Embed Size (px)
Citation preview
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 can convert PDA toM
a context-free grammar such that:G
Msimulates computations ofwith leftmost derivations
G
28
PDA
fq
x,
,
Empty stack
}{#x
Modify the PDA so that at end it empties stack and has a unique accept state
,
,
Old accept states
#,
New accept state
29
Deterministic PDAs - DPDAs
30
Deterministic PDA: DPDA
q1 q2wba ,
1q 2qwb,
Allowed transitions:
(deterministic choices)
31
Allowed transitions:
q1
q21, wb
q32, wc
q1
q21, wba
q32, wca
(deterministic choices)
32
q1
q21, wba
q32, wba
Not allowed:
(non deterministic choices)
q1
q21, wb
q32, wba
33
DPDA example
a, a
b, a q0 q1 q2 q3
b, a
, $ $
}0:{)( nbaML nn
a, a
34
}0:{)( nbaML nnThe language
is deterministic context-free
Definition:
A language is deterministic context-freeif there exists some DPDA that accepts it
L
Example:
35
Example of Non-DPDA (PDA)
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
}},{:{)( *bavvvML R
36
, $ $q1 q2
bb
aa
,
,
, q0
bb
aa
,
,
Not allowed in DPDAs
37
PDAs
Have More Power than
DPDAs
38
Deterministic
Context-FreeLanguages
(DPDA)
Context-Free
LanguagesPDAs
Since every DPDA is also a PDA
It holds that:
39
We will actually show:
We will show that there existsa context-free language which is notaccepted by any DPDA
L
LL
Deterministic
Context-FreeLanguages
(DPDA)
Context-Free
Languages(PDA)
40
The language is:
}{}{ 2nnnn babaL 0n
We will show:
L• is context-free
L• is not deterministic context-free
41
}{}{ 2nnnn babaL
Language is context-freeL
Context-free grammar for : L
21 | SSS
|11 baSS
|22 bbaSS
}{ nnba
}{ 2nnba
}{}{ 2nnnn baba
42
}{}{ 2nnnn babaL
is not deterministic context-free
Theorem:
The language
(there is no DPDA that accepts ) L
43
Proof: Assume for contradiction that
}{}{ 2nnnn babaL
is deterministic context free
Therefore:
there is a DPDA that accepts M L
44
DPDA with M
nnba nb
acceptsnnba 2
acceptsnnba
}{}{)( 2nnnn babaML
45
DPDA with M
nnba nb
}{}{)( 2nnnn babaML
Such a path exists due to determinism
M
46
The language is not context-free
}{ nnn cba
(we will prove this at a later class usingpumping lemma for context-free languages)
Fact 1:
Regular languages**ba
Context-free languagesnnba
47
The language is not context-free
}{ nnn cbaL
}){}{( 2nnnn babaL
Fact 2:
(we can prove this using pumping lemma for context-free languages)
48
We will construct a PDA that accepts:
}{ nnn cbaL
}){}{( 2nnnn babaL
which is a contradiction!
49
Modify M
M ncnnca
}{}{)( 2nnnn babaML
}{}{)( 2nnnn cacaML
Replacewith
bc
nnba nb
MDPDA
DPDA
50
A PDA that accepts }{ nnn cbaL
nnba nb
nc
M
M
nnca
Connect the final states of with the final states of M
M
51
Since is accepted by a PDA }{ nnn cbaL
it is context-free
Contradiction!
