1 pdas accept context-free languages. 2 context-free languages (grammars) languages accepted by pdas...
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1
PDAs Accept Context-Free Languages
2
Context-Free
Languages(Grammars)
LanguagesAccepted
byPDAs
Theorem:
3
Context-Free
Languages(Grammars)
LanguagesAccepted
byPDAs
Proof - Step 1:
Convert any context-free grammar to a PDA with:
GM )()( MLGL
4
Context-Free
Languages(Grammars)
LanguagesAccepted
byPDAs
Proof - Step 2:
Convert any PDA to a context-free grammar with: G
M)()( MLGL
5
Converting
Context-Free Grammarsto
PDAs
Proof - step 1
6
Context-Free
Languages(Grammars)
LanguagesAccepted
byPDAs
Convert any context-free grammar to a PDA with:
GM )()( MLGL
7
to a PDA such that:M
We will convert grammar G
M simulates leftmost derivations of G
8
q0 q1 2qS, , $ $
wA, aa,
Production in
wATerminal in
aG G
Convert grammar to PDA G M
9
nkk
mk XX
S
11
11
Grammarleftmost derivation
,$),(
$),,(
$),,(
,$),(
2
111
111
110
q
XXq
Sq
q
mnk
nkk
nkk
PDA computationSimulates grammar leftmost derivations
Leftmostvariable
10
q0 q1 2qS, , $ $
Grammar
PDA
T
TaT
bS
aSTbS
,
,
,
,
T
TaT
bS
aSTbS
bb
aa
,
,
Example
11
abab
abTab
abTb
aSTb
S
,$),(
,$),(
$),,(
$),,(
$),,(
$),,(
$),,(
$),,(
$),,(
,$),(
2
1
1
1
1
1
1
1
1
0
q
q
bbq
ababq
Tababq
Tbabq
bTbbabq
STbbabq
Sababq
ababq
Grammarderivation
PDA computation
120q q1 2q
S, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab
Time 0
b
Derivation:
13
q0 q1 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
S
Time 0
Derivation: S
14
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
a
b
ST
q1
Time 1
Derivation: aSTbS
15
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
a
b
ST
q1
Time 2
aSTbS Derivation:
16
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
bTb
q1
Time 3
Derivation: abTbaSTbS
17
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
bTb
q1
Time 4
Derivation: abTbaSTbS
18
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
b
Ta
q1
Time 5
Derivation: abTababTbaSTbS
19
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
b
Ta
q1
Time 6
Derivation: abababTababTbaSTbS
20
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
ba
q1
Time 7
Derivation: abababTababTbaSTbS
21
q0 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
b
q1
Time 8
Derivation: abababTababTbaSTbS
22
q0 q1 2qS, , $ $
T
TaT
bS
aSTbS
,
,
,
,
bb
aa
,
,
Input
Stack
$
a ab b
accept
Time 9
Derivation: abababTababTbaSTbS
23
Grammar generates string
G
wS*
PDA accepts
M
,$),(,$),( 20 qwq
In general, it can be shown that:
w wIf andOnly if
Therefore )()( MLGL
24
Therefore:
For any context-free language there is a PDA that accepts
Context-Free
Languages(Grammars)
LanguagesAccepted
byPDAs
LL
25
Converting
PDAsto
Context-Free Grammars
Proof - step 2
26
Context-Free
Languages(Grammars)
LanguagesAccepted
byPDAs
Convert any PDA to a context-free grammar with: G
M)()( MLGL
27
We will convert PDA toM
a context-free grammar such that:G
Msimulates computations ofwith leftmost derivations
G
28
Some Necessary Modifications
If necessary, modify the PDA so that:
1. The stack is never empty during computation
2. It has a single accept state and empties the stack when it accepts a string
3. Has transitions without popping
29
1. Modify the PDA so that the stack is never empty during computation
$
Stack
a
OK
$
OK NOT OK
30
$
a$
Introduce the new symbol to mark the bottom of the stack
#
$a
#$# #
31
Original PDA
$#,$
At the beginning insert into the stack#
original initial state
new initial state
32
iq jqs ,
iq jqs ,
## ,
halting state
Convert all transitions so that after popping the automaton halts $
xx,}{#x
pop $
pop$
33
PDA
fq
x,
,
Empty stack
}{#x
2. Modify the PDA so that at end it empties stack and has a unique accept state
,
,
Old accept states
#,
New accept state
34
3. Modify the PDA so that it has no transitions popping :
iq jqy ,
iq jq y,
}{#
35
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)}()(:},{{)( * wnwnbawML ba
Example of a PDA in correct form:(modifications are not necessary)
36
