Turing s Thesis. Fall Costas Busch - RPI!1

Transcription

1 Turing s Thesis Costas Busch - RPI!1

2 Turing s thesis (1930): Any computation carried out by mechanical means can be performed by a Turing Machine Costas Busch - RPI!2

3 Algorithm: An algorithm for a problem is a Turing Machine which solves the problem The algorithm describes the steps of the mechanical means This is easily translated to computation steps of a Turing machine Costas Busch - RPI!3

4 When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm Costas Busch - RPI!4

5 Variations of the Turing Machine Costas Busch - RPI!5

6 The Standard Model Infinite Tape a a b a b b c a c a Read-Write Head (Left or Right) Control Unit Deterministic Costas Busch - RPI!6

7 Variations of the Standard Model Turing machines with: Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic Different Turing Machine Classes Costas Busch - RPI!7

8 Same Power of two machine classes: both classes accept the same set of languages We will prove: each new class has the same power with Standard Turing Machine (accept Turing-Recognizable Languages) Costas Busch - RPI!8

9 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( M 2) and vice-versa Costas Busch - RPI!9

10 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 simulates M1 Costas Busch - RPI!10

11 Configurations in the Original Machine have corresponding configurations in the Simulation Machine M 2 M 1 Original Machine: d d!!! 0! 1 M 1 d n Simulation Machine:! 1 d! d!!!!! 0 dn Costas Busch - RPI!11 M 2

12 Accepting Configuration Original Machine: d f Simulation Machine: d! f the Simulation Machine and the Original Machine accept the same strings L ( M1) = L( M 2) Costas Busch - RPI!12

13 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: possible head moves Costas Busch - RPI!13

14 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 Costas Busch - RPI!14

15 Theorem: Stay-Option machines have the same power with Standard Turing machines Proof: 1. Stay-Option Machines simulate Standard Turing machines 2. Standard Turing machines simulate Stay-Option machines Costas Busch - RPI!15

16 1. Stay-Option Machines simulate Standard Turing machines Trivial: any standard Turing machine is also a Stay-Option machine Costas Busch - RPI!16

17 2. Standard Turing machines simulate Stay-Option machines We need to simulate the stay head option with two head moves, one left and one right Costas Busch - RPI!17

18 Stay-Option Machine a b, S q1 q2 Simulation in Standard Machine q 1 a b, L x x, R q 2 For every possible tape symbol x Costas Busch - RPI!18

19 For other transitions nothing changes Stay-Option Machine a b, L q1 q2 Simulation in Standard Machine a b, L q1 q2 Similar for Right moves Costas Busch - RPI!19

20 example of simulation Stay-Option Machine: q a b, S 1 q2 a a b a 1 2 q 1 b a b a q 2 1 Simulation in Standard Machine: a a b a q 1 2 b a b a q 2 3 b a b a q 3 END OF PROOF Costas Busch - RPI!20

21 Multiple Track Tape A useful trick to perform more complicated simulations One Tape a b b a a c b d track 1 track 2 One head One symbol ( a, b) Costas Busch - RPI!21

22 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 Costas Busch - RPI!22

23 Semi-Infinite Tape The head extends infinitely only to the right a b a c... Initial position is the leftmost cell When the head moves left from the border, it returns to the same position Costas Busch - RPI!23

24 Theorem: Semi-Infinite machines have the same power with Standard Turing machines Proof: 1. Standard Turing machines simulate Semi-Infinite machines 2. Semi-Infinite Machines simulate Standard Turing machines Costas Busch - RPI!24

25 1. Standard Turing machines simulate Semi-Infinite machines: # a b a c Standard Turing Machine a. insert special symbol # at left of input string b. Add a self-loop to every state (except states with no # #, R outgoing transitions) Costas Busch - RPI!25

26 2. Semi-Infinite tape machines simulate Standard Turing machines:... Standard machine... Semi-Infinite tape machine... Squeeze infinity of both directions in one direction Costas Busch - RPI!26

27 ... 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... Costas Busch - RPI!27

28 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 Costas Busch - RPI!28

29 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 For all tape symbols x L q 2 Costas Busch - RPI!29

30 Time 1 Standard machine... a b c d e... q 1 Right part Left part Semi-Infinite tape machine # d e # c b a... L q 1 Costas Busch - RPI!30

31 Time 2... Standard machine g b c d e... q 2 Right part Left part Semi-Infinite tape machine # d e # c b g... L q 2 Costas Busch - RPI!31

32 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 Costas Busch - RPI!32

33 Right part Left part Semi-Infinite tape machine # # L q 1 c Time 1 d e b g... Right part Left part # # d e c R q 1 Time 2 b g... END OF PROOF Costas Busch - RPI!33

34 The Off-Line Machine Input File a b c Input string read-only (once) Input string Appears on Control Unit input file only (state machine) Tape g read-write d e Costas Busch - RPI!34

35 Theorem: Off-Line machines have the same power with Standard Turing machines Proof: 1. Off-Line machines simulate Standard Turing machines 2. Standard Turing machines simulate Off-Line machines Costas Busch - RPI!35

36 1. Off-line machines simulate Standard Turing Machines Off-line machine: 1. Copy input file to tape! 2. Continue computation as in Standard Turing machine Costas Busch - RPI!36

37 Standard machine a b c Off-line machine Input File a b c Tape a b c 1. Copy input file to tape Costas Busch - RPI!37

38 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 Costas Busch - RPI!38

39 2. Standard Turing machines simulate Off-Line machines: Use a Standard machine with a four-track tape to keep track of the Off-line input file and tape contents Costas Busch - RPI!39

