Costas Busch - RPI1 Turing Machines

Costas Busch - RPI2 The Language Hierarchy Regular Languages Context-Free Languages ? ?

Costas Busch - RPI3 Regular Languages Context-Free Languages Languages accepted by Turing Machines

Costas Busch - RPI4 A Turing Machine Tape Read-Write head Control Unit

Costas Busch - RPI5 The Tape Read-Write head No boundaries -- infinite length The head moves Left or Right

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

Costas Busch - RPI Example: Time Time 1 1. Reads 2. Writes 3. Moves Left

Costas Busch - RPI Time Time 2 1. Reads 2. Writes 3. Moves Right

Costas Busch - RPI9 The Input String Blank symbol head Head starts at the leftmost position of the input string Input string

Costas Busch - RPI Blank symbol head Input string Remark: the input string is never empty

Costas Busch - RPI11 States & Transitions Read Write Move Left Move Right

Costas Busch - RPI12 Example: Time 1 current state

Costas Busch - RPI Time Time 2

Costas Busch - RPI Time Time 2 Example:

Costas Busch - RPI Time Time 2 Example:

Costas Busch - RPI16 Determinism Allowed Not Allowed No lambda transitions allowed Turing Machines are deterministic

Costas Busch - RPI17 Partial Transition Function Example: No transition for input symbol Allowed:

Costas Busch - RPI18 Halting The machine halts if there are no possible transitions to follow

Costas Busch - RPI19 Example: No possible transition HALT!!!

Costas Busch - RPI20 Final States Allowed Not Allowed Final states have no outgoing transitions In a final state the machine halts

Costas Busch - RPI21 Acceptance Accept Input If machine halts in a final state Reject Input If machine halts in a non-final state or If machine enters an infinite loop

Costas Busch - RPI22 Turing Machine Example A Turing machine that accepts the language:

Costas Busch - RPI23 Time 0

Costas Busch - RPI24 Time 1

Costas Busch - RPI25 Time 2

Costas Busch - RPI26 Time 3

Costas Busch - RPI27 Time 4 Halt & Accept

Costas Busch - RPI28 Rejection Example Time 0

Costas Busch - RPI29 Time 1 No possible Transition Halt & Reject

Costas Busch - RPI30 Infinite Loop Example A Turing machine for language

Costas Busch - RPI31 Time 0

Costas Busch - RPI32 Time 1

Costas Busch - RPI33 Time 2

Costas Busch - RPI34 Time 2 Time 3 Time 4 Time 5 Infinite loop

Costas Busch - RPI35 Because of the infinite loop: The final state cannot be reached The machine never halts The input is not accepted

Costas Busch - RPI36 Another Turing Machine Example Turing machine for the language

Costas Busch - RPI37 Time 0

Costas Busch - RPI38 Time 1

Costas Busch - RPI39 Time 2

Costas Busch - RPI40 Time 3

Costas Busch - RPI41 Time 4

Costas Busch - RPI42 Time 5

Costas Busch - RPI43 Time 6

Costas Busch - RPI44 Time 7

Costas Busch - RPI45 Time 8

Costas Busch - RPI46 Time 9

Costas Busch - RPI47 Time 10

Costas Busch - RPI48 Time 11

Costas Busch - RPI49 Time 12

Costas Busch - RPI50 Halt & Accept Time 13

Costas Busch - RPI51 If we modify the machine for the language we can easily construct a machine for the language Observation:

Costas Busch - RPI52 Formal Definitions for Turing Machines

Costas Busch - RPI53 Transition Function

Costas Busch - RPI54 Transition Function

Costas Busch - RPI55 Turing Machine: States Input alphabet Tape alphabet Transition function Initial state blank Final states

Costas Busch - RPI56 Configuration Instantaneous description:

Costas Busch - RPI57 Time 4Time 5 A Move:

Costas Busch - RPI58 Time 4Time 5 Time 6Time 7

Costas Busch - RPI59 Equivalent notation:

Costas Busch - RPI60 Initial configuration: Input string

Costas Busch - RPI61 The Accepted Language For any Turing Machine Initial stateFinal state

Costas Busch - RPI62 Standard Turing Machine Deterministic Infinite tape in both directions Tape is the input/output file The machine we described is the standard:

Costas Busch - RPI63 Computing Functions with Turing Machines

Costas Busch - RPI64 A function Domain: Result Region: has:

Costas Busch - RPI65 A function may have many parameters: Example: Addition function

Costas Busch - RPI66 Integer Domain Unary: Binary: Decimal: We prefer unary representation: easier to manipulate with Turing machines

Costas Busch - RPI67 Definition: A function is computable if there is a Turing Machine such that: Initial configurationFinal configuration Domain final stateinitial state For all

Costas Busch - RPI68 Initial Configuration Final Configuration A function is computable if there is a Turing Machine such that: In other words: Domain For all

Costas Busch - RPI69 Example The function is computable Turing Machine: Input string: unary Output string: unary are integers

Costas Busch - RPI70 Start initial state The 0 is the delimiter that separates the two numbers

Costas Busch - RPI71 Start Finish final state initial state

Costas Busch - RPI72 Finish final state The 0 helps when we use the result for other operations

Costas Busch - RPI73 Turing machine for function

Costas Busch - RPI74 Execution Example: Time 0 Final Result (2)

Costas Busch - RPI75 Time 0

Costas Busch - RPI76 Time 1

Costas Busch - RPI77 Time 2

Costas Busch - RPI78 Time 3

Costas Busch - RPI79 Time 4

Costas Busch - RPI80 Time 5

Costas Busch - RPI81 Time 6

Costas Busch - RPI82 Time 7

Costas Busch - RPI83 Time 8

Costas Busch - RPI84 Time 9

Costas Busch - RPI85 Time 10

Costas Busch - RPI86 Time 11

Costas Busch - RPI87 HALT & accept Time 12

Costas Busch - RPI88 Another Example The function is computable Turing Machine: Input string: unary Output string:unary is integer

Costas Busch - RPI89 Start Finish final state initial state

Costas Busch - RPI90 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

Costas Busch - RPI91 Turing Machine for

Costas Busch - RPI92 Example Start Finish

Costas Busch - RPI93 Another Example The function is computable if

Costas Busch - RPI94 Turing Machine for Input: Output: or if

Costas Busch - RPI95 Turing Machine Pseudocode: Match a 1 from with a 1 from Repeat Until all of or is matched If a 1 from is not matched erase tape, write 1 else erase tape, write 0

Costas Busch - RPI96 Combining Turing Machines

Costas Busch - RPI97 Block Diagram Turing Machine inputoutput

Costas Busch - RPI98 Example: if Comparer Adder Eraser