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