input Stack 66

Time 4 Input Stack 67

Time 5 Input Stack 68

Time 6 Input Stack 69

Time 7 Input Stack 70

Time 8 Input Stack 71

Time 9 Input Stack 72

Time 10 Input Stack accept 73

In general: Given any grammar We can construct a NPDA With 74

Constructing NPDA from grammar : For any production For any terminal 75

Grammar generates string if and only if NPDA accepts 76

Therefore: For any context-free language there is an NPDA that accepts the same language 77