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Positive Properties
of
Context-Free languages
1
Context-free languages
are closed under:
L1 is context free
L2 is context free
Union
L1  L2
is context-free
2
Union Example:
S1  aS1b | 
S2  aS2a | bS2b | 
n n
L1  {a b }
R
L2  {ww }
Union grammar:
S  S1 | S2
n n
R
L  {a b }  {ww }
3
In general:
Take any two context-free grammars G1,G2
with start variables S1, S2
The union grammar G
has start variable S  S1 | S2
and
L(G)  L(G1)  L(G2 )
is context-free
4
Context-free languages
are closed under:
Concatenation
L1 is context free
L1L2
L2 is context free
is context-free
5
Concatenation Example:
S1  aS1b | 
S2  aS2a | bS2b | 
n n
L1  {a b }
R
L2  {ww }
Concatenation grammar:
S  S1S2
n n
R
L  {a b }{ww }
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In general:
Take any two context-free grammars G1,G2
with start variables S1, S2
The concatenation grammar G
has start variable S  S1 | S2
and
L(G)  L(G1) L(G2 )
is context-free
7
Context-free languages
are closed under:
L is context free
Star-operation
* is context-free
L
8
Star operation Example:
S  aSb | 
n n
L  {a b }
Star grammar:
S1  SS1 | 
n n
L  {a b } *
9
In general:
Take any context-free grammars G
with start variable S
The concatenation grammar G1
has start variable S1  SS1 | 
and
L(G1)  L(G) *
is context-free
10
Negative Properties
of
Context-Free Languages
11
Context-free languages
intersection
are not closed under:
L1 is context free
L2 is context free
L1  L2
not necessarily
context-free
12
Counter Example:
n n m
n m m
L1  {a b c }
L2  {a b c }
Context-free:
Context-free:
S  AC
A  aAb | 
C  cC | 
n n n
L1  L2  {a b c }
S  AB
A  aA | 
B  bBb | 
NOT context-free
13
Context-free languages
complement
are not closed under:
L is context free
L
not necessarily
context-free
14
Counter Example:
n n m
n m m
L1  {a b c }
L2  {a b c }
Context-free:
Context-free:
S  AC
A  aAb | 
C  cC | 
S  AB
A  aA | 
B  bBb | 
n n n
L1  L2  L1  L2  {a b c }
NOT context-free
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Intersection
of
Context-free languages
and
Regular Languages
16
The intersection of
a context-free language and
a regular language
is a context-free language
L1 context free
L1  L2
L2 regular
context-free
17
Machine
M1
NPDA for L1
context-free
Machine M 2
DFA for
L2
regular
Construct a new NPDA machine M
that accepts L1  L2
M
simulates in parallel M1 and M 2
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NPDA M1
q1
a, b  c
DFA M 2
q2
a
p1
transition
p2
transition
NPDA M
a
,
b

c
q1, p1
q2 , p2
transition
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NPDA M1
DFA M 2
q0
p0
initial state
initial state
NPDA M
q0 , p0
Initial state
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NPDA M1
DFA M 2
p1
q1
final state
p2
final states
NPDA M
q1, p1
q1, p2
final states
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M
simulates in parallel M1 and M 2
M accepts string w if and only if
M1 accepts string w
and
M 2 accepts string w
L( M )  L( M1)  L( M 2 )
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Therefore:
L( M1)  L( M 2 ) is context-free
(since M is NPDA)
L1  L2 is context-free
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Example Applications of Regular Closure
Prove that:
n n
L  {a b : n  100}
is context-free
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We know:
n n
{a b }
is context-free
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We also know:
100 100
L1  {a
b
*
is regular
}
100 100
L1  {(a  b) }  {a
b
}
is regular
26
n n
*
100 100
L1  {(a  b) }  {a
{a b }
b
is regular
is context-free
n n
{a b }  L1
is context-free
n n
L  {a b : n  100}
is context-free
n n
 {a b }  L1
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}
Another application
Prove that:
L  {w : na  nb  nc }
is not context-free
28
If
Then
L  {w : na  nb  nc } is context-free
n n n
L  {a * b * c*}  {a b c }
context-free
regular
Is context-free
Impossible!!!
Therefore, L is not context free
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Decidable Properties
of
Context-Free Languages
30
Membership Question:
for grammar G
find if string w L(G)
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Membership Question:
for grammar G
find if string w L(G)
Membership Algorithms:
Parsers
Exhaustive search parser
CYK parsing algorithm
32
Empty Language Question:
for grammar G
find if L(G)  
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Empty Language Question:
for grammar G
find if L(G)  
Algorithm:
Remove useless variables
Check if start variable S is useless
34
Infinite Language Question:
for grammar G
find if L(G) is infinite
35
Infinite Language Question:
for grammar G
find if L(G) is infinite
Algorithm:
• Remove useless variables
• Remove unit and lambda productions
• Create the dependency graph for variables
• If there is a loop in the dependency graph
the language is infinite
36
Example:
S  AB
A  aCb | a
B  bB | bb
C  cBS
Dependency graph
infinite
A
C
S
B
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S  AB
A  aCb | a
B  bB | bb
C  cBS
S  AB  aCbB  acBSbB  acbbSbbb


2
S  acbbSbbb (acbb) S (bbb)

i
(acbb) S (bbb)
2
i
38