(since is not context-free)}{ nnn cbaL
52
Therefore:
}{}{ 2nnnn babaL
There is no DPDA that accepts it
End of Proof
Is not deterministic context free
53
Positive Propertiesof
Context-Free languages
54
Context-free languages are closed under: Union
1L is context free
2L is context free
21 LL
is context-free
Union
55
Example
|11 baSS
|| 222 bbSaaSS
Union
}{1nnbaL
}{2RwwL
21 | SSS }{}{ Rnn wwbaL
Language Grammar
56
In general:
The grammar of the union has new start variableand additional production 21 | SSS
For context-free languageswith context-free grammarsand start variables
21, LL
21, GG
21, SS
21 LL S
57
Context-free languages are closed under: Concatenation
1L is context free
2L is context free
21LL
is context-free
Concatenation
58
Example
|11 baSS
|| 222 bbSaaSS
Concatenation
}{1nnbaL
}{2RwwL
21SSS }}{{ Rnn wwbaL
Language Grammar
59
In general:
The grammar of the concatenation has new start variableand additional production 21SSS
For context-free languageswith context-free grammarsand start variables
21, LL
21, GG
21, SS
21LLS
60
Context-free languages are closed under: Star-operation
L is context free*L is context-free
Star Operation
61
|aSbS }{ nnbaL
|11 SSS *}{ nnbaL
Example
Language Grammar
Star Operation
62
In general:
The grammar of the star operation has new start variableand additional production
For context-free languagewith context-free grammarand start variable
LGS
*L1S
|11 SSS
63
Negative Propertiesof
Context-Free Languages
64
Context-free languagesare not closed under: intersection
1L is context free
2L is context free
21 LL
not necessarilycontext-free
Intersection
65
Example
}{1mnn cbaL
|
|
cCC
aAbA
ACS
Context-free:
}{2mmn cbaL
|
|
bBcB
aAA
ABS
Context-free:
}{21nnn cbaLL NOT context-free
Intersection
66
Context-free languagesare not closed under: complement
L is context free L not necessarilycontext-free
Complement
67
}{2121nnn cbaLLLL
NOT context-free
Example
}{1mnn cbaL
|
|
cCC
aAbA
ACS
Context-free:
}{2mmn cbaL
|
|
bBcB
aAA
ABS
Context-free:
Complement
68
Intersectionof
Context-free languagesand
Regular Languages
69
The intersection of a context-free language and a regular languageis a context-free language
1L context free
2L regular
21 LL
context-free
70
1Lfor for 2LNPDA
1M
DFA
2M
Construct a new NPDA machinethat accepts
Machine Machine
M21 LL
context-free regular
M simulates in parallel and 1M 2M
71
1M 2M
1q 2qcba ,
transition
1p 2pa
transition
NPDA DFA
11, pq cba ,
transition
MNPDA
22 , pq
72
1M 2M
1q 2qcb,
transition
1p
NPDA DFA
11, pq cb,
transition
MNPDA
12, pq
73
1M 2M
0q
initial state
0p
initial state
NPDA DFA
Initial state
MNPDA
00 , pq
74
1M 2M
1q
final state
1p
final states
NPDA DFA
final states
MNPDA
11, pq
2p
21, pq
75
Example:
, , ,
1, b1, a
1,d1,c
0q 1q 2q 3q
1M
}},{,},{|,|||:{ *2
*121211 dcwbawwwwwL
NPDA
context-free
76
0p
2M
*2 },{ caL
ca,
DFA
regular
77
Automaton for: }0:{21 ncaLL nn
, , ,
1, a 1,c
00, pq 01, pq 02, pq 03, pq
NPDA M
context-free
78
M simulates in parallel and 1M 2M
M accepts string w if and only if
accepts string and w1M
accepts string w2M
)()()( 21 MLMLML
In General:
79
Therefore:
)()( 21 MLML
is NPDAM
is context-free
21 LL is context-free
80
Applications of
Regular Closure
81
The intersection of a context-free language and a regular languageis a context-free language
1L context free
2L regular
21 LL
context-free
Regular Closure
82
An Application of Regular Closure
Prove that: }0,100:{ nnbaL nn
is context-free
83
}0:{ nba nn
We know:
is context-free
84
}{ 1001001 baL is regular
}{}){( 100100*1 babaL is regular
We also know:
85
}{}){( 100100*1 babaL
regular
}{ nnbacontext-free
1}{ Lba nn context-free
LnnbaLba nnnn }0,100:{}{ 1
is context-free
(regular closure)
86
Another Application of Regular Closure
Prove that: }:{ cba nnnwL
is not context-free
87
}:{ cba nnnwL
}{*}**{ nnn cbacbaL
context-free regular context-free
If is context-free
Then
Impossible!!!
Therefore, is not context free L
(regular closure)
88
Decidable Propertiesof
Context-Free Languages
89
Membership Question:
for context-free grammarfind if string
G)(GLw
Membership Algorithms: Parsers
• Exhaustive search parser
• CYK parsing algorithm
90
Empty Language Question:
for context-free grammar find if
G)(GL
Algorithm:
S
1. Remove useless variables
2. Check if start variable is useless
91
Infinite Language Question:
for context-free grammar find if is infinite
G)(GL
Algorithm:
1. Remove useless variables
2. Remove unit and productions
3. Create dependency graph for variables
4. If there is a loop in the dependency graph then the language is infinite
92
Example:
cBSC
bbbBB
aaCbA
ABS
|
|
S
A
B
C
Dependency graph Infinite language
93
cBSC
bbbBB
aaCbA
ABS
|
|
acbbSbbbacBSbBaCbBABS
ii bbbSacbb
bbbSacbbacbbSbbbS
)()(
)()( 22