Grammar Construction
A
In grammar :G
Terminals:
PDA stack symbolsVariables:
a PDA input symbols
Start Variable: Stack bottom symbol#or $
37
q 1qAa,PDA transition
Grammar production aA
38
PDA transition
Grammar production mBBaBA 21
q 1qmBBBAa 21,
39
nkk
mk XX
S
11
11
Grammarleftmost derivation
),,(
),,(
,$),(
2
111
111
q
XXq
q
mnk
nkk
PDA computation
Leftmostvariable
40
Example PDA:
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Grammar: $0$ a000 aa1
$1$ b111 bb0 $
41
abba
abba
abb
ab
a
$
$1
$
$0
$
GrammarLeftmostderivation:
PDAComputation:
),,(
,$),(
$)1,,(
,$),(
$)0,,(
,$),(
0
0
0
0
0
fq
q
aq
baq
bbaq
abbaq
42
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
$Derivation:
a ab b Time 0
43
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
$Derivation:
a ab b Time 1
Stack
$0
$0a
44
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
$Derivation:
a ab b Time 2
Stack
$
$0a $ab
45
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
$Derivation:
a ab b Time 3
Stack
$
$0a $ab
1
$1abb
46
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Derivation:
a ab b Time 4
Stack
$
$ $0a $ab $1abb$abba
47
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Derivation:
a ab b Time 5
Stack
empty
abba
$ $0a $ab $1abb$abba
48
However, this grammar conversion does not work for all PDAs:
Ab,
0q fq
$$, Aa $, AAa
Ab,1q
$,
}1:{)( nbaML nn
49
Ab,
0q fq
$$, Aa $, AAa
Ab,1q
$,
Grammar: $$ aA $aAAbA$
50
Grammar: $$ aA $aAAbA$
Bad Derivation:
)($$$ MLaabaabaaAaAS
51
The Correct Grammar Construction
)( jiAqq
In grammar :G
Terminals: Input symbols of PDA
PDA states
PDA stack symbol
Variables:
52
q 1qAa,PDA transition
Grammar production aqAq )( 1
53
PDA transition
Grammar production
)())(()( 1322211 mmmm qBqqBqqBqaqAq
q 1qmBBBAa 21,
For all possible states in PDA 12 ,, mqq
54
Start Variable: )( foZqq
Stack bottom symbol
Start state accept state
#or $
55
Example:
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
aqq )1( 00Grammar production:
56
Example:
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)$)(1(|)$)(1()$(
)$)(1(|)$)(1()$(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
Grammar productions:
57
Example:
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
Grammar production: )$( 0 fqq
58
)$)(1(|)$)(1()$(
)$)(1(|)$)(1()$(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
)1)(1(|)1)(1()1(
)1)(1(|)1)(1()1(
00000
00000000
fffff
ff
qqqqbqqqqbqq
qqqqbqqqqbqq
)$)(0(|)$)(0()$(
)$)(0(|)$)(0()$(
00000
00000000
fffff
ff
qqqqaqqqqaqq
qqqqaqqqqaqq
Resulting Grammar: ablestart vari:)$( 0 fqq
59
)0)(0(|)0)(0()0(
)0)(0(|)0)(0()0(
00000
00000000
fffff
ff
qqqqaqqqqaqq
qqqqaqqqqaqq
bqq
aqq
)0(
)1(
00
00
)$( 0 fqq
60
)$( 0 fqq
)$)(0( 000 fqqqqa
)$( 0 fqqab)$)(1( 000 fqqqqabb
)$( 0 fqqabba
abba ),,(
,$),(
$)1,,(
,$),(
$)0,,(
,$),(
0
0
0
0
0
fq
q
aq
baq
bbaq
abbaq
GrammarLeftmostderivation
PDAcomputation
61
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)$( 0 fqqDerivation:
a ab b Time 0
62
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)$( 0 fqqDerivation:
a ab b Time 1
Stack
$0
)$)(0( 000 fqqqqa
63
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)$( 0 fqqDerivation:
a ab b Time 2
Stack
$
)$)(0( 000 fqqqqa )$( 0 fqqab
64
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)$( 0 fqqDerivation:
a ab b Time 3
Stack
$
)$)(0( 000 fqqqqa )$( 0 fqqab
1
)$)(1( 000 fqqqqabb
65
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)$( 0 fqqDerivation:
a ab b Time 4
Stack
$
)$)(0( 000 fqqqqa )$( 0 fqqab)$)(1( 000 fqqqqabb )$( 0 fqqabba
66
$,0q fq
a, $ 0$a, 0 00a,1
b, $ 1$b, 1 11b, 0
)$( 0 fqqDerivation:
a ab b Time 5
Stack
)$)(0( 000 fqqqqa )$( 0 fqqab)$)(1( 000 fqqqqabb )$( 0 fqqabba abba
empty
67
In general:
wBAqq ji
)(If andOnly if
Grammar PDA
),,(),,( BqAwq ji
68
Thus:
wqq f
)$( 0
If andOnly if
Grammargenerates PDA accepts
),,(,$),( 0 fqwq
w w
69
Therefore:
For any PDAthere is a context-free grammarthat accepts the same language
Context-Free
Languages(Grammars)
LanguagesAccepted
byPDAs
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