Turing Machines. The Language Hierarchy. Context-Free Languages. Regular Languages. Courtesy Costas Busch - RPI 1

Transcription

1 Turing Machines a n b n c The anguage Hierarchy n? ww? Context-Free anguages a n b n egular anguages a * a *b* ww Courtesy Costas Busch - PI

2 a n b n c n Turing Machines anguages accepted by Turing Machines ww Context-Free anguages a n b n egular anguages a * a *b* ww Courtesy Costas Busch - PI 2

3 A Turing Machine Tape ead-write head Control Unit Courtesy Costas Busch - PI 3

4 The Tape The head moves eft or ight No boundaries -- infinite length ead-write head The head at each time step:. eads a symbol 2. Writes a symbol 3. Moves eft or ight Courtesy Costas Busch - PI 4

5 ... Time 0 Turing Machines a a c b Time a b k c.... eads 2. Writes a k 3. Moves eft Courtesy Costas Busch - PI 5

6 ... Turing Machines Time a b k c Time 2 a f k c.... eads 2. Writes b f 3. Moves ight Courtesy Costas Busch - PI 6

7 The Input String Head starts at the leftmost position of the input string Input string Blank symbol... a b a c... head emark: the input string is never empty Courtesy Costas Busch - PI 7

8 States & Transitions ead Write Move eft q a b, q2 Move ight q a b, q2 Courtesy Costas Busch - PI 8

9 Example: States & Transitions Time... a b a c... q current state q a b, q2 Time 2... a b b c... q 2 Courtesy Costas Busch - PI 9

10 Example: Time... a b a c... q a b, q2 q Time 2... a b b c... q 2 Courtesy Costas Busch - PI 0

11 Example: Time... a b a c... g, q q2 q Time 2... a b a c g... q 2 Courtesy Costas Busch - PI

12 Determinism Turing Machines are deterministic Allowed Not Allowed a b, q 2 a b, q 2 q q b d, a d, Courtesy Costas Busch - PI 2

13 Partial Transition Function Example:... a b a c... q a b, q 2 Allowed: q b d, No transition for input symbol C Courtesy Costas Busch - PI 3

14 Example: Halting The machine halts if there are no possible transitions to follow.... a b b c... q 2 q a b, b d, q 2 No possible transition HAT!!! Courtesy Costas Busch - PI 4

15 Final States q q2 Allowed q q2 Not Allowed Final states have no outgoing transitions In a final state the machine halts Courtesy Costas Busch - PI 5

16 Acceptance Accept Input If machine halts in a final state If machine halts eject Input in a non-final state or If machine enters an infinite loop Courtesy Costas Busch - PI 6

17 Turing Machine Example Turing machine for the language { a n b n } q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 7

18 Time 0 a a b b q 0 q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 8

19 Time x a b b q q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 9

20 Time 2 x a b b q q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 20

21 Time 3 x a y b q 2 q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 2

22 Time 4 x a y b q 2 q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 22

23 Time 5 x a y b q 0 q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 23

24 Time 6 x x y b q q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 24

25 Time 7 x x y b q q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 25

26 Time 8 x x y y q 2 q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 26

27 Time 9 x x y y q 2 q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 27

28 Time 0 x x y y q 0 q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 28

29 Time x x y y q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 29

30 Time 2 x x y y q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 30

31 Time 3 x x y y q 4 Halt & Accept q 4, a a, a a, a x, q0 b y, q q2 x x, Courtesy Costas Busch - PI 3

32 Observation If we modify the machine for the language { a n b n } we can easily construct { a b c } a machine for the language n n n Courtesy Costas Busch - PI 32

33 Formal Definition of Turing Machines Transition Function q a b, q2 δ ( 2 q, a) = ( q, b, ) Courtesy Costas Busch - PI 33

34 Formal Definition of Turing Machines Transition Function q c d, q2 δ ( 2 q, c) = ( q, d, ) Courtesy Costas Busch - PI 34

35 Formal Definition of Turing Machines States Input alphabet Tape alphabet M = ( Q, Σ, Γ, δ, q0,, F ) Transition function Σ Γ Γ, Σ q 0 Initial F state Q Q blank Final states Courtesy Costas Busch - PI 35

