2. Introduced by Alan Turing in 1936.
A simple mathematical model of a
computer.
Models the computing capability of a
computer.
INTRODUCING TURING MACHINES
3. DEFINATION
A Turing machine (TM) is a finite-state machine with
an infinite tape and a tape head that can read or
write one tape cell and move left or right.
It normally accepts the input string, or completes its
computation, by entering a final or accepting state.
Tape is use for input and working storage.
4. Turing Machine is represented by-
M=(Q,, Γ,δ,q0,B,F) ,
Where
Q is the finite state of states
a set of τ not including B, is the set of input symbols,
τ is the finite state of allowable tape symbols,
δ is the next move function, a mapping from Q × Γ to
Q × Γ ×{L,R}
Q0 in Q is the start state,
B a symbol of Γ is the blank,
F is the set of final states.
Representation of Turing Machine
5. THE TURING MACHINE MODEL
X1 X2 … Xi … Xn
B B …
Finite Control
R/W Head
B
Tape divided into
cells and of infinite
length
Input & Output Tape Symbols
6. TRANSITION FUNCTION
One move (denoted by |---) in a TM does the
following:
δ(q , X) = (p ,Y ,R/L)
q is the current state
X is the current tape symbol pointed by tape head
State changes from q to p
7. TURING MACHINE AS LANGUAGE ACCEPTORS
A Turing machine halts when it no longer has available
moves.
If it halts in a final state, it accepts its input, otherwise it
rejects its input.
For language accepted by M ,we define
L(M)={ w ε ∑+ : q0w |– x1qfx2 for some qf ε F , x1 ,x2ε Γ *}
8. TURING MACHINE AS TRANSDUCERS
To use a Turing machine as a transducer, treat the
entire nonblank portion of the initial tape as input
Treat the entire nonblank portion of the tape when
the machine halts as output.
A Turing machine defines a function y = f (x) for
strings x, y ε ∑* if
q0x |*– qf y
A function index is “Turing computable” if there
exists a Turing machine that can perform the above
task.
9. ID OF A TM
Instantaneous Description or ID :
X1 X2…Xi-1 q Xi Xi+1 …Xn
Means:
q is the current state
Tape head is pointing to Xi
X1X2…Xi-1XiXi+1… Xn are the current tape symbols
δ (q , Xi ) = (p ,Y , R ) is same as:
X1 X2…Xi-1 q Xi Xi+1 …Xn |---- X1 X2…Xi-1 Y p Xi+1…Xn
δ (q Xi) = (p Y L) same as:
X1 X2…Xi-1 q Xi Xi+1 …Xn |---- X1 X2…pXi-1Y Xi+1 …Xn
10. TECHNIQUES FOR TM CONSTRUCTION
Storage in the finite control
Using multiple tracks
Using Check off symbols
Shifting over
Implementing Subroutine
12. MULTITAPE TURING MACHINES
A Turing Machine with several tapes
Every Tape’s have their Controlled own R/W Head
For N- tape TM M=(Q,, Γ,δ,q0,B,F)
we define δ : Q X ΓN Q X ΓN X { L , R} N
13. For e.g., if n=2 , with the current
configuration
δ( qO ,a ,e) =(q1, x ,y, L, R)
qO
a b c d e f
Tape 1 Tape 2
q1
d y f
Tape 1 Tape 2
x b c
14. SIMULATION
Standard TM simulated by Multitape TM.
Multitape TM simulated by Standard TM
q
a b c d e f
Tape 1 Tape 2
a b C
1 B B
d e f
B 1 B
q
15. NON DETERMINISTIC TURING MACHINES
It is similar to DTM except that for any input symbol and current
state it has a number of choices
A string is accepted by a NDTM if there is a sequence of moves
that leads to a final state
The transaction function δ : Q X Γ 2 Q X Γ X { L , R}
16. Simulation:
A DTM simulated by NDTM
In straight forward way .
