Closure Properties for CFL

1
Lecture 32

• Closure Properties for CFLs
• Kleene Closure
• construction
• examples
• proof of correctness
• Others covered less thoroughly in lecture
• union, concatenation
• CFLs versus regular languages
• regular languages subset of CFL

2
Closure Properties for CFLs

• Kleene Closure

3
CFL closed under Kleene Closure

• Let L be an arbitrary CFL
• Let G1 be a CFG s.t. L(G1) L
• G1 exists by definition of L1 in CFL
• Construct CFG G2 from CFG G1
• Argue L(G2) L
• There exists CFG G2 s.t. L(G2) L
• L is a CFL

4
Visualization

• Let L be an arbitrary CFL
• Let G1 be a CFG s.t. L(G1) L
• G1 exists by definition of L1 in CFL
• Construct CFG G2 from CFG G1
• Argue L(G2) L
• There exists CFG G2 s.t. L(G2) L
• L is a CFL

CFL
5
Algorithm Specification

• Input
• CFG G1
• Output
• CFG G2 such that L(G2) (L(G1))

CFG G1
CFG G2
6
Construction

• Input
• CFG G1 (V1, S, S1, P1)
• Output
• CFG G2 (V2, S, S2, P2)
• V2 V1 union T
• T is a new symbol not in V1 or S
• S2 T
• P2 P1 union T --gt TS1 l

7
Closure Properties for CFLs

• Kleene Closure Examples

8
Example 1
V2 V1 union T T is a new symbol not in
V1 or SS2 TP2 P1 union T --gt TS l

• Input grammar
• V S
• S a,b
• S S
• P
• S --gt aa ab ba bb

• Output grammar
• V S, T
• S a,b
• Start symbol is T
• P
• T --gt TS l
• S --gt aa ab ba bb

9
Example 2
V2 V1 union T T is a new symbol not in
V1 or SS2 TP2 P1 union T --gt TS l

• Input grammar
• V S, T
• S a,b
• Start symbol is T
• P
• T --gt TS l
• S --gt aa ab ba bb

• Output grammar
• V S, T, U
• S a,b
• Start symbol is U
• P
• U --gt UT l
• T --gt TS l
• S --gt aa ab ba bb

10
Closure Properties for CFLs

• Kleene Closure Proof of Correctness

11
Is our construction correct?

• How do we prove our construction is correct?
• Informal
• Test some strings
• Review logic behind construction
• Formal
• First, show every string derived by G2 belongs to
  (L(G1))
• That is, show L(G2) is a subset of (L(G1))
• Second, show every string in (L(G1)) can be
  derived by G2
• That is, show (L(G1)) is a subset of L(G2)
• Both proofs will be inductive proofs
• Inductive proofs and recursive algorithms go well
  together

12
L(G2) is a subset of (L(G1))

• We want to prove the following
• If x in L(G2), then x is in (L(G1))
• This is equivalent to the following
• If T gtG2 x, then x is in (L(G1))
• The two statements are equivalent because
• x in L(G2) means that T gtG2 x
• We break the second statement down as follows
• If T gt1G2 x, then x is in (L(G1))
• If T gt2G2 x, then x is in (L(G1))
• If T gt3G2 x, then x is in (L(G1))
• ...

13
L(G2) is a subset of (L(G1))

• Statement to be proven
• For all n gt 1, if T gtnG2 x, then x is in
  (L(G1))
• Prove this by induction on n
• Base Case
• n 1
• The only string x such that T gt1G2 x is the
  string l
• Follows from inspection of G2
• The string l belongs to (L(G1))
• Follows from definition of Kleene Closure

14
Inductive Case

• Inductive Hypothesis
• For 1 lt j lt n, if T gtjG2 x, then x is in
  (L(G1))
• Note, this is a strong induction hypothesis
• Statement to be Proven in Inductive Case
• For n above, if T gtn1G2 x, then x is in
  (L(G1))
• Proving this statement
• Let x be an arbitrary string such that T gtn1G2
  x
• There are two possible first derivation steps
• Case 1 T gtG2 l gtnG2 x
• Case 2 T gtG2 TS gtnG2 x
• These 2 cases follow from looking at grammar G2

15
Case Analysis

• Case 1 T gtG2 l is not possible
• n gt 1 which means n1 gt 2
• Case 2 T gtG2 TS gtnG2 x
• This means x has the form uv where
• T gtltn u
• S gtltn v
• This follows because in n steps, we go from TS to
  x
• Thus, T can take at most n-1 steps to generate u
• Likewise for S generating v
• String u belongs to (L(G1))
• Follows from the inductive hypothesis

