Pushdown Automata - by Emmanuel Afriyie

8/13/2019 Pushdown Automata - by Emmanuel Afriyie

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CMSC 214 Principles of Programming Language

Pushdown automata

2nd Semester 2013-2014

Presented by EOA


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Motivation

regularexpression DFANFA

syntactic computational

CFG pushdown automaton

syntactic computational

is more powerful than


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Pushdown automata versus NFA

state control0 1 0 0

input

NFA


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Pushdown automata


input

pushdown automaton (PDA)

stack

A PDA is like an NFA with but with an infinite stack


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Pushdown automata


input

pushdown automaton (PDA)

$ 0 1

stack

1

As the PDA is reading the input, it canpush / pop symbols from the top of the stack


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Building a PDA

L = { 0 n 1 n : n 1} state control

We remember each 0

by pushing x onto the stack

read 1pop x

read 0

push x

pop $

push $

read 1pop x When we see a 1 , we

pop an x from the stack

We want to accept whenwe hit the stack bottom

Well use t$ mark bottom


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A PDA in action

L = { 0 n 1 n : n 1} state control

0 0 0 1

input

$ x x

stackx

1 1

read 0

push xread 1

read 1pop x

pop $

push $

pop x


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Notation for PDAs

read, pop / push

read 1pop x

read 0push x

pop $

push $

read 1pop x

1 , x /

0 , / x

, $ /

, / $

1 , x /

q 0

q 1

q 2

q 3


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Definition of a PDA

A pushdown automaton is (Q, , , , q 0, F) where: Q is a finite set of states ;

is the input alphabet ; is the stack alphabet

q 0 in Q is the initial state ; F Q is a set of final states;

is the transition function

: Q ( { }) ( { }) subsets of Q ( { })state input symbol pop symbol state push symbol


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Example

: Q ( { }) ( { }) subsets of Q ( { })

(q 0, , ) = {(q 1, $ )}(q 0, , $ ) = (q 0, , x ) = (q 0, 0 , ) =

...

= { 0 , 1 } = { $ , x }

1 , x /

0 , / x

, $ /

, / $

1 , x /

q 0

q 1

q 2

q 3

state input symbol pop symbol state push symbol


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The language of a PDA

A PDA is nondeterministicMultiple transitions on same pop/input allowed

Transitions may but do not have to push or pop

The language of a PDA is the set of all

strings in * that can lead the PDA to anaccepting state


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Example 1

L = { w # w R : w {0, 1}*} = {0, 1, #}

#, 0#0, 01#10 L, 01#1, 0##0 L

0, / 0

q 1

1, / 10, 0 /1, 1 /

, / $q 0 q 3, $ /q 2

#, /

write w on stack read w R from stack

= {0, 1}


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Example 2

0, / 0

q 1

1, / 1

, / $q 0 q 3, $ /

0, 0 /1, 1 /

q 2, /

L = { ww R : w *} = {0, 1}

, 00, 0110 L1, 011, 010 L

guess middle of string


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Example 3

L = { w : w = w R , w *} = {0, 1}

, 1, 00, 010, 0110 L

011 L0110110110 011010110or

x x x Rx R

0, / 0

q 1

1, / 1

, / $q 0 q 3, $ /

, /0, /1, /

0, 0 /1, 1 /

q 2

middle symbol can be , 0, or 1


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Example 4

L = {0n

1m

0m

1n

| n 0, m 0} = {0, 1}

0, / 0

q 1, /

1, / 1

q 2, / $q 0

, $ /q 5

, /

q 3

0, 1 /

, /q 4

1, 0/

0n 1m 0m 1n

0n 1m

input:

stack:


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Example 5

L = { w : w has same number 0s and 1s} = {0, 1}

Strategy:Stack keeps track of excess of 0s or 1s

If at the end, stack is empty, number is equal

0, / 0

q 1

1, / 1

, / $q 0 q 3, $ /

0, 1 / 1, 0 /


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Example 5


Invariant: In every execution of the PDA:

