Busch Complexity Lectures: Turing Machines Prof. Busch - LSU

The Language Hierarchy ? ? Context-Free Languages Regular Languages Prof. Busch - LSU

Context-Free Languages Languages accepted by Turing Machines Context-Free Languages Regular Languages Prof. Busch - LSU

A Turing Machine Tape ...... ...... Read-Write head Control Unit Prof. Busch - LSU

The Tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right Prof. Busch - LSU

The head at each transition (time step): 1. Reads a symbol ...... ...... Read-Write head The head at each transition (time step): 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right Prof. Busch - LSU

Example: Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes 3. Moves Left Prof. Busch - LSU

Time 1 ...... ...... Time 2 ...... ...... 1. Reads 2. Writes 3. Moves Right Prof. Busch - LSU

The Input String Input string Blank symbol ...... ...... head Head starts at the leftmost position of the input string Prof. Busch - LSU

States & Transitions Write Read Move Left Move Right Prof. Busch - LSU

Example: Time 1 ...... ...... current state Prof. Busch - LSU

Time 1 ...... ...... Time 2 ...... ...... Prof. Busch - LSU

Example: Time 1 ...... ...... Time 2 ...... ...... Prof. Busch - LSU

Example: Time 1 ...... ...... Time 2 ...... ...... Prof. Busch - LSU

Turing Machines are deterministic Determinism Turing Machines are deterministic Not Allowed Allowed No lambda transitions allowed Prof. Busch - LSU

Partial Transition Function Example: ...... ...... Allowed: No transition for input symbol Prof. Busch - LSU

Halting The machine halts in a state if there is no transition to follow Prof. Busch - LSU

Halting Example 1: ...... ...... No transition from HALT!!! Prof. Busch - LSU

No possible transition from and symbol Halting Example 2: ...... ...... No possible transition from and symbol HALT!!! Prof. Busch - LSU

Accepting States Not Allowed Allowed Accepting states have no outgoing transitions The machine halts and accepts Prof. Busch - LSU

Acceptance If machine halts Accept Input in an accept state string in a non-accept state or If machine enters an infinite loop Reject Input string Prof. Busch - LSU

In order to accept an input string, Observation: In order to accept an input string, it is not necessary to scan all the symbols in the string Prof. Busch - LSU

Turing Machine Example Input alphabet Accepts the language: Prof. Busch - LSU

Time 0 Prof. Busch - LSU

Time 1 Prof. Busch - LSU

Time 2 Prof. Busch - LSU

Time 3 Prof. Busch - LSU

Time 4 Halt & Accept Prof. Busch - LSU

Rejection Example Time 0 Prof. Busch - LSU

No possible Transition Time 1 No possible Transition Halt & Reject Prof. Busch - LSU

A simpler machine for same language but for input alphabet Accepts the language: Prof. Busch - LSU

Not necessary to scan input Time 0 Halt & Accept Not necessary to scan input Prof. Busch - LSU

Infinite Loop Example A Turing machine for language Prof. Busch - LSU

Time 0 Prof. Busch - LSU

Time 1 Prof. Busch - LSU

Time 2 Prof. Busch - LSU

Time 2 Time 3 Infinite loop Time 4 Time 5 Prof. Busch - LSU

Because of the infinite loop: The accepting state cannot be reached The machine never halts The input string is rejected Prof. Busch - LSU

Another Turing Machine Example Turing machine for the language Prof. Busch - LSU

Basic Idea: Match a’s with b’s: Repeat: replace leftmost a with x find leftmost b and replace it with y Until there are no more a’s or b’s If there is a remaining a or b reject Prof. Busch - LSU

Time 0 Prof. Busch - LSU

Time 1 Prof. Busch - LSU

Time 2 Prof. Busch - LSU

Time 3 Prof. Busch - LSU

Time 4 Prof. Busch - LSU

Time 5 Prof. Busch - LSU

Time 6 Prof. Busch - LSU

Time 7 Prof. Busch - LSU

Time 8 Prof. Busch - LSU

Time 9 Prof. Busch - LSU

Time 10 Prof. Busch - LSU

Time 11 Prof. Busch - LSU

Time 12 Prof. Busch - LSU

Time 13 Halt & Accept Prof. Busch - LSU

machine for the language Observation: If we modify the machine for the language we can easily construct a machine for the language Prof. Busch - LSU

Formal Definitions for Turing Machines Prof. Busch - LSU

Transition Function Prof. Busch - LSU

Transition Function Prof. Busch - LSU

Turing Machine: Input Tape alphabet alphabet States Transition Accept function Accept states Initial state blank Prof. Busch - LSU

Configuration Instantaneous description: Prof. Busch - LSU

Time 4 Time 5 A Move: (yields in one mode) Prof. Busch - LSU

Time 4 Time 5 Time 6 Time 7 A computation Prof. Busch - LSU

Equivalent notation: Prof. Busch - LSU

Initial configuration: Input string Prof. Busch - LSU

The Accepted Language For any Turing Machine Initial state Accept state Prof. Busch - LSU

Recursively Enumerable If a language is accepted by a Turing machine then we say that is: Turing Recognizable Other names used: Turing Acceptable Recursively Enumerable Prof. Busch - LSU