Busch Complexity Lectures: Turing Machines. Prof. Busch - LSU 1

Transcription

1 Busch Complexity ectures: Turing Machines Prof. Busch - SU 1

2 The anguage Hierarchy a n b n c n? ww? Context-Free anguages n b n a ww egular anguages a* a *b* Prof. Busch - SU 2

3 a n b anguages accepted by Turing Machines n c n Context-Free anguages n b n a ww ww egular anguages a* a *b* Prof. Busch - SU 3

4 Tape... A Turing Machine... Control Unit ead-write head Prof. Busch - SU 4

5 The Tape... No boundaries -- infinite length... ead-write head The head moves eft or ight Prof. Busch - SU 5

6 ead-write head The head at each transition (time step): 1. eads a symbol 2. Writes a symbol 3. Moves eft or ight Prof. Busch - SU 6

7 Example:... a Time 0 b a c Time 1 a b k c eads 2. Writes a k 3. Moves eft Prof. Busch - SU 7

8 ... Time 1 a b k c Time 2 a f k c eads 2. Writes b 3. Moves ight f Prof. Busch - SU 8

9 The Input String Input string Blank symbol... a b a c... head Head starts at the leftmost position of the input string Prof. Busch - SU 9

10 States & Transitions ead Write Move eft q a b, q2 1 Move ight q a b, q2 1 Prof. Busch - SU 10

11 Example: Time 1... a b a c... q 1 current state q a b, q2 1 Prof. Busch - SU 11

12 Time 1... a b a c... q 1 Time 2... a b b c... q 2 q a b, q2 1 Prof. Busch - SU 12

13 Example: Time 1... a b a c... q 1 Time 2... a b b c... q a b, q2 1 q 2 Prof. Busch - SU 13

14 Example: Time 1... a b a c... q 1 Time 2... a b b c g... q 2 g, q q2 1 Prof. Busch - SU 14

15 Determinism Turing Machines are deterministic Allowed Not Allowed a b, q 2 a b, q 2 q 1 q 1 b d, q 3 a d, q 3 No lambda transitions allowed Prof. Busch - SU 15

16 Example: Partial Transition Function... a b a c... q 1 q 1 a b, b d, q 2 q 3 Allowed: No transition for input symbol c Prof. Busch - SU 16

17 Halting The machine halts in a state if there is no transition to follow Prof. Busch - SU 17

18 Halting Example 1:... a b a c... q 1 q1 No transition from HAT!!! q 1 Prof. Busch - SU 18

19 Halting Example 2:... a b a c... q 1 a b, q 2 No possible transition q 1 from q1 and symbol c b d, q 3 HAT!!! Prof. Busch - SU 19

20 Accepting States q1 q2 Allowed q1 q2 Not Allowed Accepting states have no outgoing transitions The machine halts and accepts Prof. Busch - SU 20

21 Acceptance Accept Input string If machine halts in an accept state eject Input string If machine halts in a non-accept state or If machine enters an infinite loop Prof. Busch - SU 21

22 Observation: In order to accept an input string, it is not necessary to scan all the symbols in the string Prof. Busch - SU 22

23 Turing Machine Example Input alphabet Σ = { a, b} Accepts the language: a*, q 1 Prof. Busch - SU 23

24 Time 0 a a a, q 1 Prof. Busch - SU 24

25 Time 1 a a a, q 1 Prof. Busch - SU 25

26 Time 2 a a a, q 1 Prof. Busch - SU 26

27 Time 3 a a a, q 1 Prof. Busch - SU 27

28 Time 4 a a a q 1 Halt & Accept, q 1 Prof. Busch - SU 28

29 ejection Example Time 0 a b a, q 1 Prof. Busch - SU 29

30 Time 1 a b a No possible Transition Halt & eject, q 1 Prof. Busch - SU 30

31 A simpler machine for same language but for input alphabet Σ = {a } Accepts the language: a* Prof. Busch - SU 31

32 Time 0 a a a Halt & Accept Not necessary to scan input Prof. Busch - SU 32

33 Infinite oop Example A Turing machine for language a * + b( a + b) * b b,, q 1 Prof. Busch - SU 33

34 Time 0 a b a b b,, q 1 Prof. Busch - SU 34

35 Time 1 a b a b b,, q 1 Prof. Busch - SU 35

36 Time 2 a b a b b,, q 1 Prof. Busch - SU 36

37 Time 2 a b a Time 3 Time 4 Time 5 a b a a b a a b a Infinite loop Prof. Busch - SU 37

38 Because of the infinite loop: The accepting state cannot be reached The machine never halts The input string is rejected Prof. Busch - SU 38

39 Another Turing Machine Example n n Turing machine for the language { a b } n 1 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 39

40 Basic Idea: Match a s with b s: epeat: 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 - SU 40

41 Time 0 a a b b q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 41

42 Time 1 x a b b q 1 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 42

43 Time 2 x a b b q 1 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 43

44 Time 3 x a y b q 2 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 44

45 Time 4 x a y b q 2 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 45

46 Time 5 x a y b q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 46

47 Time 6 x x y b q 1 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 47

48 Time 7 x x y b q 1 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 48

49 Time 8 x x y y q 2 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 49

50 Time 9 x x y y q 2 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 50

51 Time 10 x x y y q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 51

52 Time 11 x x y y q 3 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 52

53 Time 12 x x y y q 3 q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 53

54 Time 13 x x y y q 4 Halt & Accept q 4, q 3 a x, q0 1 b y, q q2 x x, Prof. Busch - SU 54

55 Observation: If we modify the machine for the language n n { a b } we can easily construct a machine for the language { a b c } n n n Prof. Busch - SU 55

56 Formal Definitions for Turing Machines Prof. Busch - SU 56

57 Transition Function q a b, q2 1 δ ( 2 q 1, a) = ( q, b, ) Prof. Busch - SU 57

58 Transition Function q c d, q2 1 δ ( 2 q 1, c) = ( q, d, ) Prof. Busch - SU 58

59 Turing Machine: States Input alphabet Tape alphabet M = ( Q, Σ, Γ, δ, q0,, F) Transition function Initial state Accept states blank Prof. Busch - SU 59

60 Configuration c a b a q 1 Instantaneous description: ca q 1 ba Prof. Busch - SU 60

61 Time 4 Time 5 x a y b x a y b q 2 A Move: q2 xayb x q0 ayb (yields in one mode) Prof. Busch - SU 61

62 Time 4 Time 5 x a y b x a y b q 2 Time 6 Time 7 x x y b x x y b q 1 A computation q 2 xayb x q0 ayb xx q1 yb xxy q 1 q 1 b Prof. Busch - SU 62

63 q 2 xayb x q0 ayb xx q1 yb xxy q 1 b Equivalent notation: q2 xayb xxy q1 b Prof. Busch - SU 63

64 Initial configuration: w Input string w a a b b Prof. Busch - SU 64

65 The Accepted anguage For any Turing Machine M ( M ) = { w : q0 w x1 q f x 2 } Initial state Accept state Prof. Busch - SU 65

66 If a language by a Turing machine then we say that is accepted M is: Turing ecognizable Other names used: Turing Acceptable ecursively Enumerable Prof. Busch - SU 66