36 Configuration c a b a q Instantaneous description: ca q ba Courtesy Costas Busch - PI 36

37 Configuration Time Time 2 x a y b x a y b q 2 q 0 A Move: Courtesy Costas Busch - PI 37

38 Time Time 2 x a y b x a y b q 2 q 0 Time 3 Time 4 x x y b x x y b q q Courtesy Costas Busch - PI 38

39 Configuration Equivalent notation: Courtesy Costas Busch - PI 39

40 Initial configuration: Configuration q 0 w Input string w a a b b q 0 Courtesy Costas Busch - PI 40

41 The Accepted anguage For any Turing Machine M Initial state Final state Courtesy Costas Busch - PI 4

42 Computing Functions using TM A function f (w) Domain: D has: esult egion: S w D f (w) f ( w) S Courtesy Costas Busch - PI 42

43 Computing Functions using TM A function may have many parameters: Example: Addition function f ( x, y) = x + y Courtesy Costas Busch - PI 43

44 Computing Functions using TM Integer Domain Decimal: 5 Binary: 0 Unary: We prefer unary representation: easier to manipulate with Turing machines Courtesy Costas Busch - PI 44

45 Definition: f A function is computable if M there is a Turing Machine such that: Initial configuration w Final configuration f (w) q 0 initial state q f final state For all w D Domain Courtesy Costas Busch - PI 45

46 In other words: f A function is computable if M there is a Turing Machine such that: Initial Configuration Final Configuration For all w D Domain Courtesy Costas Busch - PI 46

47 Example The function f ( x, y) = x + y is computable x, y are integers Turing Machine: Input string: x0 y unary Output string: xy0 unary Courtesy Costas Busch - PI 47

48 x y Start Λ 0 Λ q 0 initial state The 0 is the delimiter that separates the two numbers Courtesy Costas Busch - PI 48

49 x y Start Λ 0 Λ q 0 initial state x + y Finish Λ 0 q f final state Courtesy Costas Busch - PI 49

50 The 0 helps when we use the result for other operations x + y Finish Λ 0 q f final state Courtesy Costas Busch - PI 50

51 Turing machine for function f ( x, y) = x + y q 0 0,, q q2 0, q 4, Courtesy Costas Busch - PI 5

52 Execution Example: Time 0 x = (2) x y 0 y = (2) q 0 Final esult x + y 0 q 4 Courtesy Costas Busch - PI 52

53 Time 0 0 q 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 53

54 Time 0 q 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 54

55 Time 2 0 q 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 55

56 Time 3 q 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 56

57 Time 4 q 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 57

58 Time 5 q 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 58

59 Time 6 q 2 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 59

60 Time 7 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 60

61 Time 8 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 6

62 Time 9 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 62

63 Time 0 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 63

64 Time 0 0, q0, q q2 0, q 4, Courtesy Costas Busch - PI 64

65 Time 2 0 q 4 0, q0, q q2 0, HAT & accept q 4, Courtesy Costas Busch - PI 65

66 Another Example The function f ( x) = 2x is computable x is integer Turing Machine: Input string: x unary Output string: xx unary Courtesy Costas Busch - PI 66

67 x Start Λ q 0 initial state 2x Finish Λ q f final state Courtesy Costas Busch - PI 67

68 Turing Machine for f ( x) = 2x $,, q0 q $, q2,, Courtesy Costas Busch - PI 68

69 Start Example Finish q 0 $,, q0 q $, q2,, Courtesy Costas Busch - PI 69

70 Another Example The function f ( x, y) = is computable x > y 0 if if x y Courtesy Costas Busch - PI 70

71 Turing Machine for f ( x, y) = if x > y 0 if x y Input: x0 y Output: or 0 Courtesy Costas Busch - PI 7

72 Combining Turing Machines Courtesy Costas Busch - PI 72

73 Block Diagram input Turing Machine output Courtesy Costas Busch - PI 73

74 Example: x + y if x > y f ( x, y) = 0 if x y x, y Adder x + y x, y Comparer x > y x y Eraser 0 Courtesy Costas Busch - PI 74