A NDTM simulated by DTM
A NDTM can be seen as one that has the ability to replicate
whenever is necessary
17. MULTIHEAD TURING MACHINE
Multihead TM has a number of heads instead of one.
Each head indepently read/ write symbols and move left / right or
keep stationery.
a b c d e f g t
Control unit
18. SIMULATION
Standard TM simulated by Multihead TM.
- Making on head active and ignore remaining head
Multihead TM simulated by standard TM.
- For k heads Using (k+1) tracks if there is..
19. .. . a b c d e f g h ….
Control Unit
…. 1 B B B B B B B ..
…. B B 1 B B B B B ..
.. B B B B 1 B B B ..
.. B B B B B B 1 B .
.. a b c d e f g h .
Head 1 Head 2 Head 3 Head 4
Multihead
TM
Multi track
TM
1st track
2nd track
3rd track
4th track
5th track
20. OFF- LINE TURING MACHINE
An Offline Turing Machine has two tapes
1. One tape is read-only and contains the input
2. The other is read-write and is initially blank.
a b c d
Control
unit
f g h i
Read- Only input
file’s tape
W/R tape
21. SIMULATION
A Standard TM simulated by Off-line TM
An Off- line TM simulated by Standard TM
a b c d
B B 1 B
f g h i
B 1 B B
Control
Unit M’
a b c d
Control
Unit M
f g h i
Read- Only input
W/R tape
22. MULTIDIMENSIONAL TURING MACHINE
A Multidimensional TM has a multidimensional tape.
For example, a two-dimensional Turing machine would read
and write on an infinite plane divided into squares, like a
checkerboard.
For a two- Dimensional Turing Machine transaction function
define as:
δ : Q X Γ Q X Γ X { L , R,U,D}
24. SIMULATION
Standard TM simulated by Multidimensional TM
Multidimensional TM simulated by Standard TM.
25. 1,-1 1,1 1,2
-1,1 -1,2
Control Unit
2-Dimensional address
shame
.. a b ….
.. 1 # 1 # 1 # 2 # …
…
Control Unit
26. TURING MACHINE WITH SEMI- INFINITE TAPE
A Turing machine may have a “semi-infinite tape”, the nonblank
input is at the extreme left end of the tape.
Turing machines with semi-infinite tape are equivalent to
Standard Turing machines.
27. SIMULATION
Semi – infinite tape simulated by two way infinite tape
$ a b c
Control Unit
28. Two way infinite tape simulated by semi -infinite tape
a b c d e f g h
$ d c b a
e f g h
Control Unit
29. TURING MACHINE WITH STATIONARY HEAD
Here TM head has one another choice of movement is
stay option , S.
we define new transaction function,
δ : Q X Γ Q X Γ X { L , R, S}
30. SIMULATION
TM with stay option can simulate a TM without stay option
by not using the stay option.
TM with stay option can simulate by a TM without stay
option by not using the stay option.
In TM with stay option: δ(q, X)= ( p , Y, S )
In TM without stay option : δ’(q, X)= ( qr , Y, R )
δ’( qr, A)= ( p , A, L ) ¥ AεΓ’
31. RECURSIVE AND RECURSIVELY ENUMERABLE
LANGUAGE
The Turing machine may
1. Halt and accept the input
2. Halt and reject the input, or
3. Never halt /loop.
Recursively Enumerable Language:
There is a TM for a language which accept every string
otherwise not..
Recursive Language:
There is a TM for a language which halt on every
string.
32. UNIVERSAL LANGUAGE AND TURING MACHINE
The universal language Lu is the set of binary strings
that encode a pair (M , w) where w is accepted by M
A Universal Turing machine (UTM) is a Turing machine
that can simulate an arbitrary Turing machine on
arbitrary input.
33. PROPERTIES OF TURING MACHINES
A Turing machine can recognize a language iff it
can be generated by a phrase-structure grammar.
The Church-Turing Thesis: A function can be
computed by an algorithm iff it can be computed by
a Turing machine.