Closure properties of recursive and r.e. languages
Theorem 1
The class of recursive languages is closed un
der
1. complementation
2. union
3. concatenation
4. Kleene star (proof similar to (3))
5. intersection, (proof similar to (2))
Note: the set of contextfree languages is not closed under com
plementation, but the set of recursive languages is closed under
complementation.
L
is CF
⇒
L
is recursive
⇒
L
0
is recursive, but
L
0
may or may
not be CF.
1
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If
L
is recursive, then
L
0
is recursive.
Proof
:
•
Let
M
= (
K,
Σ
, δ, s,
{
y, n
}
) be the TM that decides
L
.
•
We can construct a TM
M
0
that decides
L
0
by reversing
the roles of states
y
and
n
in
M
. That is:
M
0
= (
K,
Σ
, δ
0
, s,
{
y, n
}
)
δ
0
(
q, σ
) =
(
n, γ
)
if
δ
(
q, σ
) = (
y, γ
)
(
y, γ
)
if
δ
(
q, σ
) = (
n, γ
)
(
p, γ
)
if
δ
(
q, σ
) = (
p, γ
) and
p
6
=
y, p
6
=
n
Question
:
Is the complement of a recursively enumerable language always
recursively enumerable? No (example later).
2
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 Spring '10
 Prof.Tai
 Formal language, recursive languages

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