Variations of the Turing Machine

Transcription

1 Variations of the Turing Machine 1

2 The Standard Model Infinite Tape a a b a b b c a c a Read-Write Head (Left or Right) Control Unit Deterministic 2

3 Variations of the Standard Model Turing machines with: Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic 3

4 The variations form different Turing Machine Classes We want to prove: Each Class has the same power with the Standard Model 4

5 Same Power of two classes means: Both classes of Turing machines accept the same languages 5

6 Same Power of two classes means: For any machine M 1 of first class there is a machine M 2 of second class such that: L( M1) L( M2) And vice-versa 6

7 Simulation: a technique to prove same power Simulate the machine of one class with a machine of the other class First Class Original Machine Second Class Simulation Machine M 2 M1 M1 7

8 Configurations in the Original Machine correspond to configurations in the Simulation Machine Original Machine: d 0 d 1 d n Simulation Machine: d 0 d 1 dn 8

9 Final Configuration Original Machine: d f Simulation Machine: df The Simulation Machine and the Original Machine accept the same language 9

10 Turing Machines with Stay-Option The head can stay in the same position a a b a b b c a c a Left, Right, Stay L,R,S: moves 10

11 Example: Time 1 a a b a b b c a c a q 1 Time 2 b a b a b b c a c a q 2 a b, S q1 q2 11

12 Theorem: Stay-Option Machines have the same power with Standard Turing machines 12

13 Proof: Part 1: Stay-Option Machines are at least as powerful as Standard machines Proof: a Standard machine is also a Stay-Option machine (that never uses the S move) 13

14 Proof: Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof: a standard machine can simulate a Stay-Option machine 14

15 Stay-Option Machine a b, L q1 q2 Simulation in Standard Machine a b, L q1 q2 Similar for Right moves 15

16 Stay-Option Machine a b, S q1 q2 Simulation in Standard Machine a b, L q1 q2 x x, R q 3 For every symbol x 16

17 Example Stay-Option Machine: q a b, S 1 q2 1 a a b a 2 b a b a q 1 q 2 Simulation in Standard Machine: 1 a a b a 2 b a b a 3 b a b a q 1 q 2 q 3 17

18 Standard Machine--Multiple Track Tape a b b a a c b d track 1 track 2 one symbol 18

19 a b b a a c b d track 1 track 2 q 1 a b c d a c b d track 1 track 2 q 2 ( b, a) ( c, d), L q1 q2 19

20 Semi-Infinite Tape # a b a c... 20

21 Standard Turing machines simulate Semi-infinite tape machines: Trivial 21

22 Semi-infinite tape machines simulate Standard Turing machines:... Standard machine... Semi-infinite tape machine... 22

23 Standard machine... a b c d e... reference point Semi-infinite tape machine with two tracks Right part Left part # # d e c b a... 23

24 Standard machine q 1 q 2 Semi-infinite tape machine Left part Right part L q 1 L q 2 R q 1 R q 2 24

25 Standard machine a g, R q1 q2 Semi-infinite tape machine Right part R q 1 ( a, x) ( g, x), R R q 2 Left part L q 1 ( x, a) ( x, g), L L q 2 For all symbols x 25

26 Time 1... Standard machine a b c d e... q 1 Semi-infinite tape machine Right part # d e... Left part # c b a L q 1 26

27 Time 2... Standard machine g b c d e... q 2 Right part Left part Semi-infinite tape machine # # d e c b L q 2 g... 27

28 At the border: Semi-infinite tape machine Right part R q 1 (#,#) (#,#),R L q 1 Left part L q 1 (#,#) (#,#),R R q 1 28

29 Semi-infinite tape machine Right part Left part # # d e c Time 1 b g... L q 1 Right part Left part # # d e c Time 2 b g... R q 1 29

30 Theorem: Semi-infinite tape machines have the same power with Standard Turing machines 30

31 The Off-Line Machine a b c Input File read-only Control Unit Tape g read-write d e 31

32 Off-line machines simulate Standard Turing Machines: Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine 32

33 Standard machine a b c Off-line machine Input File a b c Tape a b c 1. Copy input file to tape 33

34 Standard machine a b c q 1 Off-line machine Input File a b c Tape a b c q 1 2. Do computations as in Turing machine 34

35 Standard Turing machines simulate Off-line machines: Use a Standard machine with four track tape to keep track of the Off-line input file and tape contents 35

