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### Turing Machine

### Formal Definitionsfor Turing Machines

### Computing FunctionswithTuring Machines

CSE-501

Formal Language & Automata Theory

Aug-Dec,2010

ALAK ROY.

Assistant Professor

Dept. of CSE

NIT Agartala

The Tape

No boundaries -- infinite length

......

......

Read-Write head

The head moves Left or Right

......

Read-Write head

The head at each time step:

1. Reads a symbol

2. Writes a symbol

3. Moves Left or Right

The Input String

Input string

Blank symbol

......

......

head

Head starts at the leftmost position

of the input string

Final States

Allowed

Not Allowed

- Final states have no outgoing transitions
- In a final state the machine halts

Acceptance

If machine halts

in a final state

Accept Input

If machine halts

in a non-final state

or

If machine enters

an infinite loop

Reject Input

Turing Machine Example

A Turing machine that accepts the language:

Halt & Accept

Time 0

- Because of the infinite loop:
- The final state cannot be reached
- The machine never halts
- The input is not accepted

Input

alphabet

Tape

alphabet

States

Transition

function

Final

states

Initial

state

blank

Configuration

Instantaneous description:

Input string

Standard Turing Machine

The machine we described is the standard:

- Deterministic
- Infinite tape in both directions
- Tape is the input/output file

We prefer unary representation:

easier to manipulate with Turing machines

Integer Domain

Decimal:

5

Binary:

101

Unary:

11111

A function is computable if

there is a Turing Machine such that:

Initial configuration

Final configuration

initial state

final state

For all

Domain

A function is computable if

there is a Turing Machine such that:

Initial

Configuration

Final

Configuration

For all

Domain

Example

is computable

The function

are integers

Turing Machine:

Input string:

unary

Output string:

unary

HALT & accept

Another Example – SELF STUDY

is computable

The function

is integer

Turing Machine:

Input string:

unary

Output string:

unary

- Replace every 1 with $

- Repeat:

- Find rightmost $, replace it with 1
- Go to right end, insert 1

Until no more $ remain

Any computation carried out

by mechanical means

can be performed by a Turing Machine

(1930)

A computation is mechanical

if and only if

it can be performed by a Turing Machine

There is no known model of computation

more powerful than Turing Machines

Algorithms are Turing Machines

When we say:

There exists an algorithm

We mean:

There exists a Turing Machine

that executes the algorithm

Construct a Turing Machine:

- Construct a TM to accept the set L of all strings over {0,1} ending with 010.

An Any TM Simulator

Input: < Description of some TM m, w >

Output: result of running m on w

Universal

Turing

Machine

m

Output Tape

for running

TM- m starting on tape w

w

Universal Turing Machine (UTM)

- functional notation of UTM:
U(“m”“w”) = “m(w)”

- takes 2 arguments:
- m : a description of a Turing Machine TM
- w : description of an input string w

- have 2 properties:
- UTM halts on input “m” “w” if & only if TM halts on input “w”.

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