Theory of computation Lec7 pda

Theory of Computation

Lecture 7: Push Down Automata (PDA)

Cairo UniversityFCI

Dr. Hussien SharafComputer Science [email protected]

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p.434: Regular Languages and non-regular languages

Dr. Hussien M. Sharaf

Language Defined by

Corresponding Accepting Machine

Nondeter-minism=Determinism

Language Closed Under

What Can Be Decided?

Examples of Applications

Regular expression

Finite automaton, transition graph

Yes Union,productKleene star, intersection, complement

Equivalence, emptiness, finiteness, membership

Text editors, sequential circuits, verification

Context-free grammar

Pushdown automaton

No Union, product, Kleene star

Emptiness, finiteness, membership

Parsing, compilers

Type 0 grammar

Turing machine, Post machine, Pushdown automaton

Yes Union, product, Kleene star

Not much Computers

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PDAA Pushdown Automata (PDA ) is a DFA

equipped with a single stack.


1

4

3

2

1 2 3 4

Input Tape

Tape cells

4

PDA


1. Q : A finite set of states (Q).2. Σ : A set of input alphabet.3. Γ (capital Gammma): A set of stack alphabet.4. (delta): A transition function represents the set

of transitions that moves PDA to the next states.

: Q x Σ x Γ →Q.5. q0 Q : start state.

6. F ⊆ Q is the set of final states.

ε Γ and Σ : start symbol for the stack and end symbol for the Input Tape.

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FA = “a 5-tuple “ (Q, Σ, , q0, F)

1. Q: {q0, q1, q2, …} is set of states.

2. Σ: {a, b, …} set of alphabet.3. (delta): represents the set of transitions

that FA can take between its states.: Q x Σ→Q

4. q0 Q is the start state.

5. F Q is the set of final/accepting states.


Remember and compare FA

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PDA as a deterministic machine

Its inputs are1. The current state (of its “NFA”),2. The current input symbol (or Λ),

and 3. The current symbol on top of its

stack.


1

4

3

2

Its actions are:1. Change state.2. Change stack if needed.

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Power of PDA


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Ex1 P.295 cohen Edition 2:Consider the following PDA for anbn


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Ex1 P.295:Consider the following Input


Δ

a a

Input Tape

a b b b Δ

Start

Read

Push Δa

b

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Ex1 P.295:Consider the following PDA


aa a

Input Tape

a b b b Δ

Start

Read

Push Δa

b

Δ

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a

a a

Input Tape

a b b b Δ

Start

Read

Push Δa

b

Δ

a

a

12



aa a

Input Tape

a b b b Δ

POP ab, Δ

a

Δ

a

Read

Δ

b b

Reject

Reject

13



aa a

Input Tape

a b b b Δ

POP ab, Δ

a

Δ

Read

Δ

b b

Reject

Reject

14



a a

Input Tape

a b b b Δ

POP ab, Δ

a

Δ

Read

Δ

b b

Reject

Reject

15



a a

Input Tape

a b b b Δ

Read

Δ Pop a,b

Accept

Δ

Reject

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Ex2: Even Palindrome P.304Can you list samples of words in even palindrome?


PUSH a READ2READ1

POP3

POP2

POP1START

ACCEPT

PUSH b

a

b

a

a

a

b b

b

Δ

Δ

Δ

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Ex2: Even Palindrome


Consider babbab

18



Jump to the right part

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We have just read first many blanks on the TAPE

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Assignment #5Read Example P.308 in Cohen’s book and

hand in a stack table for input 4 * 4 + 4.

The assignment must be hand written.


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PDA for Balanced strings

State diagram for

L = {0n 1n | n >= 0} Dr. Hussien M. Sharaf

0 ,ε0

q4

q1

(delta) is shown in the state diagram"ε“ or “” means ignore or don’t care for both input and stack.$ means end for both input or stack

q3

q2

0 ,00

1 ,0pop

1 ,0pop

$ , $no Action

$ , $no Action

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Thank you


Education

Theory of computation Lec7 pda