40 Input File a b c d Off-line Machine Tape e f g Standard Machine -- Four track tape # # a b c d e f g Input File head position Tape head position Costas Busch - RPI!40

41 Reference point (uses special symbol # ) # # # # a b c d e f g Input File head position Tape head position Repeat for each state transition: 1. Return to reference point 2. Find current input file symbol 3. Find current tape symbol 4. Make transition END OF PROOF Costas Busch - RPI!41

42 Multi-tape Turing Machines Control unit (state machine) Tape 1 Tape 2 a b c e f g Input string Input string appears on Tape 1 Costas Busch - RPI!42

43 Tape 1 Time 1 Tape 2 a b c e f g q1 q 1 Tape 1 Time 2 Tape 2 a g c e d g q2 q2 ( b, f ) ( g, d), L, R q1 q2 Costas Busch - RPI!43

44 Theorem: Multi-tape machines have the same power with Standard Turing machines Proof: 1. Multi-tape machines simulate Standard Turing machines 2. Standard Turing machines simulate Multi-tape machines Costas Busch - RPI!44

45 1. Multi-tape machines simulate Standard Turing Machines: Trivial: Use just one tape Costas Busch - RPI!45

46 2. Standard Turing machines simulate Multi-tape machines: Standard machine: Uses a multi-track tape to simulate the multiple tapes A tape of the Multi-tape machine corresponds to a pair of tracks Costas Busch - RPI!46

47 Multi-tape 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 Costas Busch - RPI!47

48 Reference point # # # # a b c e f g h 0 Tape 1 head position Tape 2 head position Repeat for each state transition: 1. Return to reference point 2. Find current symbol in Tape 1 3. Find current symbol in Tape 2 4. Make transition END OF PROOF Costas Busch - RPI!48

49 Same power doesn t n imply b n same speed: L = { a Standard Turing machine: Go back and forth O( n 2 ) times to match the a s with the b s 2-tape machine: n 1. Copy b to tape 2 O(n) n a } time 2. Compare on tape 1 n and b tape 2 O( n 2 ) time ( O(n) steps) ( O(n) steps) Costas Busch - RPI!49

50 Multidimensional Turing Machines 2-dimensional tape y c a b x MOVES: L,R,U,D U: up D: down HEAD Position: +2, -1 Costas Busch - RPI!50

51 Theorem: Multidimensional machines have the same power with Standard Turing machines Proof: 1. Multidimensional machines simulate Standard Turing machines 2. Standard Turing machines simulate Multi-Dimensional machines Costas Busch - RPI!51

52 1. Multidimensional machines simulate Standard Turing machines Trivial: Use one dimension Costas Busch - RPI!52

53 2. Standard Turing machines simulate Multidimensional machines Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2 Costas Busch - RPI!53

54 2-dimensional machine y c a b x a b 1 # 1 # 2 # 1 q 1 Standard Machine c # 1 q 1 symbols coordinates Costas Busch - RPI!54

55 Standard machine: Repeat for each transition followed in the 2-dimensional machine: 1. Update current symbol 2. Compute coordinates of next position 3. Go to new position END OF PROOF Costas Busch - RPI!55

56 Nondeterministic Turing Machines a b, L q 2 Choice 1 q 1 a c, R q 3 Choice 2 Allows Non Deterministic Choices Costas Busch - RPI!56

57 Time 0 a b c q 1 a b, L q 2 Time 1 Choice 1 b b c q 2 q 1 a c, R q 3 Choice 2 c b c q 3 Costas Busch - RPI!57

58 Input string w there is a computation: q0w! x q f y is accepted if Initial configuration Final Configuration Any accept state There is a computation: Costas Busch - RPI!58

59 Theorem: Nondeterministic machines have the same power with Standard Turing machines Proof: 1. Nondeterministic machines simulate Standard Turing machines 2. Standard Turing machines simulate Nondeterministic machines Costas Busch - RPI!59

60 1. Nondeterministic Machines simulate Standard (deterministic) Turing Machines Trivial: every deterministic machine is also nondeterministic Costas Busch - RPI!60

61 2. Standard (deterministic) Turing machines simulate Nondeterministic machines: Deterministic machine: Uses a 2-dimensional tape (which is equivalent to 1-dimensional tape) Stores all possible computations of the non-deterministic machine on the 2-dimensional tape Costas Busch - RPI!61

62 All possible computation paths Initial state Step 1 Step 2 reject accept infinite path Step i Step i+1 Costas Busch - RPI!62

63 The Deterministic Turing machine simulates all possible computation paths: simultaneously step-by-step in a breadth-first search fashion Costas Busch - RPI!63

64 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 # # # # # # current configuration Costas Busch - RPI!64

65 a b, q 1 a c, L R NonDeterministic machine Time 1 q 2 q 3 q 2 # # # # # # # b b c q # # 2 # # c b c # q 3 # b b c c b c q 3 Computation 1 Costas Busch - RPI!65 Deterministic machine Choice 1 Choice 2 # Computation 2

66 Deterministic Turing machine Repeat For each configuration in current step of non-deterministic machine, if there are two or more choices: 1. Replicate configuration 2. Change the state in the replicas Until either the input string is accepted or rejected in all configurations END OF PROOF Costas Busch - RPI!66

67 Remark: The simulation takes in the worst case exponential time compared to the shortest accepting path length of the nondeterministic machine Costas Busch - RPI!67