courtesy costas busch - rpi1 turing machines. courtesy costas busch - rpi2 the language hierarchy...
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- Slide 1
- Courtesy Costas Busch - RPI1 Turing Machines
- Slide 2
- Courtesy Costas Busch - RPI2 The Language Hierarchy Regular Languages Context-Free Languages ? ?
- Slide 3
- Courtesy Costas Busch - RPI3 Regular Languages Context-Free Languages Languages accepted by Turing Machines
- Slide 4
- Courtesy Costas Busch - RPI4 A Turing Machine...... Tape Read-Write head Control Unit
- Slide 5
- Courtesy Costas Busch - RPI5 The Tape...... Read-Write head No boundaries -- infinite length The head moves Left or Right
- Slide 6
- Courtesy Costas Busch - RPI6...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right
- Slide 7
- Courtesy Costas Busch - RPI7...... Example: Time 0...... Time 1 1. Reads 2. Writes 3. Moves Left
- Slide 8
- Courtesy Costas Busch - RPI8...... Time 1...... Time 2 1. Reads 2. Writes 3. Moves Right
- Slide 9
- Courtesy Costas Busch - RPI9 The Input String...... Blank symbol head Head starts at the leftmost position of the input string Input string
- Slide 10
- Courtesy Costas Busch - RPI10...... Blank symbol head Input string Remark: the input string is never empty
- Slide 11
- Courtesy Costas Busch - RPI11 States & Transitions Read Write Move Left Move Right
- Slide 12
- Courtesy Costas Busch - RPI12 Example:...... Time 1 current state
- Slide 13
- Courtesy Costas Busch - RPI13...... Time 1...... Time 2
- Slide 14
- Courtesy Costas Busch - RPI14...... Time 1...... Time 2 Example:
- Slide 15
- Courtesy Costas Busch - RPI15...... Time 1...... Time 2 Example:
- Slide 16
- Courtesy Costas Busch - RPI16 Determinism Allowed Not Allowed No lambda transitions allowed Turing Machines are deterministic
- Slide 17
- Courtesy Costas Busch - RPI17 Partial Transition Function...... Example: No transition for input symbol Allowed:
- Slide 18
- Courtesy Costas Busch - RPI18 Halting The machine halts if there are no possible transitions to follow
- Slide 19
- Courtesy Costas Busch - RPI19 Example:...... No possible transition HALT!!!
- Slide 20
- Courtesy Costas Busch - RPI20 Final States Allowed Not Allowed Final states have no outgoing transitions In a final state the machine halts
- Slide 21
- Courtesy 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
- Slide 22
- Courtesy Costas Busch - RPI22 Turing Machine Example A Turing machine that accepts the language:
- Slide 23
- Courtesy Costas Busch - RPI23 Time 0
- Slide 24
- Courtesy Costas Busch - RPI24 Time 1
- Slide 25
- Courtesy Costas Busch - RPI25 Time 2
- Slide 26
- Courtesy Costas Busch - RPI26 Time 3
- Slide 27
- Courtesy Costas Busch - RPI27 Time 4 Halt & Accept
- Slide 28
- Courtesy Costas Busch - RPI28 Rejection Example Time 0
- Slide 29
- Courtesy Costas Busch - RPI29 Time 1 No possible Transition Halt & Reject
- Slide 30
- Courtesy Costas Busch - RPI30 Infinite Loop Example A Turing machine for language
- Slide 31
- Courtesy Costas Busch - RPI31 Time 0
- Slide 32
- Courtesy Costas Busch - RPI32 Time 1
- Slide 33
- Courtesy Costas Busch - RPI33 Time 2
- Slide 34
- Courtesy Costas Busch - RPI34 Time 2 Time 3 Time 4 Time 5 Infinite loop
- Slide 35
- Courtesy Costas Busch - RPI35 Because of the infinite loop: The final state cannot be reached The machine never halts The input is not accepted
- Slide 36
- Courtesy Costas Busch - RPI36 Another Turing Machine Example Turing machine for the language
- Slide 37
- Courtesy Costas Busch - RPI37 Time 0
- Slide 38
