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Recursively Enumerable
and
R i L
1
Recursive L
Based on slides by Costas Busch [http://csc.lsu.edu/~busch]
Recursive Languages
Definition:
A recursive language L is a formal language for which there
exists a TM that will halt and accept an input string in L, and
halt and reject, otherwise.
EExample: l
The decision problem of determining whether a natural
number is a perfect square (represented by using the string
an) is decidable.
2
Recursive Languages
Let L be a recursive language and M the Turing Machine
that accepts it
For string :
If then halts in a final state
w ∈ L
w
M
If then halts in a non-final state
Costas Busch - RPI 3
w ∉ L
M
Recursively Enumerable Languages
Definition:
A language is recursively enumerable if some Turing
machine accepts it
Let L be a recursively enumerable language and M the
TTuring Machine that accepts it i M hi th t t it
For string :
If then halts in a final state
If then halts in a non-final state and loops
forever
Costas Busch - RPI 4
w ∈ L
w ∉ L
M
M
w
Non Recursively Enumerable
We will prove:
- There is a specific language which is not recursively
enumerable (not accepted by any Turing Machine)
- There is a specific language which is recursively
enumerable but not recursivebl b t t i
Costas Busch - RPI 5
Type-0 Languages
(Recursively Enum.)
Recursive
Languages
T 1
Technical Note:
Type-1 languages explicitly
exclude the null string.
All other language
families permit the null
string as a member.
(The Venn diagram is
in slight error because
of this technicality.)
Context-Free Languages
6
Type-
(Context-sensitive)
Languages
Type-
(Context-free)
Languages
Type-
(Regular)
Languages
Linear
(Context-free)
Languages
Recursively Enumerable
Non Recursively Enumerable
7
Recursive
A Language which
is not
Recursively Enumerable
Costas Busch - RPI 8
Recursively Enumerable
Non Recursively Enumerable
- We want to find a language that is Non Recursively
Enumerable
- This language is not accepted by any Turing Machine
Costas Busch - RPI 9
Non Recursively Enumerable
Consider alphabet
{ a }
Strings: a , aa , aaa , aaaa ,K
1
a
2
a
3
a
4
a
L
Costas Busch - RPI 10
Consider Turing Machines
that accept languages over alphabet { a }
They are countable:
, , , , K
1 2 3 4
M M M M
Non Recursively Enumerable
Example language accepted by
L ( M ) { aa , aaaa , aaaaaa }
i
2 4 6
L M a a a
i
i
M
Costas Busch - RPI 11
Alternative representation
1
a
2
a
3
a
4
a
5
a
6
a
7
a
i
LM
L
L
Non Recursively Enumerable
1
a
2
a
3
a
4
a
1
LM
L
0 1 0 1 L
Costas Busch - RPI 12
2
LM
3
LM
1 0 0 1 L
0 1 1 1 L
4
L M
L
1
a
2
a
3
a
4
a
1
LM
L
L
2
LM
L ( M )
L
Costas Busch - RPI 19
3
L M
0 1 1 1 L
4
L M
0 0 0 1 L
Question:?
1
M M
k
1
a
2
a
3
a
4
a
1
LM
L
L
2
LM
L ( M )
L
Costas Busch - RPI 20
3
L M
0 1 1 1 L
4
L M
0 0 0 1 L
Answer:
1
M M
k
1
1
1
a L M
a LM
k
1
a
2
a
3
a
4
a
1
LM
L
L
2
LM
L ( M )
1 0 0 1 L
Costas Busch - RPI 21
3
L M
L
4
L M
0 0 0 1 L
Question:
2
M M
k
1
a
2
a
3
a
4
a
1
LM
L
L
2
LM
L ( M )
1 0 0 1 L
Costas Busch - RPI 22
3
L M
L
4
L M
0 0 0 1 L
Answer:
2
M M
k
2
2
2
a L M
a LM
k
1
a
2
a
3
a
4
a
1
LM
L
0 1 0 1 L
2
LM
L ( M )
1 0 0 1 L
Costas Busch - RPI 23
3
L M
0 1 1 1 L
4
L M 0 0 0 1 L
Question:?
