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**ALAK ROY. Assistant Professor Dept. of CSE NIT Agartala**

Turing Machine CSE-501 Formal Language & Automata Theory Aug-Dec,2010 ALAK ROY. Assistant Professor Dept. of CSE NIT Agartala

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**The Language Hierarchy**

? ? Context-Free Languages Regular Languages

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Languages accepted by Turing Machines Context-Free Languages Regular Languages

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A Turing Machine Tape ...... ...... Read-Write head Control Unit

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**The Tape No boundaries -- infinite length ...... ......**

Read-Write head The head moves Left or Right

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...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

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Example: Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes 3. Moves Left

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Time 1 ...... ...... Time 2 ...... ...... 1. Reads 2. Writes 3. Moves Right

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**The Input String Input string Blank symbol ...... ...... head**

Head starts at the leftmost position of the input string

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Input string Blank symbol ...... ...... head Remark: the input string is never empty

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States & Transitions Write Read Move Left Move Right

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Example: Time 1 ...... ...... current state

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Time 1 ...... ...... Time 2 ...... ......

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Example: Time 1 ...... ...... Time 2 ...... ......

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Example: Time 1 ...... ...... Time 2 ...... ......

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**Determinism Not Allowed Turing Machines are deterministic Allowed**

No lambda transitions allowed

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**Partial Transition Function**

Example: ...... ...... Allowed: No transition for input symbol

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**Halting The machine halts if there are**

no possible transitions to follow

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Example: ...... ...... No possible transition HALT!!!

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**Final States Not Allowed Allowed**

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

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**Acceptance If machine halts Accept Input in a final state**

in a non-final state or If machine enters an infinite loop Reject Input

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**Turing Machine Example**

A Turing machine that accepts the language:

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Time 0

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Time 1

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Time 2

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Time 3

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Time 4 Halt & Accept

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Rejection Example Time 0

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Time 1 No possible Transition Halt & Reject

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Infinite Loop Example A Turing machine for language

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Time 0

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Time 1

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Time 2

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Time 2 Time 3 Infinite loop Time 4 Time 5

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**Because of the infinite loop:**

The final state cannot be reached The machine never halts The input is not accepted

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**Formal Definitions for Turing Machines**

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Transition Function

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Transition Function

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Turing Machine: Input alphabet Tape alphabet States Transition function Final states Initial state blank

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Configuration Instantaneous description:

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Time 4 Time 5 A Move:

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Time 4 Time 5 Time 6 Time 7

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Equivalent notation:

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**Initial configuration:**

Input string

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The Accepted Language For any Turing Machine Initial state Final state

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**Standard Turing Machine**

The machine we described is the standard: Deterministic Infinite tape in both directions Tape is the input/output file

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**Computing Functions with Turing Machines**

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A function has: Result Region: Domain:

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**A function may have many parameters:**

Example: Addition function

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Integer Domain Decimal: 5 Binary: 101 We prefer unary representation: easier to manipulate with Turing machines Unary: 11111

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Definition: A function is computable if there is a Turing Machine such that: Initial configuration Final configuration initial state final state For all Domain

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In other words: A function is computable if there is a Turing Machine such that: Initial Configuration Final Configuration For all Domain

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**Example The function is computable are integers Turing Machine:**

Input string: unary Output string: unary

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Start initial state The 0 is the delimiter that separates the two numbers

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Start initial state Finish final state

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The 0 helps when we use the result for other operations Finish final state

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**Turing machine for function**

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Execution Example: Time 0 (2) (2) Final Result

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Time 0

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Time 1

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Time 2

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Time 3

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Time 4

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Time 5

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Time 6

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Time 7

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Time 8

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Time 9

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Time 10

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Time 11

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Time 12 HALT & accept

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**Another Example – SELF STUDY**

The function is computable is integer Turing Machine: Input string: unary Output string: unary

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Start initial state Finish final state

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**Turing Machine Pseudocode for**

Replace every 1 with $ Repeat: Find rightmost $, replace it with 1 Go to right end, insert 1 Until no more $ remain

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Turing Machine for

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Example Start Finish

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Block Diagram Turing Machine input output

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Turing’s Thesis

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**Turing’s thesis: Any computation carried out by mechanical means**

can be performed by a Turing Machine (1930)

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Computer Science Law: 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

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**Definition of Algorithm:**

An algorithm for function is a Turing Machine which computes

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**Algorithms are Turing Machines**

When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm

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**Construct a Turing Machine:**

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

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**Can one TM simulate every TM?**

Can a TM simulate a TM? Yes, obviously. Can one TM simulate every TM?

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**Universal Turing Machine An Any TM Simulator**

Input: < Description of some TM m, w > Output: result of running m on w m Universal Turing Machine Output Tape for running TM- m starting on tape w w

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**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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Halting Problem

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Thank you

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