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1 Variations of the Turing Machine
2 Read-Write Head Control Unit Deterministic The Standard Model Infinite Tape (Left or Right)
3 Variations of the Standard Model Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic Turing machines with:
4 We want to prove: Each Class has the same power with the Standard Model The variations form different Turing Machine Classes
5 Same Power of two classes means: Both classes of Turing machines accept the same languages
6 Same Power of two classes means: For any machine of first class there is a machine of second class such that: And vice-versa
7 a technique to prove same powerSimulation: Simulate the machine of one class with a machine of the other class First Class Original Machine Second Class Simulation Machine
8 Configurations in the Original Machine correspond to configurations in the Simulation Machine Original Machine: Simulation Machine:
9 The Simulation Machine and the Original Machine accept the same language Original Machine: Simulation Machine: Final Configuration
10 Turing Machines with Stay-Option The head can stay in the same position Left, Right, Stay L,R,S: moves
11 Example: Time 1 Time 2
12 Stay-Option Machines have the same power with Standard Turing machines Theorem:
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)
14 Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof:a standard machine can simulate a Stay-Option machine Proof:
15 Stay-Option Machine Simulation in Standard Machine Similar for Right moves
16 Stay-Option Machine Simulation in Standard Machine For every symbol
17 Example Stay-Option Machine: 12 Simulation in Standard Machine: 123
18 Standard Machine--Multiple Track Tape track 1 track 2 one symbol
19 track 1 track 2 track 1 track 2
20 Semi-Infinite Tape
21 Standard Turing machines simulate Semi-infinite tape machines: Trivial
22 Semi-infinite tape machines simulate Standard Turing machines: Standard machine Semi-infinite tape machine
23 Standard machine Semi-infinite tape machine with two tracks reference point Right part Left part
24 Theorem:Semi-infinite tape machines have the same power with Standard Turing machines
25 The Off-Line Machine Control Unit Input File Tape read-only read-write
26 Off-line machines simulate Standard Turing Machines: Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine
27 1. Copy input file to tape Input File Tape Standard machine Off-line machine
28 2. Do computations as in Turing machine Input File Tape Standard machine Off-line machine
29 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
30 Input File Tape Off-line Machine Four track tape -- Standard Machine Input File head position Tape head position
31 Off-line machines have the same power with Standard machines Theorem:
32 Multitape Turing Machines Control unit Tape 1Tape 2 Input
33 Time 1 Time 2 Tape 1Tape 2
34 Multitape machines simulate Standard Machines: Use just one tape
35 Standard machines simulate Multitape machines: Use a multi-track tape A tape of the Multiple tape machine corresponds to a pair of tracks Standard machine:
36 Multitape Machine Tape 1Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position
37 Repeat for each state transition: Return to reference point Find current symbol in Tape 1 Find current symbol in Tape 2 Make transition Tape 1 head position Tape 2 head position Reference point
38 Theorem:Multi-tape machines have the same power with Standard Turing Machines
39 Same power doesnt imply same speed: Language Acceptance Time Standard machine Two-tape machine
40 Standard machine: Go back and forth times Two-tape machine: Copy to tape 2 Leave on tape 1 Compare tape 1 and tape 2 ( steps)
41 MultiDimensional Turing Machines Two-dimensional tape HEAD Position: +2, -1 MOVES: L,R,U,D U: up D: down
42 Multidimensional machines simulate Standard machines: Use one dimension
43 Standard machines simulate Multidimensional machines: Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2
44 symbols coordinates Two-dimensional machine Standard Machine
45 Repeat for each transition Update current symbol Compute coordinates of next position Go to new position Standard machine:
46 MultiDimensional Machines have the same power with Standard Turing Machines Theorem:
47 NonDeterministic Turing Machines Non Deterministic Choice
48 Time 0 Time 1 Choice 1 Choice 2
49 Input string is accepted if this a possible computation Initial configurationFinal Configuration Final state
50 NonDeterministic Machines simulate Standard (deterministic) Machines: Every deterministic machine is also a nondeterministic machine
51 Deterministic machines simulate NonDeterministic machines: Keeps track of all possible computations Deterministic machine:
52 Non-Deterministic Choices Computation 1
53 Non-Deterministic Choices Computation 2
54 Keeps track of all possible computations Deterministic machine: Simulation Stores computations in a two-dimensional tape
55 Time 0 NonDeterministic machine Deterministic machine Computation 1
56 Computation 1 Choice 1 Choice 2 Computation 2 NonDeterministic machine Deterministic machine Time 1
57 Theorem: NonDeterministic Machines have the same power with Deterministic machines
58 Remark: The simulation in the Deterministic machine takes time exponential time compared to the NonDeterministic machine
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