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


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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 ReadWrite Head (Left or Right) Control Unit Deterministic Costas Busch  LSU 6
7 Variations of the Standard Model Turing machines with: StayOption SemiInfinite 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 TuringRecognizable 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 viceversa 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 StayOption 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: StayOption machines have the same power with Standard Turing machines Proof: 1. StayOption Machines simulate Standard Turing machines 2. Standard Turing machines simulate StayOption machines Costas Busch  LSU 15
16 1. StayOption Machines simulate Standard Turing machines Trivial: any standard Turing machine is also a StayOption machine Costas Busch  LSU 16
17 2. Standard Turing machines simulate StayOption machines We need to simulate the stay head option with two head moves, one left and one right Costas Busch  LSU 17
18 StayOption 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 StayOption 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 StayOption 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 SemiInfinite 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: SemiInfinite machines have the same power with Standard Turing machines Proof: 1. Standard Turing machines simulate SemiInfinite machines 2. SemiInfinite Machines simulate Standard Turing machines Costas Busch  LSU 24
25 1. Standard Turing machines simulate SemiInfinite machines:... # a b a c... Standard Turing Machine SemiInfinite machine modifications a. insert special symbol # at left of input string b. Add a selfloop to every state (except states with no outgoing transitions) # #,R Costas Busch  LSU 25
26 2. SemiInfinite tape machines simulate Standard Turing machines:... Standard machine... SemiInfinite tape machine... Squeeze infinity of both directions to one direction Costas Busch  LSU 26
27 ... Standard machine a b c d e... reference point SemiInfinite 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 SemiInfinite 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 SemiInfinite 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 SemiInfinite 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 SemiInfinite tape machine # d e # c b g L q 2... Costas Busch  LSU 31
32 At the border: SemiInfinite 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 SemiInfinite 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 Multitape 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: Multitape machines have the same power with Standard Turing machines Proof: 1. Multitape machines simulate Standard Turing machines 2. Standard Turing machines simulate Multitape machines Costas Busch  LSU 36
37 1. Multitape machines simulate Standard Turing Machines: Trivial: Use one tape Costas Busch  LSU 37
38 2. Standard Turing machines simulate Multitape machines: Standard machine: Uses a multitrack tape to simulate the multiple tapes A tape of the Multitape machine corresponds to a pair of tracks Costas Busch  LSU 38
39 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 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 Multitape 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 2tape 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 2dimensional 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 MultiDimensional 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 2dimensional 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 2dimensional 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 2dimensional tape (equivalent to standard Turing machine with one tape) Stores all possible computations of the nondeterministic machine on the 2dimensional 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 stepbystep with breadthfirst search depthfirst 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 nondeterministic 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 nondeterministic 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 nondeterministic 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 neverending 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
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