meljun cortes automata lecture pushdown automata 2

8/21/2019 MELJUN CORTES Automata Lecture Pushdown Automata 2

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Theory of Computation (With Automata Theory)

* Property of STI

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Pushdown Automata

PUSHDOWN AUTOMATA

Pushdown Automata

Introduction

• A regular expression cangenerate the strings of theregular language it represents.

• A context-free grammar cangenerate the strings of thecontext-free language itrepresents.

• A finite automaton recognizesregular languages.

• The machine that recognizescontext-free languages iscalled a pushdown automaton (PDA).


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* Property of STI

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Pushdown Automata

• A PDA is similar to an NFAexcept that it has access to astack.

Representation of an NFA:

NFA

0 1 1 0

Input String


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* Property of STI

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Pushdown Automata

• Representation of a PDA:

The stack follows a last-in,first-out (LIFO) structure.

PDA

a

b

a

b

Stack

0 1 1 0

Input String


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* Property of STI

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Pushdown Automata

• Operations that can be doneon the stack are push (add asymbol to the top of the stack)and pop (remove a symbolfrom the top of the stack).

• State transitions in a PDAdepend on the current state,the incoming input symbol,and the symbol at the top of

the stack.

• The stack can give additionalmemory beyond what isavailable in an NFA (as

represented by the states).

• PDAs are more powerful thanNFAs because they canrecognize some non-regular

languages.


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* Property of STI

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Pushdown Automata

Case Study:

Consider the non-regularlanguage L1 = {0

n1n n ≥ 0}.There is no NFA that can

recognize this language but aPDA can be constructed torecognize it.

PDA Operation:

1. Read each symbol of theinput string.

2. While the input is a 0, pushit onto the stack.

3. While the input symbol is a1, pop a 0 off the stack.4. If there is no more input

symbol and the stack isempty, the input string is

accepted.


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* Property of STI

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Pushdown Automata

Formal Definition of Pushdown Automata

• A pushdown automaton is a 6-tuple (Q, , , , q0, F ), whereQ, , , and F are all finitesets, and

1. Q is the set of states,

2. is the input alphabet,

3. is the stack alphabet,

4. is the transition function,

5. q0 Q and is the startstate, and

6. F Q and is the set of final

states.


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* Property of STI

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Pushdown Automata

• The stack alphabet is the setof symbols that can be pushedonto the stack. It may or maynot be the same as the inputstring alphabet .

• Transitions in a PDA

In an NFA, transitions are

represented by:

(qi , 0) = q j

This equation indicates that if

the current state is state qi andthe input is 0, the NFA goes tostate q j . State transitions aretherefore determined by thecurrent state and the input

symbol.


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* Property of STI

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Pushdown Automata

In a PDA, transitions arerepresented by:

(qi

, 0, a) = (q j

, b)

This equation indicates that if the current state is state qi andthe input symbol is 0, and the

symbol at the top of the stackis a, the PDA goes to state q j and pushes the symbol b ontothe stack.

State transitions are thusdetermined by the currentstate, the input symbol, andthe symbol at the top of thestack.


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* Property of STI

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Pushdown Automata

• PDA State Diagrams

PDA state diagrams are similar to NFA state diagrams exceptfor the labels of the transition

edges.

Example:

The label a, b → c means thatif the input symbol is a and thesymbol read from the top of the stack is b, then the PDAgoes to state q j and it pushes

the symbol c unto the stack.

q j

a, b → c

q i


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* Property of STI

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Pushdown Automata

Take note that reading thesymbol at the top of the stackimplies a pop operation. Thelabel a, b → c implies that thesymbol b was popped off thestack.

Furthermore, any of thesymbols a, b, c in the givenlabel can also be the emptystring .

1. If a = , the PDA makesthe transition without anyinput symbol.

2. If b = , the PDA makes

the transition withoutreading any symbol fromthe stack.

3. If c = , the PDA does notpush any symbol onto the

stack.


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* Property of STI

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Pushdown Automata

• In order for the PDA todetermine if the stack isempty, a special symbol will bedesignated as the stack emptysymbol.

This symbol will be pushedonto the stack every time acomputation is started.

The symbol that will be usedhere will be the dollar sign ($ ).

So, any time the PDA seesthat the symbol at the top of the stack is $ , then the stack isconsidered empty.


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* Property of STI

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Pushdown Automata

• Sample Transitions

q j

0, a → b

q i

a

x

y

$

stack before

transition

b

x

y

$

stack after

transition

top of thestack

top of thestack

q j

0, a → a

q i

a

x

y

$

stack beforetransition

a

x

y

$

stack aftertransition

top of thestack

top of thestack


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* Property of STI

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Pushdown Automata

q j

0, → b

q i

a

x

y

$


a

x

y

$


b

top of thestack

top of thestack

q j

0, a →

q i

a

x

y

$


x

y

$


top of thestack

top of thestack


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* Property of STI

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Pushdown Automata

• It was mentioned earlier that aPDA can make a state

transition without any inputsymbol.

This implies that PDAs arenondeterministic.

q j

0, →

q i

ax

y

$


x

y

$


top of the

stacktop of the

stacka


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* Property of STI

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Pushdown Automata

• Continuation of Case Study

The PDA P 1 that canrecognize the non-regular

language L1 = {0n1n n ≥ 0} is:

q 0

q 1 q 2

q 3

, → $

0, → 0

1, 0 →

1, 0 →

, $ →


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* Property of STI

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Pushdown Automata

The given PDA can be formallydefined as P 1 = (Q, , , , q0,F ) where:

1. Q = {q0, q1, q2, q3}

2. = {0, 1}

3. = {0, $}

4. q0 is the start state.

5. F = {q0, q3}


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* Property of STI

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Pushdown Automata

6. The transition function isgiven by:

Input StackCurrent State

q0 q1 q2 q3

0

0

$

(q 1, 0)

1

0 (q 2, ) (q 2, )

$

0$ (q 3, )

(q 1, $)


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* Property of STI

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Pushdown Automata

• Example 2:


language L2 = {ww R w {0,1}*} is:

q 0

q 1 q 2

q 3

, → $

0, → 0

, →

0, 0→

, $→

1, → 1 1, 1→


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* Property of STI

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Pushdown Automata

• Example 3:


language L3 = {ai b j c k i , j , k 0and i = j or i = k } is:

q 0

, →q 1 q 4

, →q 5 q 6

, $ →

a, → a b, → c, a →

q 2 q 3

,

→

, $ →

, → $

b, a → c, →


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* Property of STI Pushdown Automata

• Exercises:

Construct a PDA that canrecognize the language L4 ={w w is composed of an equalnumber of 0s and 1s}.

Construct a PDA that canrecognize the language L5 ={02n1n n ≥ 1}.

meljun cortes automata lecture pushdown automata 2

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