1 Turing Machines. 2 The Language Hierarchy Regular Languages Context-Free Languages ? ?
59
1 Turing Machines
-
date post
20-Dec-2015 -
Category
Documents
-
view
219 -
download
0
Transcript of 1 Turing Machines. 2 The Language Hierarchy Regular Languages Context-Free Languages ? ?
1
Turing Machines
2
The Language Hierarchy
*aRegular Languages
Context-Free Languagesnnba Rww
nnn cba ww?
**ba
?
3
*aRegular Languages
Context-Free Languagesnnba Rww
nnn cba ww
**ba
Languages accepted byTuring Machines
4
A Turing Machine
............Tape
Read-Write headControl Unit
5
The Tape
............
Read-Write head
No boundaries (infinite length)
• The head moves Left or Right
6
............
Read-Write head
The head at each time step:
1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right
7
............
Example:Time 0
............Time 1
1. Reads
2. Writes
a a cb
a b k c
a
k
3. Moves Left
8
............Time 1
a b k c
............Time 2
a k cf
1. Reads
2. Writes
b
f
3. Moves Right
9
The Input String
............
Blank symbol
head
a b ca
10
The States
1q 2qLba ,
read write move Left
1q 2qRba ,
move Right
11
Example:
1q 2qRba ,
............ a b ca
Time 1
1qcurrent state
12
............ a b caTime 1
1q 2qRba ,
............ a b cbTime 2
1q
2q
13
............ a b caTime 1
1q 2qLba ,
............ a b cbTime 2
1q
2q
Example:
14
............ a b caTime 1
1q 2qRg,
............ ga b cbTime 2
1q
2q
Example:
15
Determinism
1q
2qRba ,
OK NOT ALLOWED
3qLdb ,
1q
2qRba ,
3qLda ,
•For a read symbol only one transition is allowed
•No lambda transitions allowed
16
Partial Transition Function
1q
2qRba ,
3qLdb ,
............ a b ca
1q
Example:
No transitionfor input symbol c
OK
17
Halting
The machine halts if there are no possible transitionsto take
18
Example:
............ a b ca
1q
1q
2qRba ,
3qLdb ,
No possible transition
HALT!!!
19
Final States
1q 2q OK
1q 2q NOT ALLOWED
• Final states have no outgoing transitions
• In a final state the machine halts
20
Acceptance
Accept Input If machine halts in a final state
Reject Input • If machine halts in a non-final state
• If machine enters an infinite loop
21
Turing Machine Example
A turing machine for the language *aa
0q
Raa ,
L,1q
22
0q
Raa ,
L,1q
aaTime 0
0q
a
230q
Raa ,
L,1q
aaTime 0
0q
a
aaTime 1
0q
a
240q
Raa ,
L,1q
aaTime 1
0q
a
aaTime 2
0q
a
250q
Raa ,
L,1q
aaTime 2
0q
a
aaTime 3
0q
a
260q
Raa ,
L,1q
aaTime 3
0q
a
aaTime 4
1q
a
HALT & accept
27
Rejection Example
0q
Raa ,
L,1q
baTime 0
0q
a
280q
Raa ,
L,1q
baTime 0
0q
a
baTime 1
0q
a
HALT & reject
29
Infinite Loop Example
0q
Raa ,
L,1q
Lbb ,
Turing machine for the language *aa
30
0q
Raa ,
L,1q
baTime 0
0q
a
Lbb ,
310q
Raa ,
L,1q
baTime 0
0q
a
baTime 1
0q
a
Lbb ,
320q
Raa ,
L,1q
baTime 1
0q
a
baTime 2
0q
a
Lbb ,
33
baTime 2
0q
a
baTime 3
0q
a
baTime 4
0q
a
baTime 5
0q
a
...Infinite Loop
34
Because of the infinite loopthe final state cannot be reachedand the input is not accepted
35
Another Turing Machine Example
0q
Turing machine for the language }{ nnba
1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
R,
36
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
ba
0q
a b
Time 0
bx
1q
a b
Time 1
37
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
bx
1q
a b
Time 1
bx
1q
a b
Time 2
38
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
bx
1q
a b
Time 2
yx
2q
a b
Time 3
39
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
2q
a b
Time 3
yx
2q
a b
Time 4
40
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
2q
a b
Time 4
yx
0q
a b
Time 5
41
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
0q
a b
Time 5
yx
1q
x b
Time 6
42
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
1q
x b
Time 6
yx
1q
x b
Time 7
43
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
1q
x b
Time 7
yx
2q
x yTime 8
44
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
2q
x yTime 8
yx
2q
x yTime 9
45
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
0q
x yTime 9
yx
2q
x yTime 10
46
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
0q
x yTime 10
yx
3q
x yTime 11
47
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
3q
x yTime 11
yx
3q
x yTime 12
48
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
3q
x yTime 12
yx
4q
x yTime 13
HALT & accept
49
By modifying the machine for the language }{ nnba
we can easily construct a machine for language }{ nnn cba
50
Formal Definitionsfor
Turing Machines
51
Transition Function:
1q 2qRba ,
),,(),( 21 Rbqaq
1q 2qLdc ,
),,(),( 21 Ldqcq
52
Turing Machine:
),,,,,,( 0 FqQM
states
Inputalphabet
Tapealphabet
Transitionfunction
Initialstate
blank
Finalstates
53
Configuration
ba
1q
a
Instantaneous description:
c
baqca 1
54
yx
2q
a b
Time 4
yx
0q
a b
Time 5
A move: aybqxxaybq 02
55
yx
2q
a b
Time 4
yx
0q
a b
Time 5
bqxxyybqxxaybqxxaybq 1102
yx
1q
x b
Time 6
yx
1q
x b
Time 7
56
bqxxyybqxxaybqxxaybq 1102
bqxxyxaybq 12
Equivalent notation:
57
Initial configuration: wq0
ba
0q
a b
w
Input string
58
The Accepted Language
For any Turing Machine M
}:{)( 210 xqxwqwML f
Initial state Final state
59
Standard Turing Machine
• Deterministic
• Infinite tape in both directions
•Tape is the input/output file
The machine we described is the standard:
