Transcript Closure Properties of CFL`s

6.2 Closure Properties of CFL's
Theorem 1 Context-free languages are closed under union,
concatenation and Kleene closure.
Proof Let L1= L(G1) and L2 = L(G2) be CFL’s, where
G1 = (V1, T1, P1, S1) and G2 = (V2, T2, P2, S2)
1. For union, L1L2, Construct G = (V, T, P, S),
where
P = P1P2 {SS1, SS2},
V = V1 V2 {S}
2. For concatenation, L1．L2, Construct
G = (V, T, P, S), where
P = P1P2 {SS1S2},
V = V1 V2 {S}
3. For Kleene closure, L1*, Construct
G = (V, T, P, S), where
P = P1{SS1S, S},
V = V1 {S}
Theorem 2 Context-free languages are closed under substitution.
Proof Let L be a CFL, L  *, and for each a in  let La be
a CFL.
Let L = L(G), G = (V, T, P, S), and for each a in  let La = L(Ga),
where Ga=(Va, Ta, Pa, Sa). Assume that VVa = , for all a in .
And Va Vb = , for all a≠ b in .
Construct G’=(V’, T’, P’, S), where
V’ = a in  Va  V,
T’ = a in  Ta ,
P’ = a in  Pa {Af() | A  is in P, and f(a) = Sa for each
a in }
Example 6
L = {0n1m | n ≠m } is a CFL.
Let L = L(G), G = ({S, A, B}, {0, 1}, P, S), where P
contains the following production rules:
S  0S1 | 0A | 1B, A  0A | , B  1B | .
Let L0 = L(G0), G = ({S0}, { (, ), [, ]}, P, S0), where P
contains the following production rules:
S0  S0S0 | [S0] | (S0) | [] | ()
L0 can be defined recursively as follows:
1. () and [] are in L0.
2. If R and S are in L0, then (R), [R], and RS are in L0.
And L0 contains only the strings described in 1 and 2.
Let L1 = L(G1), G = ({S1, C, D}, {c, d}, P, S1), where P
contains the following production rules:
S1  dC | cD
C  dCC | cS | c
D  cDD | dS | d
Let L1 consists of strings over {c, d} with the property that the
number of c’s is equal to the number of d’s in the strings.
If f is the substitution f(0) = L0 and f(1) = L1, then f(L) is
generated by the grammar
S  S0SS1 | S0A | S1B, A  S0A | , B  S1B | .
S0  S0S0 | [S0] | (S0) | [] | ()
S1  dC | cD
C  dCC | cS | c
D  cDD | dS | d
Corollary Context-free languages are closed under homomorphism.
Proof homomorphism is a special case of substitution.
Theorem 3 Context-free languages are closed under inverse
homomorphism.
Proof Let h : Δ* be a homomorphism and L be a CFL..
Let L = L(M), PDA M = (Q, Δ, Γ, , q0, Z0, F).
An idea of constructing a PDA M’ to accept h-1(L) is as follows:
What M’ does is the following:
On reading an input symbol a, M’ generates the string h(a) and
stores h(a) in a buffer, then simulates M on h(a). Only when the
buffer is empty, M’ is allowed to read in next input. And the
buffer holds only a suffix of h(a) for some input a at all time.
M’ accepts an input w if the buffer is empty and M enters a
final state.
Construct a PDA M’ = (Q’, , Γ, ’, [q0, ], Z0, F{}) to accept
h-1(L) as follows:
Q’ = { [q, x] | q Q and x is a suffix of some h(a) for a  .}
’ is defined as follows:
1. ’ ([q, x], , Y) contains all ([p, x], γ) if (q, , Y) contains
(p, γ). The move M’ simulates -move of M ignoring the
buffer.
2. ’ ([q, ax], , Y) contains all ([p, x], γ) if (q, a, Y) contains
(p, γ). M’ simulates moves of M on reading a Δ, then
removes a from the front of the buffer.
3. ’ ([q, ], a, Y) contains all ([q, h(a)], γ) for all a   and
Y  Γ. When the buffer is empty, M’ reads an input a and
loads the buffer with h(a).
(q, h(a), )
*(p, , ) 
([q, ,], a, )  (q, h(a), ) * ([p, ], , ).
M
M'
M'
If M accepts h(w), i.e., (q0, h(w), Z0)
for some p in F and  in Γ*, then
([q0,  ], w, Z0)
*(p, , )
M
*([p, ], , ). So M’ accepts w.
M'
We have that h-1(L(M))  L(M’)
If M’ accepts w = a1a2 … an.
([q0, ], a1a2 … an, Z0) *([p1, ], a2 … an, 1),
M'
*([p1, h(a1)], a2a3 … an, 1),
M'
*([p2, ], a2a3 … an, 2),
M'
*([p2, h(a2)], a3 … an, 2),
M'
.
.
.
*([pn, ], , n+1),
M'
where pn is a final state.
For all i, we have
(pi, h(ai), i)
 (pi+1, , i+1).
M
So, for these moves altogether we have
(q0, h(a1a2 …an), Z0)  (pn, , n+1).
M
Hence, h(a1a2…an) is in L(M). And L(M’)  h-1(L(M)}.
End of Theorem 3.
Theorem 4 Context-free languages are not closed under
intersection.
Proof L1 = {0n1n0m | n, m = 0, 1, 2, …} and
L2 = {0n1m0m | n, m = 0, 1, 2, …} are CFL’s.
L1L2 = {0n1n0n | n = 0, 1, 2, …} is not CFL.
Corollary Context-free languages are not closed under
complementation.
L1  L2 = (L1cL2c)c , where c stands for complement.
If CFL’s are closed under complement, then CFL’s are
closed under intersection.
Theorem 5 The intersection of a context-free language and a
regular language is context-free.
Proof Let L be a CFL, and let R be a regular language.
L = L(ML) for PDA ML = (QL, , Γ, L, q0, Z0, FL), and
R = L(MR) for DFA MR = (QR, , R, p0, FR).
Construct a PDA M’ for L  R by simulating ML and MR in
parallel.
M = (QLQR, , Γ, , [p0, q0], Z0, FLFR), where
 ([p, q], a, X) contains ([p’, q’], ) if and only if
R(p, a) = p’, L(q, a, X) contains (q’, ).