0, / 0

, / $q 0 q 1

1, / 1

q 3, $ /

0, 1 / 1, 0 /

#1 #0 on stack = #1 #0 in input so far

If w is not in L , it must be rejected


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Example 5


Property: In some execution of the PDA:

0, / 0

, / $q 0 q 1

1, / 1

q 3, $ /

0, 1 / 1, 0 /

stack consists only of 0s or only of 1s (or )

If w is in L , some execution will accept


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Example 5


0, / 0

, / $q 0 q 1

1, / 1

q 3, $ /

0, 1 / 1, 0 /

w = 001110 read stack0 $00 $001 $01 $1 $10 $


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Example 6

= { w : w has two 0-blocks with same number of 0s

Strategy: Detect start of first 0-blockPush 0s on stack

01011, 001011001, 10010101001 01001000, 01111allowed not allowed

Detect start of second 0-blockPop 0s from stack


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Example 6

1 Detect start of first 0-block 2 Push 0s on stack

3 Detect start of second 0-block 4 Pop 0s from stack

11

1

0, 1

0, 0 /

0, / 0

0, / 0q 0 q 1

0, 1

1


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1 2

3

4

Example 6

= { w : w has two 0-blocks with same number of 0s

q 2

0, / 0

q 30, / 0

1, / q 4

1, /

1, /

0, /1, /

q 5

0, 0 /

0, /

, / $q 0 q 1

1, /

1, /

, / $

1, /q 6

0, /1, /

q 7

, $ /

, $ /


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CFG PDA conversions


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CFGs and PDAs

L has a context-free grammar if and only if itis accepted by some pushdown automaton.

context-free grammar pushdown automaton


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A convention

When we have a sequence of transitions like:

We will abbreviate it like this:

x , a / bcdq 0 q 1

, / cq 0 q 1

x , a / b , / d

popa

, then pushb

,c

, andd

replace a by bcd on stack


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Converting a CFG to a PDA

Idea: Use PDA to simulate derivations A 0 A1 A BB # A 0 A1 00 A11 00 B11 00#11

write start variable $Areplace production in reverse $1A0pop terminal and match $1A

, / A

0, 0 /

, A / 1A0

00#11

00#11

0#11

replace production in reverse $11A0, A / 1A0 0#11pop terminal and match $11A0, 0 / #11

replace production in reverse $11B, A / B #11

PDA control: stack: input:


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Converting a CFG to a PDA

A 0 A1 A BB #

CFG

0 , 0 /1 , 1 /# , # /

, A / 1A0, A / B, B / #

q 0 , /$A , $ /

q 1 q 2

$A

00#11

$1A0

00#11

$1A

0 0#11

$11A0

0 0#11

$11A

00 #11

$11B

00 #11

$11#

00 #11

$11

00# 11

$1

00#1 1

$

00#11

inputstack

A 0 A1 00 A11 00 B11 00#11


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General PDA to CFG conversion

q 0 q 1 q 2, / $S

a , a /for every terminal a

, A / k ... 1for every production A 1... k

, $ /


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Andrej Bogdanov

http://www.cse.cuhk.edu.hk/~andrejb/csc3130 The Chinese University of Hong Kong

REFERENCES

F. D. Lewishttp://www.cs.uky.edu/~lewis/texts/theory/automata/pd-auto.pdf Departmentof Computer Science -University of Kentucky
http://www.cs.uky.edu/~lewis/texts/theory/automata/pd-auto.pdfhttp://www.cs.uky.edu/~lewis/texts/theory/automata/pd-auto.pdfhttp://www.cs.uky.edu/~lewis/texts/theory/automata/pd-auto.pdfhttp://www.cs.uky.edu/~lewis/texts/theory/automata/pd-auto.pdf

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Pushdown Automata - by Emmanuel Afriyie