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##### Turing Machines

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**The Language Hierarchy**? ? Context-Free Languages Regular Languages**Languages accepted by**Turing Machines Context-Free Languages Regular Languages**A Turing Machine**Tape ...... ...... Read-Write head Control Unit**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**Example:**Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes 3. Moves Left**Time 1**...... ...... Time 2 ...... ...... 1. Reads 2. Writes 3. Moves Right**The Input String**Input string Blank symbol ...... ...... head Head starts at the leftmost position of the input string**States & Transitions**Write Read Move Left Move Right**Example:**Time 1 ...... ...... current state**Time 1**...... ...... Time 2 ...... ......**Example:**Time 1 ...... ...... Time 2 ...... ......**Example:**Time 1 ...... ...... Time 2 ...... ......**Determinism**Turing Machines are deterministic Not Allowed Allowed No epsilon transitions allowed**Partial Transition Function**Example: ...... ...... Allowed: No transition for input symbol**Halting**The machine haltsif there are no possible transitions to follow**Example:**...... ...... No possible transition HALT!!!**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 language a***Time 4**Halt & Accept**Rejection Example**Time 0**Time 1**No possible Transition Halt & Reject**Infinite Loop Example**Another Turing machine for language a*and is this one correct???**Time 2**Time 3 Time 4 Time 5 ... Infinite Loop**Because of the infinite loop:**• The final state cannot be reached • The machine never halts • The input is not accepted**Another Turing Machine Example**Turing machine for the language**Time 13**Halt & Accept**Observation:**If we modify the machine for the language we can easily construct a machine for the language