- Courtesy Costas Busch - RPI38 Time 1
- Slide 39
- Courtesy Costas Busch - RPI39 Time 2
- Slide 40
- Courtesy Costas Busch - RPI40 Time 3
- Slide 41
- Courtesy Costas Busch - RPI41 Time 4
- Slide 42
- Courtesy Costas Busch - RPI42 Time 5
- Slide 43
- Courtesy Costas Busch - RPI43 Time 6
- Slide 44
- Courtesy Costas Busch - RPI44 Time 7
- Slide 45
- Courtesy Costas Busch - RPI45 Time 8
- Slide 46
- Courtesy Costas Busch - RPI46 Time 9
- Slide 47
- Courtesy Costas Busch - RPI47 Time 10
- Slide 48
- Courtesy Costas Busch - RPI48 Time 11
- Slide 49
- Courtesy Costas Busch - RPI49 Time 12
- Slide 50
- Courtesy Costas Busch - RPI50 Halt & Accept Time 13
- Slide 51
- Courtesy Costas Busch - RPI51 If we modify the machine for the language we can easily construct a machine for the language Observation:
- Slide 52
- Courtesy Costas Busch - RPI52 Formal Definitions for Turing Machines
- Slide 53
- Courtesy Costas Busch - RPI53 Transition Function
- Slide 54
- Courtesy Costas Busch - RPI54 Transition Function
- Slide 55
- Courtesy Costas Busch - RPI55 Turing Machine: States Input alphabet Tape alphabet Transition function Initial state blank Final states
- Slide 56
- Courtesy Costas Busch - RPI56 Configuration Instantaneous description:
- Slide 57
- Courtesy Costas Busch - RPI57 Time 4Time 5 A Move:
- Slide 58
- Courtesy Costas Busch - RPI58 Time 4Time 5 Time 6Time 7
- Slide 59
- Courtesy Costas Busch - RPI59 Equivalent notation:
- Slide 60
- Courtesy Costas Busch - RPI60 Initial configuration: Input string
- Slide 61
- Courtesy Costas Busch - RPI61 The Accepted Language For any Turing Machine Initial stateFinal state
- Slide 62
- Courtesy 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:
- Slide 63
- Courtesy Costas Busch - RPI63 Computing Functions with Turing Machines
- Slide 64
- Courtesy Costas Busch - RPI64 A function Domain: Result Region: has:
- Slide 65
- Courtesy Costas Busch - RPI65 A function may have many parameters: Example: Addition function
- Slide 66
- Courtesy Costas Busch - RPI66 Integer Domain Unary: Binary: Decimal: 11111 101 5 We prefer unary representation: easier to manipulate with Turing machines
- Slide 67
- Courtesy 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
- Slide 68
- Courtesy 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
- Slide 69
- Courtesy Costas Busch - RPI69 Example The function is computable Turing Machine: Input string: unary Output string: unary are integers
- Slide 70
- Courtesy Costas Busch - RPI70 Start initial state The 0 is the delimiter that separates the two numbers
- Slide 71
- Courtesy Costas Busch - RPI71 Start Finish final state initial state
- Slide 72
- Courtesy Costas Busch - RPI72 Finish final state The 0 helps when we use the result for other operations
- Slide 73
- Courtesy Costas Busch - RPI73 Turing machine for function
- Slide 74
- Courtesy Costas Busch - RPI74 Execution Example: Time 0 Final Result (2)
- Slide 75
- Courtesy Costas Busch - RPI75 Time 0
- Slide 76
- Courtesy Costas Busch - RPI76 Time 1
- Slide 77
- Courtesy Costas Busch - RPI77 Time 2
- Slide 78
- Courtesy Costas Busch - RPI78 Time 3
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- Courtesy Costas Busch - RPI79 Time 4
- Slide 80
- Courtesy Costas Busch - RPI80 Time 5
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- Courtesy Costas Busch - RPI81 Time 6
- Slide 82
- Courtesy Costas Busch - RPI82 Time 7
- Slide 83
- Courtesy Costas Busch - RPI83 Time 8
- Slide 84
- Courtesy Costas Busch - RPI84 Time 9
- Slide 85
- Courtesy Costas Busch - RP