3
M M
k
1
a
2
a
3
a
4
a
1
LM
L
0 1 0 1 L
2
LM
L ( M )
1 0 0 1 L
Costas Busch - RPI 24
3
L M
0 1 1 1 L
4
L M 0 0 0 1 L
Answer:
3
M M
k
3
3
3
a L M
a LM
k
Non Recursively Enumerable
Similarly:
k i
M ≠ M for any i
Because either:
Costas Busch - RPI 25
i
i
k
i
a LM
a LM
i
i
k
i
a LM
a LM
or
Non Recursively Enumerable
Therefore, the machine cannot exist. Therefore, the
language is not recursively enumerable
Observation:
There is no algorithm that describes (otherwise
ld b t d b T i M hi )
k
M
L
L
would be accepted by some Turing Machine)
Costas Busch - RPI 26
Recursively Enumerable
Non Recursively Enumerable
L
27
Recursive
A Language which is
Recursively Enumerable
and not Recursive
Costas Busch - RPI 28
and not Recursive
We want to find a language which
Is recursively
enumerable
But not
recursive
Costas Busch - RPI 29
There is a Turing
Machine that accepts
the language
The machine
doesn’t halt
on some input
We will prove that the language
i
i i
L = a a ∈ LM
Is recursively enumerable but not recursive
Costas Busch - RPI 30
Therefore:
L is recursive
But we know:
Costas Busch - RPI 37
L
is not recursively enumerable
thus, not recursive
CONTRADICTION!!!!
Therefore, is not recursive
L
Recursively Enumerable
Non Recursively Enumerable
L
L
38
Recursive
Turing acceptable languages
and
Enumeration Procedures
Costas Busch - RPI 39
Enumeration Procedures
We will prove:
If a language is recursive then there is an
enumeration procedure for it
(weak result)
Costas Busch - RPI 40
A language is recursively enumerable if and only if
there is an enumeration procedure for it
(strong result)
Theorem:
If a language is recursive then there is an
enumeration procedure for it
L
Costas Busch - RPI 41
Proof:
M
M
Enumeration Machine
Costas Busch - RPI 42
M
M
Accepts
L
Enumerates all
strings of input alphabet
If the alphabet is then can enumerate
strings as follows:
{ a , b }
M
a
b
aa
ab
Costas Busch - RPI 43
ba
bb
aaa
aab
Enumeration procedure
Repeat:
M
generates a string
M
w
checks if w ∈ L
Costas Busch - RPI 44
M checks if w ∈ L
YES: print w to output
NO: ignore
w
End of Proof
Example:
M
a
b
aa
L ( M )
b
L ={ b , ab , bb , aaa ,....}
Enumeration
Output
b
Costas Busch - RPI 45
aa
ab
ba
bb
aaa
aab
ab
bb
aaa
ab
bb
aaa
Theorem:
If language is recursively enumerable then
there is an enumeration procedure for it
L
Costas Busch - RPI 46
Proof:
M
M
Enumeration Machine
Costas Busch - RPI 47
M
M
Accepts L
Enumerates all
strings of input alphabet
If the alphabet is then can enumerate
strings as follows:
{ a , b } M
a
b
aa
Costas Busch - RPI 48
ab
ba
bb
aaa
aab
Proof:
w
Input Tape
Machine that
accepts L
Costas Busch - RPI 55
Enumerator
for L
Compare
Turing machine that accepts L
Repeat:
Using the enumerator,
For input string w
Costas Busch - RPI 56
generate the next string of
L
Compare generated string with w
If same, accept and exit loop
End of Proof
We have proven:
A language is recursively enumerable if and only if
there is an enumeration procedure for it
Costas Busch - RPI 57
Technical Note:
Type-1 languages explicitly
exclude the null string.
All other language
families permit the null
string as a member.
(The Venn diagram is
in slight error because
of this technicality.)
Context-Free Languages
Type-0 Languages
(Recursively Enum.)
Recursive
Languages
T 1
Non Recursively Enumerator)
58
Type-
(Regular)
Languages
Type-
(Context-sensitive)
Languages
Type-
(Context-free)
Languages
Linear
(Context-free)
Languages