16
Concluding Case 2 T gtG2 TS gtnG2 x

• String v belongs to L(G1)
• Follows from S gt G2 v and
• Our construction insures that all strings derived
  from S in L(G2) are also in L(G1)
• Conclude x belongs to (L(G1))
• x uv where u belongs to (L(G1)) and v belongs
  to L(G1)
• By definition of Kleene closure, this means xuv
  belongs to (L(G1))
• Wrapping up inductive case
• In all possible derivations of x, we have shown
  that x belongs to (L(G1))
• Thus, we have proven the inductive case
• Conclusion
• By the principle of mathematical induction, we
  have shown that L(G2) is a subset of (L(G1))

17
(L(G1)) is a subset of L(G2)

• We want to prove the following
• If x is in (L(G1)), then x is in L(G2)
• This is equivalent to the following
• If x is in (L(G1)), then T gtG2 x
• The two statements are equivalent because
• x in L(G2) means that T gtG2 x
• We break the second statement down as follows
• If x is in (L(G1))0, then T gtG2 x
• If x is in (L(G1))1, then T gtG2 x
• If x is in (L(G1))2, then T gtG2 x
• ...

18
(L(G1)) is a subset of L(G2)

• Statement to be proven
• For all n gt 0, if x is in (L(G1))n, then x is in
  L(G2)
• Prove this by induction on n
• Base Case
• n 0
• The only string x such that x is in (L(G1))0 is
  the string l
• Follows from definition of (L(G1))0
• The string l belongs to L(G2)
• Follows from production T --gt l

19
Inductive Case

• Inductive Hypothesis
• For 0 lt j lt n, if x is in (L(G1))j, then T
  gtG2 x
• Note, this is a strong induction hypothesis
• Statement to be Proven in Inductive Case
• For n above, if x is in (L(G1))n1, then T gtG2
  x
• Proving this statement
• Let x be an arbitrary string in (L(G1))n1
• This means x uv where
• u is a string in (L(G1))n
• v is a string in L(G1)
• This follows from definition of (L(G1))n1 and
  that ngt0

20
Deriving x

• x uv where
• u is a string in (L(G1))n
• v is a string in L(G1)
• T gt G2 TS gt G2 uS gt G2 uv x
• The first derivation step is possible from our
  grammar
• The second sequence of derivation steps follows
  from T gt G2 u
• This follows from applying the induction
  hypothesis
• Third sequence of derivation steps follows from S
  gt G2 v
• This follows from the fact that v is in L(G1) and
  
• Our construction includes all productions from G1
• Thus T gt G2 x
• The inductive case follows
• The result is proven by the principle of
  mathematical induction

21
Construction for Set Union

• Input
• CFG G1 (V1, S, S1, P1)
• CFG G2 (V2, S, S2, P2)
• Output
• CFG G3 (V3, S, S3, P3)
• V3 V1 union V2 union T
• Variable renaming to insure no names shared
  between V1 and V2
• T is a new symbol not in V1 or V2 or S
• S3 T
• P3 P1 union P2 union T --gt S1 S2

22
Construction for Set Concatenation

• Input
• CFG G1 (V1, S, S1, P1)
• CFG G2 (V2, S, S2, P2)
• Output
• CFG G3 (V3, S, S3, P3)
• V3 V1 union V2 union T
• Variable renaming to insure no names shared
  between V1 and V2
• T is a new symbol not in V1 or V2 or S
• S3 T
• P3 P1 union P2 union T --gt S1S2

23
CFLs and regular languages
24
CFL Closure Properties

• CFLs are closed under Kleene closure
• Just proven, proof also in book
• CFLs are closed under set union
• Proof in book
• CFLs are closed under set concatenation
• Proof in book
• What can we conclude from these 3 results?
• It follows that regular languages are a subset of
  CFLs

25
Regular languages subset of CFL

• Recursive definition of regular languages
• Base Case
• , l, a, b are regular languages over
  a,b
• P, PS --gt l, PS --gt a, PS --gt b
• Inductive Case
• If L and L are are regular languages, then L,
  LL, L union L are regular languages
• Use previous constructions to see that these
  resulting languages are also context-free

26
Other CFL Closure Properties

• We will show that CFLs are NOT closed under many
  other set operations
• Examples include
• set complement
• set intersection
• set difference

27
Language class hierarchy
REG