[PDF] Pushdown Automata (()PDA) - Washington State University





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Homework 6 Solutions

Give pushdown automata that recognize the following languages. Give both a drawing and 6-tuple specification for each PDA. (a) A = { w ∈ {0 1} 



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Homework 6 Solutions

Give pushdown automata that recognize the following languages. Give both a drawing and 6-tuple specification for each PDA. (a) A = { w ? {0 1} 



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A pushdown automaton(PDA) is essentially a finite automaton with a stack Example PDA accepting Initially the symbol the stack 0is on Acceptance can be by final state or empty stack = 01 ?0: 0 Stack Input string: 0011 Current input A PDA can be defined by a 7-tuple ?? 0 0 : A finite set of states



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16 A two-way pushdown automaton may move on its input tape in two directions As usual for two-way automata we assume that the begin and end of the input tape is marked by special symbols In this way the automaton can recognize those positions Describe a two-way pda for each of the following languages (a) f anbncn j n 2 N g (easy)



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How to convert an final state PDA into an empty stack PDA? P F==> P N construction Main idea: Whenever P F reaches a final state just make an -transition into a new end state clear out the stack and acceptnew end state clear out the stack and accept



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Solutions for CSE303 Homework 5 1 Construct nondeterministic pushdown automata (npda) that accept the following regular languages Note: Observe that all the languages are regular languages so the solutions are essentially NFA’s (ornpda’s with inactive stack) For all the languagesfis the ?nal state (a)L1=L(aaa?b)



Searches related to pushdown automata examples solutions filetype:pdf

• A DPDA is simply a pushdown automata without non-determinism – i e no epsilon transitions or transitions to multiple states on same input – Only one state at a time • DPDA not as powerful a non-deterministic PDA – This machine accepts a class of languages somewhere between regular languages and context-free languages

Pushdown Automata

PDA

Reading: Chapter 6

1

PDA - the automata for CFLs

What is?

FA to Reg Lang

PDA is to CFL

FA to Reg Lang PDA is to CFL

PDA == [

-NFA + "a stack" ]

Wh t k?

Wh y a s t ac k? -NFA Input stringAccept/reject 2

A stack filled with "stack symbols"

Pushdown Automata -Definition

A PDA P := ( Q,,, į,q

0 ,Z 0 ,F ):

Q: states of the -NFA

: input alphabet : stack symbols

į: transition function

q 0 : start state Z

Initial stack top s mbol

Z 0

Initial

stack top s y mbol

F: Final/accepting states

3 old stateStack top input symb.new state(s)new Stack top(s)

į : Q x x =>Q x

į: The Transition Function

į(q,a,X) = {(p,Y), ...}

1. state transition from q to p 2. a is the next input symbolX is the current stack top symbol q a X p Y 3. X is the current stack top symbol 4.

Y is the replacement for X;

it is in * (a string of stack symbols)

Set Y =

for:

Pop(X)

q p

Y= ? Action

i Y=

Pop(X)

i. Set Y for:

Pop(X)

ii.

If Y=X: stack top is

unchanged iii.

If Y=Z

1 Z 2 ...Z k : X is popped and is replaced by Yin i Y=

Pop(X)

ii) Y=X Pop(X)

Push(X)iii)

Y Z 1 Z 2 Z k

Pop(X)

and is replaced by Y in reverse order (i.e., Z 1 will be the new stack top) iii) YZ 1 Z 2 Z k

Pop(X)Push(Z

k

Push(Z

k-1

Push(Z

2 4

Push(Z

2

Push(Z

1

Example

Let L wwr = {ww R | w is in (0+1)*}

CFG for L

wwr

S==> 0S0 | 1S1 |

PDA for L

wwr

P := ( Q,, , į,q

0 ,Z 0 ,F ) = ( {q 0 , q 1 , q 2 },{0,1},{0,1,Z 0 },į,q 0 ,Z 0 ,{q 2 5

Initial state of the PDA:

q St k

PDA for L

wwr Z 0 q 0 St ac k top 1.

į(q

0 ,0, Z 0 )={(q 0 ,0Z 0 2.

į(q

0 ,1, Z 0 )={(q 0 ,1Z 0

į(q

0

0) {(q

0 0)}

First symbol push on stack

3.

į(q

0 0 0) {(q 0 0 0)} 4.

į(q

0 ,0, 1)={(q 0 ,01)} 5.

į(q

0 ,1, 0)={(q 0 ,10)} 6.

į(q

0 ,1, 1)={(q 0 ,11)}

Grow the stack by pushing

new symbols on top of old( t) 7.

į(q

0 , , 0)={(q 1 , 0)} 8.

į(q

0 , , 1)={(q 1 , 1)} 9

į(q

0 Z 0 q 1 Z 0 w-par t)

Switch to popping mode, nondeterministicall

y (boundary between w and w R 9

į(q

0 Z 0 q 1 Z 0 10.

į(q

1 ,0, 0)={(q 1 11.

į(q

1 ,1, 1)={(q 1 (boundary between w and w

Shrink the stack by popping matching

s y mbols w R p art 6 12.

į(q

1 , , Z 0 )={(q 2 , Z 0 y( p)

Enter acceptance state

PDA as a state diagram

į(q

i ,a, X)={(q j ,Y)}

Next input

Currentstack

StackTop

input symbol

Current

state stacktop

TopReplacement(w/ string Y)

q i q j a, X / Y Next state 7

PDA for L

wwr : Transition Diagram 0, Z 0 /0Z 0 1, Z 0 /1Z 0

Grow stack

Pop stack for

= {0, 1} = {Z 0 , 0, 1} Q={q q q 1, Z 0 /1Z 0

0, 0/00

0, 1/01

1, 0/10

1, 1/110, 0/

1, 1/ Pop stack for matching symbols Q {q 0 q 1 q 2 q 0 q 1 q 2 , Z 0 /Z 0 , 0/0 , Z 0 /Z 0 G , Z 0 /Z 0 , 1/1quotesdbs_dbs12.pdfusesText_18
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