36 Input File a b c d Off-line Machine Tape e f g Four track tape -- Standard Machine # # a b c d e f g Input File head position Tape head position 36

37 Reference point # # a b c d e f g Input File head position Tape head position Repeat for each state transition: Return to reference point Find current input file symbol Find current tape symbol Make transition 37

38 Theorem: Off-line machines have the same power with Stansard machines 38

39 Multitape Turing Machines Control unit Tape 1 Tape 2 a b c Input e f g 39

40 Tape 1 Time 1 Tape 2 a b c e f g q1 q1 a g c Time 2 e d g q2 q2 ( b, f ) ( g, d), L, R q1 q2 40

41 Multitape machines simulate Standard Machines: Use just one tape 41

42 Standard machines simulate Multitape machines: Standard machine: Use a multi-track tape A tape of the Multiple tape machine corresponds to a pair of tracks 42

43 Multitape Machine Tape 1 Tape 2 a b c e f g h Standard machine with four track tape a b c e f g h 0 Tape 1 head position Tape 2 head position 43

44 Reference point # # # # a b c e f g h 0 Tape 1 head position Tape 2 head position Repeat for each state transition: Return to reference point Find current symbol in Tape 1 Find current symbol in Tape 2 Make transition 44

45 Theorem: Multi-tape machines have the same power with Standard Turing Machines 45

46 Same power doesn t imply same speed: Language L { a n b n } Standard machine Acceptance Time 2 n Two-tape machine n 46

47 L { a n b n } Standard machine: Go back and forth 2 n times Two-tape machine: n Copy b to tape 2 n Leave a on tape 1 Compare tape 1 and tape 2 ( n steps) ( n steps) ( n steps) 47

48 MultiDimensional Turing Machines Two-dimensional tape y c a b x MOVES: L,R,U,D U: up D: down HEAD Position: +2, -1 48

49 Multidimensional machines simulate Standard machines: Use one dimension 49

50 Standard machines simulate Multidimensional machines: Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2 50

51 Two-dimensional machine y c a b x a b 1 # 1 # 2 # 1 q 1 Standard Machine c # 1 q 1 symbols coordinates 51

52 tandard machine: Repeat for each transition Update current symbol Compute coordinates of next position Go to new position 52

53 Theorem: MultiDimensional Machines have the same power with Standard Turing Machines 53

54 NonDeterministic Turing Machines a b, L q 2 q 1 a c, R q 3 Non Deterministic Choice 54

55 a b, L q 2 Time 0 q 1 a b c a c, R q 3 q 1 Time 1 Choice 1 Choice 2 b b c c b c q 2 q 3 55

56 Input string w is accepted if this a possible computation q 0 w x q f y Initial configuration Final Configuration Final state 56

57 NonDeterministic Machines simulate Standard (deterministic) Machines: Every deterministic machine is also a nondeterministic machine 57

58 Deterministic machines simulate NonDeterministic machines: Deterministic machine: Keeps track of all possible computations 58

59 Non-Deterministic Choices q 1 q 2 q 3 q 4 Computation 1 q 5 q6 q7 59

60 Non-Deterministic Choices q 1 q 2 q 3 q 4 q 5 Computation 2 q6 q7 60

61 Simulation Deterministic machine: Keeps track of all possible computations Stores computations in a two-dimensional tape 61

62 NonDeterministic machine a b, q 1 a c, L R q 2 q 3 Time 0 a b c q 1 Deterministic machine # # # # # # # a b c # # q 1 # # # # # # Computation 1 62

63 a b, q 1 a c, L R NonDeterministic machine Time 1 q 2 q 2 # # # # # # # b b c # q 2 # # # c b c # # q 3 # b b c c b c q 3 q 3 Deterministic machine Choice 1 Choice 2 Computation 1 Computation 2 63

64 Repeat Execute a step in each computation: If there are two or more choices in current computation: 1. Replicate configuration 2. Change the state in the replica 64

65 Theorem: NonDeterministic Machines have the same power with Deterministic machines 65

66 Remark: The simulation in the Deterministic machine takes time exponential time compared to the NonDeterministic machine 66

67 Polynomial Time in NonDeterministic Machine: NP-Time Polynomial Time in Deterministic Machine: P-Time Fundamental Problem: P = NP? 67