Busch Complexity Lectures: Turing s Thesis. Costas Busch - LSU 1

Transcription

1 Busch Complexity Lectures: Turing s Thesis Costas Busch - LSU 1

2 Turing s thesis (1930): Any computation carried out by mechanical means can be performed by a Turing Machine Costas Busch - LSU 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 - LSU 3

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

5 Variations of the Turing Machine Costas Busch - LSU 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 - LSU 6

7 Variations of the Standard Model Turing machines with: Stay-Option Semi-Infinite Tape Multitape Multidimensional Nondeterministic Different Turing Machine Classes Costas Busch - LSU 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 - LSU 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( M2) and vice-versa Costas Busch - LSU 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 - LSU 10

11 Configurations in the Original Machine have corresponding configurations in the Simulation Machine M 2 M 1 M 1 Original Machine: d 0 d 1 " d n Simulation Machine: dʹ 0 d 1 ʹ dn Costas Busch - LSU 11 M 2 " ʹ

12 Accepting Configuration Original Machine: d f Simulation Machine: dʹf the Simulation Machine and the Original Machine accept the same strings L ( M1) = L( M2) Costas Busch - LSU 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 - LSU 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 - LSU 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 - LSU 15

16 1. Stay-Option Machines simulate Standard Turing machines Trivial: any standard Turing machine is also a Stay-Option machine Costas Busch - LSU 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 - LSU 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 - LSU 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 - LSU 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 - LSU 20

21 A useful trick: Multiple Track Tape helps for more complicated simulations One Tape a b b a a c b d track 1 track 2 One head One symbol ( a, b) It is a standard Turing machine, but each tape alphabet symbol describes a pair of actual useful symbols Costas Busch - LSU 21

22 a b b a a c b d track 1 track 2 q 1 a c a b track 1 b d c d track 2 q 2 ( b, a) ( c, d), L q1 q2 Costas Busch - LSU 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 back to leftmost position Costas Busch - LSU 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 - LSU 24

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

26 2. Semi-Infinite tape machines simulate Standard Turing machines:... Standard machine... Semi-Infinite tape machine... Squeeze infinity of both directions to one direction Costas Busch - LSU 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 - LSU 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 - LSU 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 ( x, a) ( x, g), L L q 1 For all tape symbols x L q 2 Costas Busch - LSU 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 - LSU 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 - LSU 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 - LSU 32

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

34 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 - LSU 34

35 Tape 1 Time 1 Tape 2 a b c e f g q1 q1 Tape 1 Time 2 Tape 2 a g c e d g q2 q2 ( b, f ) ( g, d), L, R q1 q2 Costas Busch - LSU 35

36 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 - LSU 36

37 1. Multi-tape machines simulate Standard Turing Machines: Trivial: Use one tape Costas Busch - LSU 37

38 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 - LSU 38

39 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 - LSU 39

40 Reference point # # # # a b c e f g h 0 Tape 1 head position Tape 2 head position Repeat for each Multi-tape state transition: 1. Return to reference point 2. Find current symbol in Track 1 and update 3. Return to reference point 4. Find current symbol in Tape 2 and update END OF PROOF Costas Busch - LSU 40

41 Same power doesn t imply same speed: n n L = { a b } Standard Turing machine: Go back and forth O( n 2 ) times to match the a s with the b s 2-tape machine: O(n) time 1. Copy n b to tape 2 a n 2. Compare on tape 1 n and b tape 2 O( n 2 ) time ( O(n) steps) ( O(n) steps) Costas Busch - LSU 41

42 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 - LSU 42

43 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 - LSU 43

44 1. Multidimensional machines simulate Standard Turing machines Trivial: Use one dimension Costas Busch - LSU 44

45 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 - LSU 45

46 2-dimensional machine y c a b x a b 1 # 1 # 2 # 1 q 1 Standard Machine c # 1 q 1 symbol coordinates Costas Busch - LSU 46

47 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 - LSU 47

48 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 - LSU 48

49 Time 0 a b c Time 1 q 1 Choice 1 a b, L q 2 b b c q 2 q 1 Choice 2 a c, R q 3 c b c q 3 Costas Busch - LSU 49

50 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 - LSU 50

51 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 - LSU 51

52 1. Nondeterministic Machines simulate Standard (deterministic) Turing Machines Trivial: every deterministic machine is also nondeterministic Costas Busch - LSU 52

53 2. Standard (deterministic) Turing machines simulate Nondeterministic machines: Deterministic machine: Uses a 2-dimensional tape (equivalent to standard Turing machine with one tape) Stores all possible computations of the non-deterministic machine on the 2-dimensional tape Costas Busch - LSU 53

54 All possible computation paths Initial state Step 1 Step 2 reject accept infinite path Step i Step i+1 Costas Busch - LSU 54

55 The Deterministic Turing machine simulates all possible computation paths: simultaneously step-by-step with breadth-first search depth-first may result getting stuck at exploring an infinite path before discovering the accepting path Costas Busch - LSU 55

56 NonDeterministic machine a b, q 1 a c, L R q 2 a b c q 1 q 3 Deterministic machine # # # # # # # a b c # # q 1 # # # # # # Time 0 current configuration Costas Busch - LSU 56

57 a b, q 1 a c, L R NonDeterministic machine Time 1 q 2 q 2 q 3 q 3 Deterministic machine # # # # # # # b b c q # # 2 # # c b c # # q 3 # b b c c b c Computation 1 Costas Busch - LSU 57 Choice 1 Choice 2 Computation 2

58 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 Costas Busch - LSU 58

59 If the non-deterministic machine accepts the input string: The deterministic machine accepts and halts too The simulation takes in the worst case exponential time compared to the shortest length of an accepting path Costas Busch - LSU 59

60 If the non-deterministic machine does not accept the input string: 1. The simulation halts if all paths reach a halting state OR 2. The simulation never terminates if there is a never-ending path (infinite loop) In either case the deterministic machine rejects too (1. by halting or 2. by simulating the infinite loop) END OF PROOF Costas Busch - LSU 60