PRESENTATION ON TURING MACHINE
PREPARED BY:
DEEPJYOTI KALITA
CS-16 (3RD SEM)
MSC COMPUTER SCIENCE
Introduced by Alan Turing in 1936.
A simple mathematical model of a computer.
Models the computing capability of a computer.
INTRODUCING TURING MACHINES
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.
Turing Machine is represented by- M=(Q,, Γ,δ,q0,B,F) ,
WhereQ 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
THE TURING MACHINE MODEL
X1 X2 … Xi …Xn
B B …
Finite ControlR/W Head
B
Tape divided into cells and of
infinite length
Input & Output Tape Symbols
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
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 *}
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.
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
TECHNIQUES FOR TM CONSTRUCTION
Storage in the finite control
Using multiple tracks
Using Check off symbols
Shifting over
Implementing Subroutine
VARIATIONS OF TURING MACHINES
Multitape Turing Machines
Non deterministic Turing machines
Multihead Turing Machines
Off-line Turing machines
Multidimensional Turing machines
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
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
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 C1 B Bd e fB 1 B
q
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}
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
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
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..
.. .
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
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
Controlunit
f g
h
i
Read- Only input file’s tape
W/R tape
SIMULATION
A Standard TM simulated by Off-line TM
An Off- line TM simulated by Standard TM
a b c d B B 1 Bf g h iB 1 B B
ControlUnit M’
a b
c d
ControlUnit M
f g
h
i
Read- Only input
W/R tape
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}
1,-1
1,1
1,2
-1,1
-1,2
Control Unit
2-Dimensional address shame
SIMULATION
Standard TM simulated by Multidimensional TM
Multidimensional TM simulated by Standard TM.
1,-1
1,1
1,2
-1,1
-1,2
Control Unit
2-Dimensional address shame
.. a b ….
.. 1 # 1 # 1 # 2 # ……
Control Unit
TURING MACHINE WITH SEMI- INFINITE TAPE
A Turing machine may have a “semi-infinite tape”, the nonblankinput is at the extreme left end of the tape.
Turing machines with semi-infinite tape are equivalent toStandard Turing machines.
SIMULATION
Semi – infinite tape simulated by two way infinite tape
$ a b c
Control Unit
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
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}
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’
RECURSIVE AND RECURSIVELY ENUMERABLE LANGUAGE
The Turing machine may1. Halt and accept the input2. Halt and reject the input, or3. 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.
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.
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.
THANKS
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